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We investigated the applicability and efficiency of the MLMC approach to the Henry-like problem with uncertain porosity, permeability and recharge. These uncertain parameters were modelled by random fields with three independent random variables. Permeability is a function of porosity. Both functions are time-dependent, have multi-scale behaviour and are defined for two layers. The numerical solution for each random realisation was obtained using the well-known ug4 parallel multigrid solver. The number of random samples required at each level was estimated by calculating the decay of the variances and the computational cost for each level. The MLMC method was used to compute the expected value and variance of several QoIs, such as the solution at a few preselected points $(t,\bx)$, the solution integrated over a small subdomain, and the time evolution of the freshwater integral. We have found that some QoIs require only 2-3 mesh levels and samples from finer meshes would not significantly improve the result. Other QoIs require more grid levels.

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litvinenko_Gamm2023.pdf

Saltwater intrusion occurs when sea levels rise and saltwater moves onto the land. Usually, this occurs during storms, high tides, droughts, or when saltwater penetrates freshwater aquifers and raises the groundwater table. Since groundwater is an essential nutrition and irrigation resource, its salinization may lead to catastrophic consequences. Many acres of farmland may be lost because they can become too wet or salty to grow crops. Therefore, accurate modeling of different scenarios of saline flow is essential to help farmers and researchers develop strategies to improve the soil quality and decrease saltwater intrusion effects.
Saline flow is density-driven and described by a system of time-dependent nonlinear partial differential equations (PDEs). It features convection dominance and can demonstrate very complicated behavior.
As a specific model, we consider a Henry-like problem with uncertain permeability and porosity.
These parameters may strongly affect the flow and transport of salt.

Uncertain_Henry_problem-poster.pdf

We investigated the applicability and efficiency of the MLMC approach for the Henry-like problem with uncertain porosity, permeability, and recharge. These uncertain parameters were modeled by random fields with three independent random variables. The numerical solution for each random realization was obtained using the well-known ug4 parallel multigrid solver. The number of required random samples on each level was estimated by computing the decay of the variances and computational costs for each level. We also computed the expected value and variance of the mass fraction in the whole domain, the evolution of the pdfs, the solutions at a few preselected points $(t,\bx)$, and the time evolution of the freshwater integral value. We have found that some QoIs require only 2-3 of the coarsest mesh levels, and samples from finer meshes would not significantly improve the result. Note that a different type of porosity may lead to a different conclusion.
The results show that the MLMC method is faster than the QMC method at the finest mesh. Thus, sampling at different mesh levels makes sense and helps to reduce the overall computational cost.

Litvinenko_Poster_Henry_22May.pdf

We consider a class of density-driven flow problems. We are particularly interested in the problem of the salinization of coastal aquifers. We consider the Henry saltwater intrusion problem with uncertain porosity, permeability, and recharge parameters as a test case.
The reason for the presence of uncertainties is the lack of knowledge, inaccurate measurements,
and inability to measure parameters at each spatial or time location. This problem is nonlinear and time-dependent. The solution is the salt mass fraction, which is uncertain and changes in time. Uncertainties in porosity, permeability, recharge, and mass fraction are modeled using random fields. This work investigates the applicability of the well-known multilevel Monte Carlo (MLMC) method for such problems. The MLMC method can reduce the total computational and storage costs. Moreover, the MLMC method runs multiple scenarios on different spatial and time meshes and then estimates the mean value of the mass fraction.
The parallelization is performed in both the physical space and stochastic space. To solve every deterministic scenario, we run the parallel multigrid solver ug4 in a black-box fashion.
We use the solution obtained from the quasi-Monte Carlo method as a reference solution.

Uncertainty quantification of groundwater contamination

The document summarizes research on quantifying uncertainty in groundwater contamination modeling. It discusses using stochastic methods and surrogate models to estimate how uncertainties in geological parameters, like porosity, propagate and influence quantities of interest in groundwater flow and contaminant transport simulations. Numerical experiments were conducted in 2D and 3D domains using parallel computing to analyze mean concentrations, variances, and other statistics over time under different uncertainty scenarios.

Poster_density_driven_with_fracture_MLMC.pdf

Poster to be presented at Stochastic Numerics and Statistical Learning: Theory and Applications Workshop 2024, Kaust, Saudi Arabia, https://cemse.kaust.edu.sa/stochnum/events/event/snsl-workshop-2024.
In this work we have considered a setting that mimics the Henry problem \cite{Simpson2003,Simpson04_Henry}, modeling seawater intrusion into a 2D coastal aquifer. The pure water recharge from the ``land side'' resists the salinisation of the aquifer due to the influx of saline water through the ``sea side'', thereby achieving some equilibrium in the salt concentration. In our setting, following \cite{GRILLO2010}, we consider a fracture on the sea side that significantly increases the permeability of the porous medium.
The flow and transport essentially depend on the geological parameters of the porous medium, including the fracture. We investigated the effects of various uncertainties on saltwater intrusion. We assumed uncertainties in the fracture width, the porosity of the bulk medium, its permeability and the pure water recharge from the land side. The porosity and permeability were modeled by random fields, the recharge by a random but periodic intensity and the thickness by a random variable. We calculated the mean and variance of the salt mass fraction, which is also uncertain.
The main question we investigated in this work was how well the MLMC method can be used to compute statistics of different QoIs. We found that the answer depends on the choice of the QoI. First, not every QoI requires a hierarchy of meshes and MLMC. Second, MLMC requires stable convergence rates for $\EXP{g_{\ell} - g_{\ell-1}}$ and $\Var{g_{\ell} - g_{\ell-1}}$. These rates should be independent of $\ell$. If these convergence rates vary for different $\ell$, then it will be hard to estimate $L$ and $m_{\ell}$, and MLMC will either not work or be suboptimal. We were not able to get stable convergence rates for all levels $\ell=1,\ldots,5$ when the QoI was an integral as in \eqref{eq:integral_box}. We found that for $\ell=1,\ldots 4$ and $\ell=5$ the rate $\alpha$ was different. Further investigation is needed to find the reason for this. Another difficulty is the dependence on time, i.e. the number of levels $L$ and the number of sums $m_{\ell}$ depend on $t$. At the beginning the variability is small, then it increases, and after the process of mixing salt and fresh water has stopped, the variance decreases again.
The number of random samples required at each level was estimated by calculating the decay of the variances and the computational cost for each level. These estimates depend on the minimisation function in the MLMC algorithm.
To achieve the efficiency of the MLMC approach presented in this work, it is essential that the complexity of the numerical solution of each random realisation is proportional to the number of grid vertices on the grid levels.

Density Driven Groundwater Flow with Uncertain Porosity and Permeability

In this work, we solved the density driven groundwater flow problem with uncertain porosity and permeability. An accurate solution of this time-dependent and non-linear problem is impossible because of the presence of natural uncertainties in the reservoir such as porosity and permeability.
Therefore, we estimated the mean value and the variance of the solution, as well as the propagation of uncertainties from the random input parameters to the solution.
We started by defining the Elder-like problem. Then we described the multi-variate polynomial approximation (\gPC) approach and used it to estimate the required statistics of the mass fraction.
Utilizing the \gPC method allowed us
to reduce the computational cost compared to the classical quasi Monte Carlo method.
\gPC assumes that the output function $\sol(t,\bx,\thetab)$ is square-integrable and smooth w.r.t uncertain input variables $\btheta$.
Many factors, such as non-linearity, multiple solutions, multiple stationary states, time dependence and complicated solvers, make the investigation of the convergence of the \gPC method a non-trivial task.
We used an easy-to-implement, but only sub-optimal \gPC technique to quantify the uncertainty. For example, it is known that by increasing the degree of global polynomials (Hermite, Langange and similar), Runge's phenomenon appears. Here, probably local polynomials, splines or their mixtures would be better. Additionally, we used an easy-to-parallelise quadrature rule, which was also only suboptimal. For instance, adaptive choice of sparse grid (or collocation) points \cite{ConradMarzouk13,nobile-sg-mc-2015,Sudret_sparsePCE,CONSTANTINE12,crestaux2009polynomial} would be better, but we were limited by the usage of parallel methods. Adaptive quadrature rules are not (so well) parallelisable. In conclusion, we can report that: a) we developed a highly parallel method to quantify uncertainty in the Elder-like problem; b) with the \gPC of degree 4 we can achieve similar results as with the \QMC method.
In the numerical section we considered two different aquifers - a solid parallelepiped and a solid elliptic cylinder. One of our goals was to see how the domain geometry influences the formation, the number and the shape of fingers.
Since the considered problem is nonlinear,
a high variance in the porosity may result in totally different solutions; for instance, the number of fingers, their intensity and shape, the propagation time, and the velocity may vary considerably.
The number of cells in the presented experiments varied from $241{,}152$ to $15{,}433{,}728$ for the cylindrical domain and from $524{,}288$ to $4{,}194{,}304$ for the parallelepiped. The maximal number of parallel processing units was $600\times 32$, where $600$ is the number of parallel nodes and $32$ is the number of computing cores on each node. The total computing time varied from 2 hours for the coarse mesh to 24 hours for the finest mesh.

Propagation of Uncertainties in Density Driven Groundwater Flow

Major Goal: estimate risks of the pollution in a subsurface flow.
How?: we solve density-driven groundwater flow with uncertain porosity and permeability.
We set up density-driven groundwater flow problem,
review stochastic modeling and stochastic methods, use UG4 framework (https://gcsc.uni-frankfurt.de/simulation-and-modelling/ug4),
model uncertainty in porosity and permeability,
2D and 3D numerical experiments.

Talk Alexander Litvinenko on SIAM GS Conference in Houston

This document summarizes a talk on solving density-driven groundwater flow problems with uncertain porosity and permeability coefficients. The major goal is to estimate pollution risks in subsurface flows. The presentation covers: (1) setting up the groundwater flow problem; (2) reviewing stochastic modeling methods; (3) modeling uncertainty in porosity and permeability; (4) numerical methods to solve deterministic problems; and (5) 2D and 3D numerical experiments. The experiments demonstrate computing statistics of contaminant concentration and its propagation under uncertain parameters.

litvinenko_Gamm2023.pdf

Saltwater intrusion occurs when sea levels rise and saltwater moves onto the land. Usually, this occurs during storms, high tides, droughts, or when saltwater penetrates freshwater aquifers and raises the groundwater table. Since groundwater is an essential nutrition and irrigation resource, its salinization may lead to catastrophic consequences. Many acres of farmland may be lost because they can become too wet or salty to grow crops. Therefore, accurate modeling of different scenarios of saline flow is essential to help farmers and researchers develop strategies to improve the soil quality and decrease saltwater intrusion effects.
Saline flow is density-driven and described by a system of time-dependent nonlinear partial differential equations (PDEs). It features convection dominance and can demonstrate very complicated behavior.
As a specific model, we consider a Henry-like problem with uncertain permeability and porosity.
These parameters may strongly affect the flow and transport of salt.

Uncertain_Henry_problem-poster.pdf

We investigated the applicability and efficiency of the MLMC approach for the Henry-like problem with uncertain porosity, permeability, and recharge. These uncertain parameters were modeled by random fields with three independent random variables. The numerical solution for each random realization was obtained using the well-known ug4 parallel multigrid solver. The number of required random samples on each level was estimated by computing the decay of the variances and computational costs for each level. We also computed the expected value and variance of the mass fraction in the whole domain, the evolution of the pdfs, the solutions at a few preselected points $(t,\bx)$, and the time evolution of the freshwater integral value. We have found that some QoIs require only 2-3 of the coarsest mesh levels, and samples from finer meshes would not significantly improve the result. Note that a different type of porosity may lead to a different conclusion.
The results show that the MLMC method is faster than the QMC method at the finest mesh. Thus, sampling at different mesh levels makes sense and helps to reduce the overall computational cost.

Litvinenko_Poster_Henry_22May.pdf

We consider a class of density-driven flow problems. We are particularly interested in the problem of the salinization of coastal aquifers. We consider the Henry saltwater intrusion problem with uncertain porosity, permeability, and recharge parameters as a test case.
The reason for the presence of uncertainties is the lack of knowledge, inaccurate measurements,
and inability to measure parameters at each spatial or time location. This problem is nonlinear and time-dependent. The solution is the salt mass fraction, which is uncertain and changes in time. Uncertainties in porosity, permeability, recharge, and mass fraction are modeled using random fields. This work investigates the applicability of the well-known multilevel Monte Carlo (MLMC) method for such problems. The MLMC method can reduce the total computational and storage costs. Moreover, the MLMC method runs multiple scenarios on different spatial and time meshes and then estimates the mean value of the mass fraction.
The parallelization is performed in both the physical space and stochastic space. To solve every deterministic scenario, we run the parallel multigrid solver ug4 in a black-box fashion.
We use the solution obtained from the quasi-Monte Carlo method as a reference solution.

Uncertainty quantification of groundwater contamination

The document summarizes research on quantifying uncertainty in groundwater contamination modeling. It discusses using stochastic methods and surrogate models to estimate how uncertainties in geological parameters, like porosity, propagate and influence quantities of interest in groundwater flow and contaminant transport simulations. Numerical experiments were conducted in 2D and 3D domains using parallel computing to analyze mean concentrations, variances, and other statistics over time under different uncertainty scenarios.

Poster_density_driven_with_fracture_MLMC.pdf

Poster to be presented at Stochastic Numerics and Statistical Learning: Theory and Applications Workshop 2024, Kaust, Saudi Arabia, https://cemse.kaust.edu.sa/stochnum/events/event/snsl-workshop-2024.
In this work we have considered a setting that mimics the Henry problem \cite{Simpson2003,Simpson04_Henry}, modeling seawater intrusion into a 2D coastal aquifer. The pure water recharge from the ``land side'' resists the salinisation of the aquifer due to the influx of saline water through the ``sea side'', thereby achieving some equilibrium in the salt concentration. In our setting, following \cite{GRILLO2010}, we consider a fracture on the sea side that significantly increases the permeability of the porous medium.
The flow and transport essentially depend on the geological parameters of the porous medium, including the fracture. We investigated the effects of various uncertainties on saltwater intrusion. We assumed uncertainties in the fracture width, the porosity of the bulk medium, its permeability and the pure water recharge from the land side. The porosity and permeability were modeled by random fields, the recharge by a random but periodic intensity and the thickness by a random variable. We calculated the mean and variance of the salt mass fraction, which is also uncertain.
The main question we investigated in this work was how well the MLMC method can be used to compute statistics of different QoIs. We found that the answer depends on the choice of the QoI. First, not every QoI requires a hierarchy of meshes and MLMC. Second, MLMC requires stable convergence rates for $\EXP{g_{\ell} - g_{\ell-1}}$ and $\Var{g_{\ell} - g_{\ell-1}}$. These rates should be independent of $\ell$. If these convergence rates vary for different $\ell$, then it will be hard to estimate $L$ and $m_{\ell}$, and MLMC will either not work or be suboptimal. We were not able to get stable convergence rates for all levels $\ell=1,\ldots,5$ when the QoI was an integral as in \eqref{eq:integral_box}. We found that for $\ell=1,\ldots 4$ and $\ell=5$ the rate $\alpha$ was different. Further investigation is needed to find the reason for this. Another difficulty is the dependence on time, i.e. the number of levels $L$ and the number of sums $m_{\ell}$ depend on $t$. At the beginning the variability is small, then it increases, and after the process of mixing salt and fresh water has stopped, the variance decreases again.
The number of random samples required at each level was estimated by calculating the decay of the variances and the computational cost for each level. These estimates depend on the minimisation function in the MLMC algorithm.
To achieve the efficiency of the MLMC approach presented in this work, it is essential that the complexity of the numerical solution of each random realisation is proportional to the number of grid vertices on the grid levels.

Density Driven Groundwater Flow with Uncertain Porosity and Permeability

In this work, we solved the density driven groundwater flow problem with uncertain porosity and permeability. An accurate solution of this time-dependent and non-linear problem is impossible because of the presence of natural uncertainties in the reservoir such as porosity and permeability.
Therefore, we estimated the mean value and the variance of the solution, as well as the propagation of uncertainties from the random input parameters to the solution.
We started by defining the Elder-like problem. Then we described the multi-variate polynomial approximation (\gPC) approach and used it to estimate the required statistics of the mass fraction.
Utilizing the \gPC method allowed us
to reduce the computational cost compared to the classical quasi Monte Carlo method.
\gPC assumes that the output function $\sol(t,\bx,\thetab)$ is square-integrable and smooth w.r.t uncertain input variables $\btheta$.
Many factors, such as non-linearity, multiple solutions, multiple stationary states, time dependence and complicated solvers, make the investigation of the convergence of the \gPC method a non-trivial task.
We used an easy-to-implement, but only sub-optimal \gPC technique to quantify the uncertainty. For example, it is known that by increasing the degree of global polynomials (Hermite, Langange and similar), Runge's phenomenon appears. Here, probably local polynomials, splines or their mixtures would be better. Additionally, we used an easy-to-parallelise quadrature rule, which was also only suboptimal. For instance, adaptive choice of sparse grid (or collocation) points \cite{ConradMarzouk13,nobile-sg-mc-2015,Sudret_sparsePCE,CONSTANTINE12,crestaux2009polynomial} would be better, but we were limited by the usage of parallel methods. Adaptive quadrature rules are not (so well) parallelisable. In conclusion, we can report that: a) we developed a highly parallel method to quantify uncertainty in the Elder-like problem; b) with the \gPC of degree 4 we can achieve similar results as with the \QMC method.
In the numerical section we considered two different aquifers - a solid parallelepiped and a solid elliptic cylinder. One of our goals was to see how the domain geometry influences the formation, the number and the shape of fingers.
Since the considered problem is nonlinear,
a high variance in the porosity may result in totally different solutions; for instance, the number of fingers, their intensity and shape, the propagation time, and the velocity may vary considerably.
The number of cells in the presented experiments varied from $241{,}152$ to $15{,}433{,}728$ for the cylindrical domain and from $524{,}288$ to $4{,}194{,}304$ for the parallelepiped. The maximal number of parallel processing units was $600\times 32$, where $600$ is the number of parallel nodes and $32$ is the number of computing cores on each node. The total computing time varied from 2 hours for the coarse mesh to 24 hours for the finest mesh.

Propagation of Uncertainties in Density Driven Groundwater Flow

Major Goal: estimate risks of the pollution in a subsurface flow.
How?: we solve density-driven groundwater flow with uncertain porosity and permeability.
We set up density-driven groundwater flow problem,
review stochastic modeling and stochastic methods, use UG4 framework (https://gcsc.uni-frankfurt.de/simulation-and-modelling/ug4),
model uncertainty in porosity and permeability,
2D and 3D numerical experiments.

Talk Alexander Litvinenko on SIAM GS Conference in Houston

This document summarizes a talk on solving density-driven groundwater flow problems with uncertain porosity and permeability coefficients. The major goal is to estimate pollution risks in subsurface flows. The presentation covers: (1) setting up the groundwater flow problem; (2) reviewing stochastic modeling methods; (3) modeling uncertainty in porosity and permeability; (4) numerical methods to solve deterministic problems; and (5) 2D and 3D numerical experiments. The experiments demonstrate computing statistics of contaminant concentration and its propagation under uncertain parameters.

Talk litvinenko gamm19

Talk presented on GAMM 2019 Conference in Vienna, Austria.
Parallel algorithm for uncertainty quantification in the density driven subsurface flow. Estimate risks of subsurface flow pollution.

Flood routing by kinematic wave model

In this study the kinematic wave equation has been solved numerically using the modified Lax
explicit finite difference scheme (MLEFDS) and used for flood routing in a wide prismatic channel and a nonprismatic
channel. Two flood waves, one sinusoidal wave and one exponential wave, have been imposed at the
upstream boundary of the channel in which the flow is initially uniform. Six different schemes have been
introduced and used to compute the routing parameter, the wave celerity c. Two of these schemes are based on
constant depth and use constant celerity throughout the computation process. The rest of the schemes are based
on local depths and give celerity dependent on time and space. The effects of the routing parameter c on the
travel time of flood wave, the subsidence of the flood peak and the conservation flood flow volume have been
studied. The results seem to indicate that there is a minimal loss/gain of flow volume whatever the scheme is.
While it is confirmed that neither of the schemes is 100% volume conservative, it is found that the scheme
Kinematic Wave Model-2 (KWM-II) gives the most accurate result giving only 0.1% error in perspective of
volume conservation. The results obtained in this study are in good qualitative agreement with those obtained in
other similar studies.

Adaptive Restore algorithm & importance Monte Carlo

Talk given at the SMC 2024 workshop, ICMS, Edinburgh, Scotland

120715 - LMAJdS paper - HydroVision 2012 presentation - 14 pages

1) The document presents Von Karman 3D procedures as a generalization of Westergaard 2D formulae for estimating hydrodynamic forces on dams during earthquakes.
2) Westergaard's 2D formulae have limitations and can produce singularities, while Von Karman's approach is non-periodic and non-singular with easier generalization to 3D.
3) Numerical examples show that accounting for 3D effects through Von Karman procedures can increase estimated overturning moments on dams by up to 7% compared to 2D Westergaard formulae.

Efficient Simulations for Contamination of Groundwater Aquifers under Uncerta...

Efficient Simulations for Contamination of Groundwater Aquifers under Uncerta...Alexander Litvinenko

1. Solved time-dependent density driven flow problem with uncertain porosity and permeability in 2D and 3D
2. Computed propagation of uncertainties in porosity into the mass fraction.
3. Computed the mean, variance, exceedance probabilities, quantiles, risks.
4. Such QoIs as the number of fingers, their size, shape, propagation time can be unstable
5. For moderate perturbations, our gPCE surrogate results are similar to qMC results.
6. Used highly scalable solver on up to 800 computing nodes,
Low-rank tensor approximation (Introduction)

1. Motivation: why do we need low-rank tensors
2. Tensors of the second order (matrices)
3. CP, Tucker and tensor train tensor formats
4. Many classical kernels have (or can be approximated in ) low-rank tensor format
5. Post processing: Computation of mean, variance, level sets, frequency

QMC: Transition Workshop - Applying Quasi-Monte Carlo Methods to a Stochastic...

QMC: Transition Workshop - Applying Quasi-Monte Carlo Methods to a Stochastic...The Statistical and Applied Mathematical Sciences Institute

We study an elliptic eigenvalue problem, with a random coefficient that can be parametrised by infinitely-many stochastic parameters. The physical motivation is the criticality problem for a nuclear reactor: in steady state the fission reaction can be modeled by an elliptic eigenvalue
problem, and the smallest eigenvalue provides a measure of how close the reaction is to equilibrium -- in terms of production/absorption of neutrons. The coefficients are allowed to be random to model the uncertainty of the composition of materials inside the reactor, e.g., the
control rods, reactor structure, fuel rods etc.
The randomness in the coefficient also results in randomness in the eigenvalues and corresponding eigenfunctions. As such, our quantity of interest is the expected value, with
respect to the stochastic parameters, of the smallest eigenvalue, which we formulate as an integral over the infinite-dimensional parameter domain. Our approximation involves three steps: truncating the stochastic dimension, discretizing the spatial domain using finite elements and approximating the now finite but still high-dimensional integral.
To approximate the high-dimensional integral we use quasi-Monte Carlo (QMC) methods. These are deterministic or quasi-random quadrature rules that can be proven to be very efficient for the numerical integration of certain classes of high-dimensional functions. QMC methods have previously been applied to linear functionals of the solution of a similar elliptic source problem; however, because of the nonlinearity of eigenvalues the existing analysis of the integration error
does not hold in our case.
We show that the minimal eigenvalue belongs to the spaces required for QMC theory, outline the approximation algorithm and provide numerical results.pRO

The document discusses modeling a flood in a river using 1D Saint-Venant equations. It presents the mass and momentum equations, describes common routing methods varying in complexity, and provides details on explicit and implicit numerical schemes to solve the equations. It also explains the HEC-1 hydrologic routing method and provides sample input parameters and results from a test model run.

Solving the Poisson Equation

This document describes a numerical study involving the solution of Poisson's equation using the finite volume method. It presents results from solving Laplace's equation on square domains with mixed boundary conditions, as well as solving the pressure Poisson equation derived from incompressible flow equations. Gaussian elimination, successive over-relaxation, and second-order accuracy are discussed. Numerical experiments demonstrate that over-relaxation improves convergence and the method achieves second-order accuracy based on grid refinement studies.

Computing f-Divergences and Distances of High-Dimensional Probability Density...

Computing f-Divergences and Distances of High-Dimensional Probability Density...Alexander Litvinenko

Poster presented on Stochastic Numerics and Statistical Learning: Theory and Applications Workshop in KAUST, Saudi Arabia.
The task considered here was the numerical computation of characterising statistics of
high-dimensional pdfs, as well as their divergences and distances,
where the pdf in the numerical implementation was assumed discretised on some regular grid.
Even for moderate dimension $d$, the full storage and computation with such objects become very quickly infeasible.
We have demonstrated that high-dimensional pdfs,
pcfs, and some functions of them
can be approximated and represented in a low-rank tensor data format.
Utilisation of low-rank tensor techniques helps to reduce the computational complexity
and the storage cost from exponential $\C{O}(n^d)$ to linear in the dimension $d$, e.g.
O(d n r^2) for the TT format. Here $n$ is the number of discretisation
points in one direction, r<n is the maximal tensor rank, and d the problem dimension.
The particular data format is rather unimportant,
any of the well-known tensor formats (CP, Tucker, hierarchical Tucker, tensor-train (TT)) can be used,
and we used the TT data format. Much of the presentation and in fact the central train
of discussion and thought is actually independent of the actual representation.
In the beginning it was motivated through three possible ways how one may
arrive at such a representation of the pdf. One was if the pdf was given in some approximate
analytical form, e.g. like a function tensor product of lower-dimensional pdfs with a
product measure, or from an analogous representation of the pcf and subsequent use of the
Fourier transform, or from a low-rank functional representation of a high-dimensional
RV, again via its pcf.
The theoretical underpinnings of the relation between pdfs and pcfs as well as their
properties were recalled in Section: Theory, as they are important to be preserved in the
discrete approximation. This also introduced the concepts of the convolution and of
the point-wise multiplication Hadamard algebra, concepts which become especially important if
one wants to characterise sums of independent RVs or mixture models,
a topic we did not touch on for the sake of brevity but which follows very naturally from
the developments here. Especially the Hadamard algebra is also
important for the algorithms to compute various point-wise functions in the sparse formats.
Talk at SciCADE2013 about "Accelerated Multiple Precision ODE solver base on ...

Talk at SciCADE2013 about "Accelerated Multiple Precision ODE solver base on ...Shizuoka Inst. Science and Tech.

We have implemented a multiple precision ODE solver based on high-order fully implicit Runge-Kutta(IRK) methods. This ODE solver uses any order Gauss type formulas, and can be accelerated by using (1) MPFR as multiple precision floating-point arithmetic library, (2) real tridiagonalization supported in SPARK3, of linear equations to be solved in simplified Newton method as inner iteration, (3) mixed precision iterative refinement method\cite{mixed_prec_iterative_ref}, (4) parallelization with OpenMP, and (5) embedded formulas for IRK methods. In this talk, we describe the reason why we adopt such accelerations, and show the efficiency of the ODE solver through numerical experiments such as Kuramoto-Sivashinsky equation.Sparse data formats and efficient numerical methods for uncertainties in nume...

Sparse data formats and efficient numerical methods for uncertainties in nume...Alexander Litvinenko

Description of methodologies and overview of numerical methods, which we used for modeling and quantification of uncertainties in numerical aerodynamicsQMC: Transition Workshop - Density Estimation by Randomized Quasi-Monte Carlo...

QMC: Transition Workshop - Density Estimation by Randomized Quasi-Monte Carlo...The Statistical and Applied Mathematical Sciences Institute

We examine the effectiveness of randomized quasi Monte Carlo (RQMC) to improve the convergence rate of the mean integrated square error, compared with crude Monte Carlo (MC), when estimating the density of a random variable X defined as a function over the s-dimensional unit cube (0,1)^s. We consider histograms and kernel density estimators. We show both theoretically and empirically that RQMC estimators can achieve faster convergence rates in
some situations.
This is joint work with Amal Ben Abdellah, Art B. Owen, and Florian Puchhammer.Spillover Dynamics for Systemic Risk Measurement Using Spatial Financial Time...

Spillover Dynamics for Systemic Risk Measurement Using Spatial Financial Time Series Models. Andre Lucas. Amsterdam - June, 25 2015. European Financial Management Association 2015 Annual Meetings.

GDQ SIMULATION FOR FLOW AND HEAT TRANSFER OF A NANOFLUID OVER A NONLINEARLY S...

This paper presents the generalized differential quadrature (GDQ) simulation for analysis of a nanofluid
over a nonlinearly stretching sheet. The obtained governing equations of flow and heat transfer are
discretized by GDQ method and then are solved by Newton-Raphson method. The effects of stretching
parameter, Brownian motion number (Nb), Thermophoresis number (Nt) and Lewis number (Le), on the
concentration distribution and temperature distribution are evaluated. The obtained results exhibit that

Anomalous Diffusion Through Homopolar Membrane: One-Dimensional Model_ Crimso...

Anomalous Diffusion Through Homopolar Membrane: One-Dimensional Model_ Crimso...Crimsonpublishers-Mechanicalengineering

Anomalous Diffusion Through Homopolar Membrane: One-Dimensional Model by Guilherme Garcia Gimenez and Adélcio C Oliveira* in Evolutions in Mechanical Engineering
Presentation

1) The document describes the development, verification, and validation of a responsive boundary model.
2) Verification was performed using manufactured solutions and grid convergence studies to evaluate spatial and temporal discretization errors.
3) Validation compared model predictions of helium plume interferometry data to experimental measurements, analyzing sources of error and sensitivity.
4) Results showed good agreement between computation and experiment, demonstrating the model captured important physical phenomena.

Hierarchical matrices for approximating large covariance matries and computin...

Hierarchical matrices for approximating large covariance matries and computin...Alexander Litvinenko

I do overview of methods for computing KL expansion. For tensor grid one can use high-dimensional Fourier, for non-tensor --- hierarchical matrices.Mathematical models for a chemical reactor

This document presents a mathematical model for the concentration of a chemical in a reactor. It examines both steady state and time-dependent models. For steady state, the model is an ordinary differential equation that can be solved analytically. For time dependence, the model is a partial differential equation that requires numerical solution. Two numerical methods are presented: an implicit finite difference method and the finite element method.

My Prize Winning Physics Poster from 2006

This poster was created in LaTeX on a Dell Inspiron laptop with a Linux Fedora Core 4 operating system. The background image and the animation snapshots are dxf meshes of elastic waveform solutions, rendered on a Windows machine using 3D Studio Max.

litvinenko_Intrusion_Bari_2023.pdf

1. Investigated efficiency of MLMC for Henry problem with
uncertain porosity, permeability, and recharge.
2. Uncertainties are modeled by random fields.
3. MLMC could be much faster than MC, 3200 times faster !
4. The time dependence is challenging.
Remarks:
1. Check if MLMC is needed.
2. The optimal number of samples depends on the point (t;x)
3. An advanced MLMC may give better estimates of L and m`.

Litvinenko_RWTH_UQ_Seminar_talk.pdf

Here the interest is mainly to compute characterisations like the entropy,
the Kullback-Leibler divergence, more general $f$-divergences, or other such characteristics based on
the probability density. The density is often not available directly,
and it is a computational challenge to just represent it in a numerically
feasible fashion in case the dimension is even moderately large. It
is an even stronger numerical challenge to then actually compute said characteristics
in the high-dimensional case.
The task considered here was the numerical computation of characterising statistics of
high-dimensional pdfs, as well as their divergences and distances,
where the pdf in the numerical implementation was assumed discretised on some regular grid.
We have demonstrated that high-dimensional pdfs,
pcfs, and some functions of them
can be approximated and represented in a low-rank tensor data format.
Utilisation of low-rank tensor techniques helps to reduce the computational complexity
and the storage cost from exponential $\C{O}(n^d)$ to linear in the dimension $d$, e.g.\
$O(d n r^2 )$ for the TT format. Here $n$ is the number of discretisation
points in one direction, $r<<n$ is the maximal tensor rank, and $d$ the problem dimension.

Talk litvinenko gamm19

Talk presented on GAMM 2019 Conference in Vienna, Austria.
Parallel algorithm for uncertainty quantification in the density driven subsurface flow. Estimate risks of subsurface flow pollution.

Flood routing by kinematic wave model

In this study the kinematic wave equation has been solved numerically using the modified Lax
explicit finite difference scheme (MLEFDS) and used for flood routing in a wide prismatic channel and a nonprismatic
channel. Two flood waves, one sinusoidal wave and one exponential wave, have been imposed at the
upstream boundary of the channel in which the flow is initially uniform. Six different schemes have been
introduced and used to compute the routing parameter, the wave celerity c. Two of these schemes are based on
constant depth and use constant celerity throughout the computation process. The rest of the schemes are based
on local depths and give celerity dependent on time and space. The effects of the routing parameter c on the
travel time of flood wave, the subsidence of the flood peak and the conservation flood flow volume have been
studied. The results seem to indicate that there is a minimal loss/gain of flow volume whatever the scheme is.
While it is confirmed that neither of the schemes is 100% volume conservative, it is found that the scheme
Kinematic Wave Model-2 (KWM-II) gives the most accurate result giving only 0.1% error in perspective of
volume conservation. The results obtained in this study are in good qualitative agreement with those obtained in
other similar studies.

Adaptive Restore algorithm & importance Monte Carlo

Talk given at the SMC 2024 workshop, ICMS, Edinburgh, Scotland

120715 - LMAJdS paper - HydroVision 2012 presentation - 14 pages

1) The document presents Von Karman 3D procedures as a generalization of Westergaard 2D formulae for estimating hydrodynamic forces on dams during earthquakes.
2) Westergaard's 2D formulae have limitations and can produce singularities, while Von Karman's approach is non-periodic and non-singular with easier generalization to 3D.
3) Numerical examples show that accounting for 3D effects through Von Karman procedures can increase estimated overturning moments on dams by up to 7% compared to 2D Westergaard formulae.

Efficient Simulations for Contamination of Groundwater Aquifers under Uncerta...

Efficient Simulations for Contamination of Groundwater Aquifers under Uncerta...Alexander Litvinenko

1. Solved time-dependent density driven flow problem with uncertain porosity and permeability in 2D and 3D
2. Computed propagation of uncertainties in porosity into the mass fraction.
3. Computed the mean, variance, exceedance probabilities, quantiles, risks.
4. Such QoIs as the number of fingers, their size, shape, propagation time can be unstable
5. For moderate perturbations, our gPCE surrogate results are similar to qMC results.
6. Used highly scalable solver on up to 800 computing nodes,
Low-rank tensor approximation (Introduction)

1. Motivation: why do we need low-rank tensors
2. Tensors of the second order (matrices)
3. CP, Tucker and tensor train tensor formats
4. Many classical kernels have (or can be approximated in ) low-rank tensor format
5. Post processing: Computation of mean, variance, level sets, frequency

QMC: Transition Workshop - Applying Quasi-Monte Carlo Methods to a Stochastic...

QMC: Transition Workshop - Applying Quasi-Monte Carlo Methods to a Stochastic...The Statistical and Applied Mathematical Sciences Institute

We study an elliptic eigenvalue problem, with a random coefficient that can be parametrised by infinitely-many stochastic parameters. The physical motivation is the criticality problem for a nuclear reactor: in steady state the fission reaction can be modeled by an elliptic eigenvalue
problem, and the smallest eigenvalue provides a measure of how close the reaction is to equilibrium -- in terms of production/absorption of neutrons. The coefficients are allowed to be random to model the uncertainty of the composition of materials inside the reactor, e.g., the
control rods, reactor structure, fuel rods etc.
The randomness in the coefficient also results in randomness in the eigenvalues and corresponding eigenfunctions. As such, our quantity of interest is the expected value, with
respect to the stochastic parameters, of the smallest eigenvalue, which we formulate as an integral over the infinite-dimensional parameter domain. Our approximation involves three steps: truncating the stochastic dimension, discretizing the spatial domain using finite elements and approximating the now finite but still high-dimensional integral.
To approximate the high-dimensional integral we use quasi-Monte Carlo (QMC) methods. These are deterministic or quasi-random quadrature rules that can be proven to be very efficient for the numerical integration of certain classes of high-dimensional functions. QMC methods have previously been applied to linear functionals of the solution of a similar elliptic source problem; however, because of the nonlinearity of eigenvalues the existing analysis of the integration error
does not hold in our case.
We show that the minimal eigenvalue belongs to the spaces required for QMC theory, outline the approximation algorithm and provide numerical results.pRO

The document discusses modeling a flood in a river using 1D Saint-Venant equations. It presents the mass and momentum equations, describes common routing methods varying in complexity, and provides details on explicit and implicit numerical schemes to solve the equations. It also explains the HEC-1 hydrologic routing method and provides sample input parameters and results from a test model run.

Solving the Poisson Equation

This document describes a numerical study involving the solution of Poisson's equation using the finite volume method. It presents results from solving Laplace's equation on square domains with mixed boundary conditions, as well as solving the pressure Poisson equation derived from incompressible flow equations. Gaussian elimination, successive over-relaxation, and second-order accuracy are discussed. Numerical experiments demonstrate that over-relaxation improves convergence and the method achieves second-order accuracy based on grid refinement studies.

Computing f-Divergences and Distances of High-Dimensional Probability Density...

Computing f-Divergences and Distances of High-Dimensional Probability Density...Alexander Litvinenko

Poster presented on Stochastic Numerics and Statistical Learning: Theory and Applications Workshop in KAUST, Saudi Arabia.
The task considered here was the numerical computation of characterising statistics of
high-dimensional pdfs, as well as their divergences and distances,
where the pdf in the numerical implementation was assumed discretised on some regular grid.
Even for moderate dimension $d$, the full storage and computation with such objects become very quickly infeasible.
We have demonstrated that high-dimensional pdfs,
pcfs, and some functions of them
can be approximated and represented in a low-rank tensor data format.
Utilisation of low-rank tensor techniques helps to reduce the computational complexity
and the storage cost from exponential $\C{O}(n^d)$ to linear in the dimension $d$, e.g.
O(d n r^2) for the TT format. Here $n$ is the number of discretisation
points in one direction, r<n is the maximal tensor rank, and d the problem dimension.
The particular data format is rather unimportant,
any of the well-known tensor formats (CP, Tucker, hierarchical Tucker, tensor-train (TT)) can be used,
and we used the TT data format. Much of the presentation and in fact the central train
of discussion and thought is actually independent of the actual representation.
In the beginning it was motivated through three possible ways how one may
arrive at such a representation of the pdf. One was if the pdf was given in some approximate
analytical form, e.g. like a function tensor product of lower-dimensional pdfs with a
product measure, or from an analogous representation of the pcf and subsequent use of the
Fourier transform, or from a low-rank functional representation of a high-dimensional
RV, again via its pcf.
The theoretical underpinnings of the relation between pdfs and pcfs as well as their
properties were recalled in Section: Theory, as they are important to be preserved in the
discrete approximation. This also introduced the concepts of the convolution and of
the point-wise multiplication Hadamard algebra, concepts which become especially important if
one wants to characterise sums of independent RVs or mixture models,
a topic we did not touch on for the sake of brevity but which follows very naturally from
the developments here. Especially the Hadamard algebra is also
important for the algorithms to compute various point-wise functions in the sparse formats.
Talk at SciCADE2013 about "Accelerated Multiple Precision ODE solver base on ...

Talk at SciCADE2013 about "Accelerated Multiple Precision ODE solver base on ...Shizuoka Inst. Science and Tech.

We have implemented a multiple precision ODE solver based on high-order fully implicit Runge-Kutta(IRK) methods. This ODE solver uses any order Gauss type formulas, and can be accelerated by using (1) MPFR as multiple precision floating-point arithmetic library, (2) real tridiagonalization supported in SPARK3, of linear equations to be solved in simplified Newton method as inner iteration, (3) mixed precision iterative refinement method\cite{mixed_prec_iterative_ref}, (4) parallelization with OpenMP, and (5) embedded formulas for IRK methods. In this talk, we describe the reason why we adopt such accelerations, and show the efficiency of the ODE solver through numerical experiments such as Kuramoto-Sivashinsky equation.Sparse data formats and efficient numerical methods for uncertainties in nume...

Sparse data formats and efficient numerical methods for uncertainties in nume...Alexander Litvinenko

Description of methodologies and overview of numerical methods, which we used for modeling and quantification of uncertainties in numerical aerodynamicsQMC: Transition Workshop - Density Estimation by Randomized Quasi-Monte Carlo...

QMC: Transition Workshop - Density Estimation by Randomized Quasi-Monte Carlo...The Statistical and Applied Mathematical Sciences Institute

We examine the effectiveness of randomized quasi Monte Carlo (RQMC) to improve the convergence rate of the mean integrated square error, compared with crude Monte Carlo (MC), when estimating the density of a random variable X defined as a function over the s-dimensional unit cube (0,1)^s. We consider histograms and kernel density estimators. We show both theoretically and empirically that RQMC estimators can achieve faster convergence rates in
some situations.
This is joint work with Amal Ben Abdellah, Art B. Owen, and Florian Puchhammer.Spillover Dynamics for Systemic Risk Measurement Using Spatial Financial Time...

Spillover Dynamics for Systemic Risk Measurement Using Spatial Financial Time Series Models. Andre Lucas. Amsterdam - June, 25 2015. European Financial Management Association 2015 Annual Meetings.

GDQ SIMULATION FOR FLOW AND HEAT TRANSFER OF A NANOFLUID OVER A NONLINEARLY S...

This paper presents the generalized differential quadrature (GDQ) simulation for analysis of a nanofluid
over a nonlinearly stretching sheet. The obtained governing equations of flow and heat transfer are
discretized by GDQ method and then are solved by Newton-Raphson method. The effects of stretching
parameter, Brownian motion number (Nb), Thermophoresis number (Nt) and Lewis number (Le), on the
concentration distribution and temperature distribution are evaluated. The obtained results exhibit that

Anomalous Diffusion Through Homopolar Membrane: One-Dimensional Model_ Crimso...

Anomalous Diffusion Through Homopolar Membrane: One-Dimensional Model_ Crimso...Crimsonpublishers-Mechanicalengineering

Anomalous Diffusion Through Homopolar Membrane: One-Dimensional Model by Guilherme Garcia Gimenez and Adélcio C Oliveira* in Evolutions in Mechanical Engineering
Presentation

1) The document describes the development, verification, and validation of a responsive boundary model.
2) Verification was performed using manufactured solutions and grid convergence studies to evaluate spatial and temporal discretization errors.
3) Validation compared model predictions of helium plume interferometry data to experimental measurements, analyzing sources of error and sensitivity.
4) Results showed good agreement between computation and experiment, demonstrating the model captured important physical phenomena.

Hierarchical matrices for approximating large covariance matries and computin...

Hierarchical matrices for approximating large covariance matries and computin...Alexander Litvinenko

I do overview of methods for computing KL expansion. For tensor grid one can use high-dimensional Fourier, for non-tensor --- hierarchical matrices.Mathematical models for a chemical reactor

This document presents a mathematical model for the concentration of a chemical in a reactor. It examines both steady state and time-dependent models. For steady state, the model is an ordinary differential equation that can be solved analytically. For time dependence, the model is a partial differential equation that requires numerical solution. Two numerical methods are presented: an implicit finite difference method and the finite element method.

My Prize Winning Physics Poster from 2006

This poster was created in LaTeX on a Dell Inspiron laptop with a Linux Fedora Core 4 operating system. The background image and the animation snapshots are dxf meshes of elastic waveform solutions, rendered on a Windows machine using 3D Studio Max.

Talk litvinenko gamm19

Talk litvinenko gamm19

Flood routing by kinematic wave model

Flood routing by kinematic wave model

Adaptive Restore algorithm & importance Monte Carlo

Adaptive Restore algorithm & importance Monte Carlo

120715 - LMAJdS paper - HydroVision 2012 presentation - 14 pages

120715 - LMAJdS paper - HydroVision 2012 presentation - 14 pages

Efficient Simulations for Contamination of Groundwater Aquifers under Uncerta...

Efficient Simulations for Contamination of Groundwater Aquifers under Uncerta...

Low-rank tensor approximation (Introduction)

Low-rank tensor approximation (Introduction)

QMC: Transition Workshop - Applying Quasi-Monte Carlo Methods to a Stochastic...

QMC: Transition Workshop - Applying Quasi-Monte Carlo Methods to a Stochastic...

pRO

pRO

Solving the Poisson Equation

Solving the Poisson Equation

Computing f-Divergences and Distances of High-Dimensional Probability Density...

Computing f-Divergences and Distances of High-Dimensional Probability Density...

Talk at SciCADE2013 about "Accelerated Multiple Precision ODE solver base on ...

Talk at SciCADE2013 about "Accelerated Multiple Precision ODE solver base on ...

Sparse data formats and efficient numerical methods for uncertainties in nume...

Sparse data formats and efficient numerical methods for uncertainties in nume...

QMC: Transition Workshop - Density Estimation by Randomized Quasi-Monte Carlo...

QMC: Transition Workshop - Density Estimation by Randomized Quasi-Monte Carlo...

Spillover Dynamics for Systemic Risk Measurement Using Spatial Financial Time...

Spillover Dynamics for Systemic Risk Measurement Using Spatial Financial Time...

GDQ SIMULATION FOR FLOW AND HEAT TRANSFER OF A NANOFLUID OVER A NONLINEARLY S...

GDQ SIMULATION FOR FLOW AND HEAT TRANSFER OF A NANOFLUID OVER A NONLINEARLY S...

Anomalous Diffusion Through Homopolar Membrane: One-Dimensional Model_ Crimso...

Anomalous Diffusion Through Homopolar Membrane: One-Dimensional Model_ Crimso...

Presentation

Presentation

Hierarchical matrices for approximating large covariance matries and computin...

Hierarchical matrices for approximating large covariance matries and computin...

Mathematical models for a chemical reactor

Mathematical models for a chemical reactor

My Prize Winning Physics Poster from 2006

My Prize Winning Physics Poster from 2006

litvinenko_Intrusion_Bari_2023.pdf

1. Investigated efficiency of MLMC for Henry problem with
uncertain porosity, permeability, and recharge.
2. Uncertainties are modeled by random fields.
3. MLMC could be much faster than MC, 3200 times faster !
4. The time dependence is challenging.
Remarks:
1. Check if MLMC is needed.
2. The optimal number of samples depends on the point (t;x)
3. An advanced MLMC may give better estimates of L and m`.

Litvinenko_RWTH_UQ_Seminar_talk.pdf

Here the interest is mainly to compute characterisations like the entropy,
the Kullback-Leibler divergence, more general $f$-divergences, or other such characteristics based on
the probability density. The density is often not available directly,
and it is a computational challenge to just represent it in a numerically
feasible fashion in case the dimension is even moderately large. It
is an even stronger numerical challenge to then actually compute said characteristics
in the high-dimensional case.
The task considered here was the numerical computation of characterising statistics of
high-dimensional pdfs, as well as their divergences and distances,
where the pdf in the numerical implementation was assumed discretised on some regular grid.
We have demonstrated that high-dimensional pdfs,
pcfs, and some functions of them
can be approximated and represented in a low-rank tensor data format.
Utilisation of low-rank tensor techniques helps to reduce the computational complexity
and the storage cost from exponential $\C{O}(n^d)$ to linear in the dimension $d$, e.g.\
$O(d n r^2 )$ for the TT format. Here $n$ is the number of discretisation
points in one direction, $r<<n$ is the maximal tensor rank, and $d$ the problem dimension.

Litv_Denmark_Weak_Supervised_Learning.pdf

This document proposes a method for weakly supervised regression on uncertain datasets. It combines graph Laplacian regularization and cluster ensemble methodology. The method solves an auxiliary minimization problem to determine the optimal solution for predicting uncertain parameters. It is tested on artificial data to predict target values using a mixture of normal distributions with labeled, inaccurately labeled, and unlabeled samples. The method is shown to outperform a simplified version by reducing mean Wasserstein distance between predicted and true values.

Computing f-Divergences and Distances of\\ High-Dimensional Probability Densi...

Computing f-Divergences and Distances of\\ High-Dimensional Probability Densi...Alexander Litvinenko

Talk presented on SIAM IS 2022 conference.
Very often, in the course of uncertainty quantification tasks or
data analysis, one has to deal with high-dimensional random variables (RVs)
(with values in $\Rd$). Just like any other RV,
a high-dimensional RV can be described by its probability density (\pdf) and/or
by the corresponding probability characteristic functions (\pcf),
or a more general representation as
a function of other, known, random variables.
Here the interest is mainly to compute characterisations like the entropy, the Kullback-Leibler, or more general
$f$-divergences. These are all computed from the \pdf, which is often not available directly,
and it is a computational challenge to even represent it in a numerically
feasible fashion in case the dimension $d$ is even moderately large. It
is an even stronger numerical challenge to then actually compute said characterisations
in the high-dimensional case.
In this regard, in order to achieve a computationally feasible task, we propose
to approximate density by a low-rank tensor.Low rank tensor approximation of probability density and characteristic funct...

Low rank tensor approximation of probability density and characteristic funct...Alexander Litvinenko

This document summarizes a presentation on computing divergences and distances between high-dimensional probability density functions (pdfs) represented using tensor formats. It discusses:
1) Motivating the problem using examples from stochastic PDEs and functional representations of uncertainties.
2) Computing Kullback-Leibler divergence and other divergences when pdfs are not directly available.
3) Representing probability characteristic functions and approximating pdfs using tensor decompositions like CP and TT formats.
4) Numerical examples computing Kullback-Leibler divergence and Hellinger distance between Gaussian and alpha-stable distributions using these tensor approximations.Identification of unknown parameters and prediction of missing values. Compar...

Identification of unknown parameters and prediction of missing values. Compar...Alexander Litvinenko

H-matrix approximation of large Mat\'{e}rn covariance matrices, Gaussian log-likelihoods.
Identifying unknown parameters and making predictions
Comparison with machine learning methods.
kNN is easy to implement and shows promising results.
Computation of electromagnetic fields scattered from dielectric objects of un...

Computation of electromagnetic fields scattered from dielectric objects of un...Alexander Litvinenko

This document describes using the Continuation Multi-Level Monte Carlo (CMLMC) method to compute electromagnetic fields scattered from dielectric objects of uncertain shapes. CMLMC optimally balances statistical and discretization errors using fewer samples on fine meshes and more on coarse meshes. The method is tested by computing scattering cross sections for randomly perturbed spheres under plane wave excitation and comparing results to the unperturbed sphere. Computational costs and errors are analyzed to demonstrate the efficiency of CMLMC for this scattering problem with uncertain geometry.Identification of unknown parameters and prediction with hierarchical matrice...

Identification of unknown parameters and prediction with hierarchical matrice...Alexander Litvinenko

We compare four numerical methods for the prediction of missing values in four different datasets.
These methods are 1) the hierarchical maximum likelihood estimation (H-MLE), and three machine learning (ML) methods, which include 2) k-nearest neighbors (kNN), 3) random forest, and 4) Deep Neural Network (DNN).
From the ML methods, the best results (for considered datasets) were obtained by the kNN method with three (or seven) neighbors.
On one dataset, the MLE method showed a smaller error than the kNN method, whereas, on another, the kNN method was better.
The MLE method requires a lot of linear algebra computations and works fine on almost all datasets. Its result can be improved by taking a smaller threshold and more accurate hierarchical matrix arithmetics. To our surprise, the well-known kNN method produces similar results as H-MLE and worked much faster.Computation of electromagnetic fields scattered from dielectric objects of un...

Computation of electromagnetic fields scattered from dielectric objects of un...Alexander Litvinenko

Computational tools for characterizing electromagnetic scattering from objects with uncertain shapes are needed in various applications ranging from remote sensing at microwave frequencies to Raman spectroscopy at optical frequencies. Often, such computational tools use the Monte Carlo (MC) method to sample a parametric space describing geometric uncertainties. For each sample, which corresponds to a realization of the geometry, a deterministic electromagnetic solver computes the scattered fields. However, for an accurate statistical characterization the number of MC samples has to be large. In this work, to address this challenge, the continuation multilevel Monte Carlo (\CMLMC) method is used together with a surface integral equation solver.
The \CMLMC method optimally balances statistical errors due to sampling of
the parametric space, and numerical errors due to the discretization of the geometry using a hierarchy of discretizations, from coarse to fine.
The number of realizations of finer discretizations can be kept low, with most samples
computed on coarser discretizations to minimize computational cost.
Consequently, the total execution time is significantly reduced, in comparison to the standard MC scheme.Application of parallel hierarchical matrices for parameter inference and pre...

Application of parallel hierarchical matrices for parameter inference and pre...Alexander Litvinenko

We developed efficient linear algebra method (parallel hierarchical matrices) to do parameter inference and prediction of missing valuesComputation of electromagnetic fields scattered from dielectric objects of un...

Computation of electromagnetic fields scattered from dielectric objects of un...Alexander Litvinenko

Computational tools for characterizing electromagnetic scattering from objects with uncertain shapes are needed in various applications ranging from remote sensing at microwave frequencies to Raman spectroscopy at optical frequencies. Often, such computational tools use the Monte Carlo (MC) method to sample a parametric space describing geometric uncertainties. For each sample, which corresponds to a realization of the geometry, a deterministic electromagnetic solver computes the scattered fields. However, for an accurate statistical characterization the number of MC samples has to be large. In this work, to address this challenge, the continuation multilevel Monte Carlo (\CMLMC) method is used together with a surface integral equation solver.
The \CMLMC method optimally balances statistical errors due to sampling of
the parametric space, and numerical errors due to the discretization of the geometry using a hierarchy of discretizations, from coarse to fine.
The number of realizations of finer discretizations can be kept low, with most samples
computed on coarser discretizations to minimize computational cost.
Consequently, the total execution time is significantly reduced, in comparison to the standard MC scheme.Simulation of propagation of uncertainties in density-driven groundwater flow

Consider stochastic modelling of the density-driven subsurface flow in 3D. This talk was presented by Dmitry Logashenko on the IMG conference in Kunming, China, August 2019.

Approximation of large covariance matrices in statistics

Large data sets result large dense matrices, say with 2.000.000 rows and columns. How to work with such large matrices? How to approximate them? How to compute log-likelihood? determination? inverse? All answers are in this work.

Semi-Supervised Regression using Cluster Ensemble

This document summarizes a semi-supervised regression method that combines graph Laplacian regularization with cluster ensemble methodology. It proposes using a weighted averaged co-association matrix from the cluster ensemble as the similarity matrix in graph Laplacian regularization. The method (SSR-LRCM) finds a low-rank approximation of the co-association matrix to efficiently solve the regression problem. Experimental results on synthetic and real-world datasets show SSR-LRCM achieves significantly better prediction accuracy than an alternative method, while also having lower computational costs for large datasets. Future work will explore using a hierarchical matrix approximation instead of low-rank.

Computation of electromagnetic_fields_scattered_from_dielectric_objects_of_un...

Computation of electromagnetic_fields_scattered_from_dielectric_objects_of_un...Alexander Litvinenko

Tools for electromagnetic scattering from objects with uncertain shapes are needed in various applications.
We develop numerical methods for predicting radar and scattering cross sections (RCS and SCS) of complex targets.
To reduce cost of Monte Carlo (MC) we offer modified multilevel MC (CMLMC) method.Overview of sparse and low-rank matrix / tensor techniques

A short overview about sparse and low-rank matrix/tensor techniques, which could be used in spatio-temporal and environmental statistics

Application of Parallel Hierarchical Matrices in Spatial Statistics and Param...

Application of Parallel Hierarchical Matrices in Spatial Statistics and Param...Alexander Litvinenko

Part 1: Parallel H-matrices in spatial statistics
1. Motivation: improve statistical model
2. Tools: Hierarchical matrices [Hackbusch 1999]
3. Matern covariance function and joint Gaussian likelihood
4. Identification of unknown parameters via maximizing Gaussian
log-likelihood
5. Implementation with HLIBPro.Tucker tensor analysis of Matern functions in spatial statistics

1. Motivation: improve statistical models
2. Motivation: disadvantages of matrices
3. Tools: Tucker tensor format
4. Tensor approximation of Matern covariance function via FFT
5. Typical statistical operations in Tucker tensor format
6. Numerical experiments

litvinenko_Intrusion_Bari_2023.pdf

litvinenko_Intrusion_Bari_2023.pdf

Litvinenko_RWTH_UQ_Seminar_talk.pdf

Litvinenko_RWTH_UQ_Seminar_talk.pdf

Litv_Denmark_Weak_Supervised_Learning.pdf

Litv_Denmark_Weak_Supervised_Learning.pdf

Computing f-Divergences and Distances of\\ High-Dimensional Probability Densi...

Computing f-Divergences and Distances of\\ High-Dimensional Probability Densi...

Low rank tensor approximation of probability density and characteristic funct...

Low rank tensor approximation of probability density and characteristic funct...

Identification of unknown parameters and prediction of missing values. Compar...

Identification of unknown parameters and prediction of missing values. Compar...

Computation of electromagnetic fields scattered from dielectric objects of un...

Computation of electromagnetic fields scattered from dielectric objects of un...

Identification of unknown parameters and prediction with hierarchical matrice...

Identification of unknown parameters and prediction with hierarchical matrice...

Computation of electromagnetic fields scattered from dielectric objects of un...

Computation of electromagnetic fields scattered from dielectric objects of un...

Application of parallel hierarchical matrices for parameter inference and pre...

Application of parallel hierarchical matrices for parameter inference and pre...

Computation of electromagnetic fields scattered from dielectric objects of un...

Computation of electromagnetic fields scattered from dielectric objects of un...

Simulation of propagation of uncertainties in density-driven groundwater flow

Simulation of propagation of uncertainties in density-driven groundwater flow

Approximation of large covariance matrices in statistics

Approximation of large covariance matrices in statistics

Semi-Supervised Regression using Cluster Ensemble

Semi-Supervised Regression using Cluster Ensemble

Computation of electromagnetic_fields_scattered_from_dielectric_objects_of_un...

Computation of electromagnetic_fields_scattered_from_dielectric_objects_of_un...

Overview of sparse and low-rank matrix / tensor techniques

Overview of sparse and low-rank matrix / tensor techniques

Application of Parallel Hierarchical Matrices in Spatial Statistics and Param...

Application of Parallel Hierarchical Matrices in Spatial Statistics and Param...

Tucker tensor analysis of Matern functions in spatial statistics

Tucker tensor analysis of Matern functions in spatial statistics

14 Template Contractual Notice - EOT Application

EOT Application, Contractual Notice

Low power architecture of logic gates using adiabatic techniques

The growing significance of portable systems to limit power consumption in ultra-large-scale-integration chips of very high density, has recently led to rapid and inventive progresses in low-power design. The most effective technique is adiabatic logic circuit design in energy-efficient hardware. This paper presents two adiabatic approaches for the design of low power circuits, modified positive feedback adiabatic logic (modified PFAL) and the other is direct current diode based positive feedback adiabatic logic (DC-DB PFAL). Logic gates are the preliminary components in any digital circuit design. By improving the performance of basic gates, one can improvise the whole system performance. In this paper proposed circuit design of the low power architecture of OR/NOR, AND/NAND, and XOR/XNOR gates are presented using the said approaches and their results are analyzed for powerdissipation, delay, power-delay-product and rise time and compared with the other adiabatic techniques along with the conventional complementary metal oxide semiconductor (CMOS) designs reported in the literature. It has been found that the designs with DC-DB PFAL technique outperform with the percentage improvement of 65% for NOR gate and 7% for NAND gate and 34% for XNOR gate over the modified PFAL techniques at 10 MHz respectively.

5214-1693458878915-Unit 6 2023 to 2024 academic year assignment (AutoRecovere...

Bigdata of technology

CSM Cloud Service Management Presentarion

Usefull PPT

Technical Drawings introduction to drawing of prisms

Method of technical Drawing of prisms,and cylinders.

Iron and Steel Technology Roadmap - Towards more sustainable steelmaking.pdf

Iron and Steel Technology towards Sustainable Steelmaking

International Conference on NLP, Artificial Intelligence, Machine Learning an...

International Conference on NLP, Artificial Intelligence, Machine Learning and Applications (NLAIM 2024) offers a premier global platform for exchanging insights and findings in the theory, methodology, and applications of NLP, Artificial Intelligence, Machine Learning, and their applications. The conference seeks substantial contributions across all key domains of NLP, Artificial Intelligence, Machine Learning, and their practical applications, aiming to foster both theoretical advancements and real-world implementations. With a focus on facilitating collaboration between researchers and practitioners from academia and industry, the conference serves as a nexus for sharing the latest developments in the field.

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- 1. Uncertainty quantification in the coastal aquifers using Multi Level Monte Carlo Alexander Litvinenko, joint work with D. Logashenko, R. Tempone, E. Vasilyeva, G. Wittum RWTH Aachen and KAUST
- 2. Overview Problem: Henry saltwater intrusion (nonlinear and time-dependent, describes a two-phase subsurface flow) Input uncertainty: porosity, permeability, and recharge (model by random fields) Solution: the salt mass fraction (uncertain and time-dependent) Method: Multi Level Monte Carlo (MLMC) method Deterministic solver: parallel multigrid solver ug4 1 / 27
- 3. Henry problem 1. How long can wells be used? 2. Where is the largest uncertainty? 3. Freshwater exceedance probability? 4. What is the mean scenario and its variations? 5. What are the extreme scenarios? 6. How do the uncertainties change over time? taken from https://www.mdpi.com/2073-4441/10/2/230 2 / 27
- 4. Henry problem settings The mass conservation laws for the entire liquid phase and salt yield the following equations ∂t(φρ) + ∇ · (ρq) = 0, ∂t(φρc) + ∇ · (ρcq − ρD∇c) = 0, where φ(x,ξ) is porosity, x ∈ D, c(t,x) mass fraction of the salt, ρ = ρ(c) density of the liquid phase, and D(t,x) molecular diffusion tensor. For q(t,x) velocity, we assume Darcy’s law: q = − K µ (∇p − ρg), where p = p(t,x) is the hydrostatic pressure, K permeability, µ = µ(c) viscosity of the liquid phase, and g gravity. 3 / 27
- 5. Henry problem settings To compute: c and p. Comput. domain: D × [0,T]. We set ρ(c) = ρ0 + (ρ1 − ρ0)c, and D = φDI, I.C.: c|t=0 = 0, B.C.: c|x=2 = 1, p|x=2 = −ρ1gy. c|x=0 = 0, ρq · ex|x=0 = q̂in. We model φ by a random field and assume: K = KI, K = K(φ), and Kozeny–Carman-like dependence K(φ) = κ · φ3 1 − φ2 , (1) where κ is a scalar. Discretisation: vertex-centered finite volume, implicit Euler Methods: Newton method, BiCGStab preconditioned with the geometric multigrid method (V-cycle), ILUβ-smoothers. 4 / 27
- 6. Porosity and solution of the Henry problem q̂in = 6.6 · 10−2 kg/s c = 0 c = 1 p = −ρ1gy 0 −1 m 2 m y x D := [0,2] × [−1,0]; a realization of c(t,x) with streamlines of the velocity field q; porosity φ(ξ∗ ) ∈ (0.18,0.59); permeability K ∈ (1.8e − 10,4.4e − 9) 5 / 27
- 7. Expectation and variance of the mass fraction c E[c] ∈ [0,0.35); Var[c] ∈ [0.0,0.04) 6 / 27
- 8. What can we compute? QoIs: c in the whole domain, c at a point, or integral values (the freshwater/saltwater integrals): QFW(t,ω) := Z x∈D I(c(t,x,ω) ≤ 0.012178)dx, (2) Qs(t,ω) := Z x∈D c(t,x,ω)ρ(t,x,ω)dx, (3) Q9(t,ω) := Z x∈∆9 c(t,x,ω)ρ(t,x,ω)dx, (4) where ∆9 := [x9 − 0.1,x9 + 0.1] × [y9 − 0.1,y9 + 0.1]. 7 / 27
- 9. Multi Level Monte Carlo (MLMC) method Spatial and temporal grid hierarchies D0,D1,...,DL, T0,T1,...,TL; n0 = 512, n` ≈ n0 · 16`, τ`+1 = 1 4τ`, r`+1 = 4r` and r` = r04`. Approx. error: kc − ch,τk2 = O(h + τ) = O(n−1/2 + r−1) Computation complexity on level ` is s` = O(n`r`) = O(43`γ n0 · r0) 8 / 27
- 10. Multi Level Monte Carlo (MLMC) method MLMC approximates E[g] ≈ E[gL] using the following telescopic sum: E[gL] = E[g0] + L X `=1 E[g` − g`−1] ≈ ≈ m−1 0 m0 X i=1 g (0,i) 0 + L X `=1 m−1 ` m X̀ i=1 (g (`,i) ` − g (`,i) `−1 ) . Let Y` := m−1 ` Pm` i=1(g (`,i) ` − g (`,i) `−1 ), where g−1 ≡ 0, so that E[Y`] := E[g0], ` = 0 E[g` − g`−1], ` > 0 . (5) 9 / 27
- 11. MLMC notation Denote by Y:= PL `=0 Y` the multilevel estimator of E[g] based on L + 1 levels and m` independent samples on level `, where ` = 0,...,L. Denote V0 := V [g0] and for ` ≥ 1, and V` := V [g` − g`−1], ` ≥ 1. The standard theory states: E[Y] = E[gL], V [Y] = PL `=0 m−1 ` V`. The cost of the multilevel estimator Y is S := PL `=0 m`s`. See details in Giles’18 or Teckentrup’s PhD Thesis, 2013 10 / 27
- 12. Minimization problem For a fixed variance V [Y] = ε2/2, the cost S is minimized by choosing as m` the solution of the optimization problem: F(m0,...,mL) := L X `=0 m`s` + µ2 V` m` obtain m` = 2ε−2 r V` s` L X i=0 p Visi The total complexity is S := 2ε−2 L X `=0 p V`s` 2 11 / 27
- 13. The mean squared error (MSE) Is used to measure the quality of the multilevel estimator: MSE := E h (Y − E[g])2 i = V [Y] + (E[Y] − E[g])2 , (6) where Y is what we computed via MLMC, and E[g] what actually should be computed. To achieve MSE ≤ ε2 for some prescribed tolerance ε, we ensure that both (E[Y] − E[g])2 = (E[gL − g])2 ≤ 1 2ε2 . (7) and V [Y] ≤ 1 2ε2 (8) 12 / 27
- 14. Theorem Consider a fixed t = t∗. Suppose positive constants α,β,γ > 0 exist such that α ≥ 1 2 min(β,γd̂), and |E[g` − g]| ≤ c14−α` (9a) V` ≤ c24−β` (9b) s` ≤ c34d̂γ` . (9c) Then, for any accuracy ε < e−1, a constant c4 > 0 and a sequence of realizations {m`}L `=0 exist, such that MSE := E h (Y − E[g])2 i < ε2 , and the computational cost is S = c4ε−2, β > d̂γ c4ε−2 (log(ε))2 , β = d̂γ c4ε − 2+ d̂γ−β α , β d̂γ. 13 / 27
- 15. Modeling of porosity and recharge: We assume two horizontal layers: y ∈ (−0.8,0] (the upper layer) and y ∈ [−1,−0.8] (the lower layer). The porosity inside each layer is uncertain and is modeled as: φ(x,ξ) = 0.35 · C0(ξ1) · C1(ξ1,ξ2) · C2(ξ1,ξ2) where C0(ξ1) = ( 1.2 · (1 + 0.2ξ1) if y −0.8 1 if y ≥ −0.8 C1(ξ1,ξ2) = 1 + 0.15(ξ2cos(πx/2) − ξ2sin(2πy) + ξ1cos(2πx)) C2(ξ1,ξ2) = 1 + 0.2(ξ1sin(64πx) + ξ2sin(32πy)) Recharge q̂in = −6.6 · 10−2 (1 + 0.5 · ξ3)(1 + sinπt 40), where ξ1, ξ2, and ξ3 are sampled independently and uniformly in [−1,1]. 14 / 27
- 16. Examples: two porosity and two permeability realisations 1st row: porosity φ1 ∈ (0.29,0.49) and φ2 ∈ (0.21,0.54). 2nd row: permeability K1 ∈ (5.9 · 10−10,3.25 · 10−9) and K2 ∈ (1.97 · 10−10,4.5 · 10−9). 15 / 27
- 17. Mean and variance on different levels Comparison of mean values E[c(t,x9,y9)]; and variances Var[c](t,x9,y9) computed on levels 0,1,2,3. We observe: (on the left) that the results obtained on the coarsest scale are not so accurate. All other scales produce more or less similar results. (on the right) Each new finer scale gives better and better results. 16 / 27
- 18. MLMC: weak and strong convergence (left) The mean value E[g` − g`−1] and (right) the variance value V [g` − g`−1] as a function of time for t ∈ [τ,48τ], ` = 1,2,3. For every time point we observe convergence in the mean and in the variance. The amplitude is decreasing. 17 / 27
- 19. QoI is the integral value over D9 ; 100 realisations of g1 − g0, g2 − g1, g3 − g2, QoI g` is the integral value Q9(t,ω) computed over a subdomain around 9th point, t ∈ [τ,48τ]. 18 / 27
- 20. Complexity on each mesh level ` ` n`, ( n` n`−1 ) r`, ( r` r`−1 ) τ` Computing times (s`), ( s` s`−1 ) average min. max. 0 153 94 64 0.6 0.5 0.7 1 2145 (14) 376 (4) 16 7.1 (14) 7 9 2 33153 (15) 1504 (4) 4 253 (36) 246 266 3 525825 (16) 6016 (4) 1 11110 (44) 9860 15507 Here: r` is the number of time steps , τ` is a time step τ` = 6016/r`, #ndofs= n`, average, minimal, and maximal computing times on each level `. The numbers in brackets (column 2 and 3) confirm the theory that the method has order one w.r.t to h and order one w.r.t. the time step τ. 19 / 27
- 21. Rates of the weak and strong convergences (left) Weak (α = 0.94, ζ1 = 3.2) and (right) strong (β = 1.7, ζ2 = 4.8) convergences in log-scale computed for levels 0,1,2,3 (horizontal axis). The QoI is a subdomain integral of c over D9, a domain around point (x,y)9 = (1.65,−0.75). Now the identified convergence rates (red color) can be used to estimate L and all m`. 20 / 27
- 22. Comparison MLMC vs MC ε 0.1 0.05 0.01 ε2 0.01 0.0025 0.0001 MC cost 2.0 · 103 2.8 · 105 3.1 · 108 MLMC cost 6.4 · 101 1.06 · 103 8.9 · 104 required L 2 3 4 m0,m1,m2,m3 44,5,0,0 362,43,3,0 16672,1990,120,4 21 / 27
- 23. Comparison of MC and MLMC for different ε Good news: 1. MLMC (red line) is much faster than MC (blue line) 2. MLMC theory (dashed violet line) fits to the MLMC numerics (red line) 22 / 27
- 24. Conclusion 1. Investigated efficiency of MLMC for Henry problem with uncertain porosity, permeability, and recharge. 2. Uncertainties are modeled by random fields. 3. MLMC could be much faster than MC, 3200 times faster ! 4. The time dependence is challenging. Remarks: 1. Check if MLMC is needed. 2. The optimal number of samples depends on the point (t,x) 3. An advanced MLMC may give better estimates of L and m`. Future work: 1. Consider a more complicated/multiscale/realistic porosity and geometry 2. Incorporate known experimental and measurement data to reduce uncertainties. 23 / 27
- 25. More advanced and realistic computing domain A. Schneider et al., Modeling saltwater intrusion scenarios for a coastal aquifer at the German North Sea, E3S Web of Conferences 54, 00031 (2018) Sandelermöns: 3d hydrogeological model (30x vertically exaggerated) with coarse grid, rivers, pumping wells and recharge map (right). 24 / 27
- 26. More advanced and realistic computing domain A. Schneider et al., Modeling saltwater intrusion scenarios for a coastal aquifer at the German North Sea, E3S Web of Conferences 54, 00031 (2018) Situation of the Sandelermöns model area including the wells of the three waterworks and area of saline groundwater. 25 / 27
- 27. Literature 1. A. Litvinenko, D. Logashenko, R. Tempone, E. Vasilyeva, G. Wittum, Uncertainty quantification in coastal aquifers using the multilevel Monte Carlo method, arXiv:2302.07804, 2023 2. A. Litvinenko, D. Logashenko, R. Tempone, G. Wittum, D. Keyes, Propagation of Uncertainties in Density-Driven Flow, In: Bungartz, HJ., Garcke, J., Pflüger, D. (eds) Sparse Grids and Applications - Munich 2018. LNCSE, Vol. 144, pp 121-126, Springer, Cham. https://doi.org/10.1007/978-3-030-81362-8_52023 3. A .Litvinenko, D. Logashenko, R. Tempone, G. Wittum, D. Keyes, Solution of the 3D density-driven groundwater flow problem with uncertain porosity and permeability GEM-International Journal on Geomathematics Vol. 11, pp 1-29, 2020 4. A Litvinenko, AC Yucel, H Bagci, J Oppelstrup, E Michielssen, R Tempone, Computation of electromagnetic fields scattered from objects with uncertain shapes using multilevel Monte Carlo method, IEEE Journal on Multiscale and Multiphysics Computational Techniques, Vol. 4, pp 37-50, 2019. 5. H.G. Matthies, E. Zander, B.V. Rosić, et al. Parameter estimation via conditional expectation: a Bayesian inversion. Adv. Model. and Simul. in Eng. Sci. 3, 24 (2016). https://doi.org/10.1186/s40323-016-0075-7 26 / 27
- 28. Acknowledgments We thank the KAUST HPC support team for assistance with Shaheen II and for the project k1051. This work was supported by the Alexander von Humboldt foundation. 27 / 27