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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.

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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.

litvinenko_Henry_Intrusion_Hong-Kong_2024.pdf

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.

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_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.

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.

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.

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.

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.

litvinenko_Henry_Intrusion_Hong-Kong_2024.pdf

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.

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_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.

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.

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.

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.

Bayesian inference on mixtures

This document discusses Bayesian inference on mixtures models. It covers several key topics:
1. Density approximation and consistency results for mixtures as a way to approximate unknown distributions.
2. The "scarcity phenomenon" where the posterior probabilities of most component allocations in mixture models are zero, concentrating on just a few high probability allocations.
3. Challenges with Bayesian inference for mixtures, including identifiability issues, label switching, and complex combinatorial calculations required to integrate over all possible component allocations.

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.

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.kactl.pdf

This document contains information about data structures and algorithms taught at KTH Royal Institute of Technology. It includes code templates for a contest, descriptions and implementations of common data structures like an order statistic tree and hash map, as well as summaries of mathematical and algorithmic concepts like trigonometry, probability theory, and Markov chains.

Litvinenko low-rank kriging +FFT poster

We combined: low-rank tensor techniques and FFT to compute kriging, estimate variance, compute conditional covariance. We are able to solve 3D problems with very high resolution

Response Surface in Tensor Train format for Uncertainty Quantification

We apply low-rank Tensor Train format to solve PDEs with uncertain coefficients. First, we approximate uncertain permeability coefficient in TT format, then the operator and then apply iterations to solve stochastic Galerkin system.

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.Possible applications of low-rank tensors in statistics and UQ (my talk in Bo...

Possible applications of low-rank tensors in statistics and UQ (my talk in Bo...Alexander Litvinenko

Just some ideas how low-rank matrices/tensors can be useful in spatial and environmental statistics, where one usually has to deal with very large dataComputing 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.
The Multivariate Gaussian Probability Distribution

The document discusses the multivariate Gaussian probability distribution. It defines the distribution and provides its probability density function. It then discusses various properties including: functions of Gaussian variables such as linear transformations and addition; the characteristic function and how to calculate moments; marginalization and conditional distributions. It also provides some tips and tricks for working with Gaussian distributions including how to calculate products.

2018 MUMS Fall Course - Mathematical surrogate and reduced-order models - Ral...

2018 MUMS Fall Course - Mathematical surrogate and reduced-order models - Ral...The Statistical and Applied Mathematical Sciences Institute

This document discusses uncertainty propagation techniques for determining statistics of model outputs given uncertain model inputs. It covers analytic approaches for linear models, perturbation methods for nonlinear models, and direct sampling methods. It also discusses computing moments using stochastic spectral methods like stochastic Galerkin with polynomial chaos. The document provides an example of applying perturbation and sampling methods to a nonlinear oscillator model with uncertain parameters. It compares the results from both approaches to the true natural frequency. Finally, it discusses uncertainty quantification for a HIV model and the use of prediction intervals in nuclear power plant design.Adaptive Restore algorithm & importance Monte Carlo

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

A series of maximum entropy upper bounds of the differential entropy

A series of maximum entropy upper bounds of the differential entropy
https://arxiv.org/abs/1612.02954

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.

QMC Program: Trends and Advances in Monte Carlo Sampling Algorithms Workshop,...

QMC Program: Trends and Advances in Monte Carlo Sampling Algorithms Workshop,...The Statistical and Applied Mathematical Sciences Institute

This talk was presented as part of the Trends and Advances in Monte Carlo Sampling Algorithms Workshop.Solutions Manual for Friendly Introduction To Numerical Analysis 1st Edition ...

Full download : https://goo.gl/VJs7j8 Solutions Manual for Friendly Introduction To Numerical Analysis 1st Edition by Bradie

Hypocenter

This document describes using a probabilistic approach to determine the hypocentral coordinates (location and origin time) of a seismic event based on arrival time data from multiple seismic stations. It provides the mathematical framework and shows example Mathematica code for implementing the approach. The code defines probability densities for the arrival time data and location priors, calculates predicted arrival times as a function of location/time, and combines these into a posterior probability density function for the hypocentral coordinates and origin time. Graphs of the marginal spatial and temporal probability densities are generated to visualize the solution.

Computational Information Geometry on Matrix Manifolds (ICTP 2013)

Computational Information Geometry
on Matrix Manifolds
http://cdsagenda5.ictp.trieste.it/full_display.php?ida=a12193

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.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

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.

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.

Bayesian inference on mixtures

This document discusses Bayesian inference on mixtures models. It covers several key topics:
1. Density approximation and consistency results for mixtures as a way to approximate unknown distributions.
2. The "scarcity phenomenon" where the posterior probabilities of most component allocations in mixture models are zero, concentrating on just a few high probability allocations.
3. Challenges with Bayesian inference for mixtures, including identifiability issues, label switching, and complex combinatorial calculations required to integrate over all possible component allocations.

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.

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.kactl.pdf

This document contains information about data structures and algorithms taught at KTH Royal Institute of Technology. It includes code templates for a contest, descriptions and implementations of common data structures like an order statistic tree and hash map, as well as summaries of mathematical and algorithmic concepts like trigonometry, probability theory, and Markov chains.

Litvinenko low-rank kriging +FFT poster

We combined: low-rank tensor techniques and FFT to compute kriging, estimate variance, compute conditional covariance. We are able to solve 3D problems with very high resolution

Response Surface in Tensor Train format for Uncertainty Quantification

We apply low-rank Tensor Train format to solve PDEs with uncertain coefficients. First, we approximate uncertain permeability coefficient in TT format, then the operator and then apply iterations to solve stochastic Galerkin system.

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.Possible applications of low-rank tensors in statistics and UQ (my talk in Bo...

Possible applications of low-rank tensors in statistics and UQ (my talk in Bo...Alexander Litvinenko

Just some ideas how low-rank matrices/tensors can be useful in spatial and environmental statistics, where one usually has to deal with very large dataComputing 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.
The Multivariate Gaussian Probability Distribution

The document discusses the multivariate Gaussian probability distribution. It defines the distribution and provides its probability density function. It then discusses various properties including: functions of Gaussian variables such as linear transformations and addition; the characteristic function and how to calculate moments; marginalization and conditional distributions. It also provides some tips and tricks for working with Gaussian distributions including how to calculate products.

2018 MUMS Fall Course - Mathematical surrogate and reduced-order models - Ral...

2018 MUMS Fall Course - Mathematical surrogate and reduced-order models - Ral...The Statistical and Applied Mathematical Sciences Institute

This document discusses uncertainty propagation techniques for determining statistics of model outputs given uncertain model inputs. It covers analytic approaches for linear models, perturbation methods for nonlinear models, and direct sampling methods. It also discusses computing moments using stochastic spectral methods like stochastic Galerkin with polynomial chaos. The document provides an example of applying perturbation and sampling methods to a nonlinear oscillator model with uncertain parameters. It compares the results from both approaches to the true natural frequency. Finally, it discusses uncertainty quantification for a HIV model and the use of prediction intervals in nuclear power plant design.Adaptive Restore algorithm & importance Monte Carlo

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

A series of maximum entropy upper bounds of the differential entropy

A series of maximum entropy upper bounds of the differential entropy
https://arxiv.org/abs/1612.02954

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.

QMC Program: Trends and Advances in Monte Carlo Sampling Algorithms Workshop,...

QMC Program: Trends and Advances in Monte Carlo Sampling Algorithms Workshop,...The Statistical and Applied Mathematical Sciences Institute

This talk was presented as part of the Trends and Advances in Monte Carlo Sampling Algorithms Workshop.Solutions Manual for Friendly Introduction To Numerical Analysis 1st Edition ...

Full download : https://goo.gl/VJs7j8 Solutions Manual for Friendly Introduction To Numerical Analysis 1st Edition by Bradie

Hypocenter

This document describes using a probabilistic approach to determine the hypocentral coordinates (location and origin time) of a seismic event based on arrival time data from multiple seismic stations. It provides the mathematical framework and shows example Mathematica code for implementing the approach. The code defines probability densities for the arrival time data and location priors, calculates predicted arrival times as a function of location/time, and combines these into a posterior probability density function for the hypocentral coordinates and origin time. Graphs of the marginal spatial and temporal probability densities are generated to visualize the solution.

Computational Information Geometry on Matrix Manifolds (ICTP 2013)

Computational Information Geometry
on Matrix Manifolds
http://cdsagenda5.ictp.trieste.it/full_display.php?ida=a12193

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.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

Bayesian inference on mixtures

Bayesian inference on mixtures

Propagation of Uncertainties in Density Driven Groundwater Flow

Propagation of Uncertainties in Density Driven Groundwater Flow

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

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

kactl.pdf

kactl.pdf

Litvinenko low-rank kriging +FFT poster

Litvinenko low-rank kriging +FFT poster

Response Surface in Tensor Train format for Uncertainty Quantification

Response Surface in Tensor Train format for Uncertainty Quantification

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

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

Possible applications of low-rank tensors in statistics and UQ (my talk in Bo...

Possible applications of low-rank tensors in statistics and UQ (my talk in Bo...

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

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

The Multivariate Gaussian Probability Distribution

The Multivariate Gaussian Probability Distribution

2018 MUMS Fall Course - Mathematical surrogate and reduced-order models - Ral...

2018 MUMS Fall Course - Mathematical surrogate and reduced-order models - Ral...

Adaptive Restore algorithm & importance Monte Carlo

Adaptive Restore algorithm & importance Monte Carlo

A series of maximum entropy upper bounds of the differential entropy

A series of maximum entropy upper bounds of the differential entropy

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

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

QMC Program: Trends and Advances in Monte Carlo Sampling Algorithms Workshop,...

QMC Program: Trends and Advances in Monte Carlo Sampling Algorithms Workshop,...

Solutions Manual for Friendly Introduction To Numerical Analysis 1st Edition ...

Solutions Manual for Friendly Introduction To Numerical Analysis 1st Edition ...

Hypocenter

Hypocenter

Computational Information Geometry on Matrix Manifolds (ICTP 2013)

Computational Information Geometry on Matrix Manifolds (ICTP 2013)

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

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

Tucker tensor analysis of Matern functions in spatial statistics

Tucker tensor analysis of Matern functions in spatial statistics

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.

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.

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.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

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.

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,
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.Density Driven Groundwater Flow with Uncertain Porosity and Permeability

Density Driven Groundwater Flow with Uncertain Porosity and Permeability

Litvinenko_RWTH_UQ_Seminar_talk.pdf

Litvinenko_RWTH_UQ_Seminar_talk.pdf

Litv_Denmark_Weak_Supervised_Learning.pdf

Litv_Denmark_Weak_Supervised_Learning.pdf

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...

Low-rank tensor approximation (Introduction)

Low-rank tensor approximation (Introduction)

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

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

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

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...

Level 3 NCEA - NZ: A Nation In the Making 1872 - 1900 SML.ppt

The History of NZ 1870-1900.
Making of a Nation.
From the NZ Wars to Liberals,
Richard Seddon, George Grey,
Social Laboratory, New Zealand,
Confiscations, Kotahitanga, Kingitanga, Parliament, Suffrage, Repudiation, Economic Change, Agriculture, Gold Mining, Timber, Flax, Sheep, Dairying,

Geography as a Discipline Chapter 1 __ Class 11 Geography NCERT _ Class Notes...

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INDIA`S OWN LITERARY GENIUS MR.KHUSHWANT SINGH WAS TRULY A VERY BRAVE SOUL AND WAS AWARDED WITH THE MAGIC OF WORDS BY GOD.

Gender and Mental Health - Counselling and Family Therapy Applications and In...

A proprietary approach developed by bringing together the best of learning theories from Psychology, design principles from the world of visualization, and pedagogical methods from over a decade of training experience, that enables you to: Learn better, faster!

BÀI TẬP BỔ TRỢ TIẾNG ANH LỚP 8 - CẢ NĂM - FRIENDS PLUS - NĂM HỌC 2023-2024 (B...

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https://app.box.com/s/nrwz52lilmrw6m5kqeqn83q6vbdp8yzpHYPERTENSION - SLIDE SHARE PRESENTATION.

IT WILL BE HELPFULL FOR THE NUSING STUDENTS
IT FOCUSED ON MEDICAL MANAGEMENT AND NURSING MANAGEMENT.
HIGHLIGHTS ON HEALTH EDUCATION.

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Educational Technology in the Health Sciences

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Skimbleshanks-The-Railway-Cat by T S Eliot

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Andreas Schleicher presents PISA 2022 Volume III - Creative Thinking - 18 Jun...

Andreas Schleicher, Director of Education and Skills at the OECD presents at the launch of PISA 2022 Volume III - Creative Minds, Creative Schools on 18 June 2024.

Temple of Asclepius in Thrace. Excavation results

The temple and the sanctuary around were dedicated to Asklepios Zmidrenus. This name has been known since 1875 when an inscription dedicated to him was discovered in Rome. The inscription is dated in 227 AD and was left by soldiers originating from the city of Philippopolis (modern Plovdiv).

مصحف القراءات العشر أعد أحرف الخلاف سمير بسيوني.pdf

مصحف أحرف الخلاف للقراء العشرةأعد أحرف الخلاف بالتلوين وصلا سمير بسيوني غفر الله له

Oliver Asks for More by Charles Dickens (9)

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A Visual Guide to 1 Samuel | A Tale of Two Hearts

These slides walk through the story of 1 Samuel. Samuel is the last judge of Israel. The people reject God and want a king. Saul is anointed as the first king, but he is not a good king. David, the shepherd boy is anointed and Saul is envious of him. David shows honor while Saul continues to self destruct.

Level 3 NCEA - NZ: A Nation In the Making 1872 - 1900 SML.ppt

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Geography as a Discipline Chapter 1 __ Class 11 Geography NCERT _ Class Notes...

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CIS 4200-02 Group 1 Final Project Report (1).pdf

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Gender and Mental Health - Counselling and Family Therapy Applications and In...

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A Visual Guide to 1 Samuel | A Tale of Two Hearts

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- 1. Density driven flow in fractured porous media. Uncertainty quantification via MLMC. Alexander Litvinenko1 , D. Logashenko2 , R. Tempone1,2 , G. Wittum2 1 RWTH Aachen, Germany, 2 KAUST, Saudi Arabia litvinenko@uq.rwth-aachen.de Abstract Problem: Henry-like saltwater intrusion in fractured media (nonlinear, time- dependent, two-phase flow) Input uncertainty: fracture width, porosity, permeability, and recharge (mod- elled 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 Questions: 1.How does fracture affect flow? 2.Where is the largest uncertainty? 3.What is the mean scenario and its variations? 4.What are the extreme scenarios? 5.How do the uncertainties change over time? q̂in c = 0 c = 1 p = −ρ1gy 0 −1 m 2 m y x (2, −0.5) (1, −0.7) 1 2 3 4 5 6 (left) Henry problem with a fracture. (right) Mass fraction cm (red for cm = 1) 1. Henry-like problem settings Denote: the whole domain D, porous medium M ⊂ D, fracture porosity ϕm and permeability Km, salt mass fraction cm(t, x) and pressure pm(t, x). The flow in M satisfies conservation laws: ∂t(ϕmρm) + ∇ · (ρmqm) = 0 ∂t(ϕmρmcm) + ∇ · (ρmcmqm − ρmDm∇cm) = 0 x ∈ M (1) with the Darcy’s law for the velocity: qm = − Km µm (∇pm − ρmg), x ∈ M , (2) where ρm = ρ(cm) and µm = µ(cm) indicate the density and the viscosity of the liquid phase, Dm(t, x) denotes the molecular diffusion and mechanical dispersion tensor. The fracture is assumed to be filled with a porous medium, too. The fracture is represented by a surface S ⊂ D, M ∪ S = D, M ∩ S = ∅. ∂t(ϕfϵρf) + ∇S · (ϵρfqf) + Q (1) fn + Q (2) fn = 0 ∂t(ϕfϵρfcf) + ∇S · (ϵρfcfqf − ϵρfDf∇S cf) + P (1) fn + P (2) fn = 0 x ∈ S , (3) where ϵ is the fracture width. The Darcy velocity along the fracture is qf = − Kf µf (∇S pf − ρfg), x ∈ S . (4) The terms Q (k) fn and P (k) fn , k ∈ {1, 2}, are the mass fluxes through the faces S (k) of the fracture: Q (k) fn := ρ(c (k) m )q (k) fn P (k) fn := ρ(c (k) m )c (k) upwindq (k) fn − ρ(c (k) m )D (k) fn c (k) m − cf ϵ/2 x ∈ S (k), (5) The uncertain width of the fracture, the recharge, and the porosity are modeled as follows, ξ1, ξ2, ξ3 ∈ U[−1, 1], ϵ(ξ1) = 0.01 · ((1 − 0.01) · ξ1 + (1 + 0.01))/2, (6) q̂x(t, ξ3) = 3.3 · 10−6 · (1 + 0.1ξ3)(1 + 0.1 sin(πt/40)), (7) ϕm(x, y, ξ2) = 0.35 · (1 + 0.02 · (ξ2 cos(πx/2) + ξ2 sin(2πy)). (8) To compute: cm. 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. Methods: Newton method, BiCGStab, preconditioned with the geometric multigrid method (V-cycle), ILUβ- smoothers and Gaussian elimination. Multi Level Monte Carlo (MLMC) method Spatial and temporal grid hierarchies D0, D1, . . . , DL, and T0, T1, . . . , TL; n0 = 512, nℓ ≈ n0 · 4ℓ, τℓ+1 = 1 2τℓ, number of time steps rℓ+1 = 2rℓ or rℓ = r02ℓ. Computation complexity is sℓ = O(nℓrℓ) = O(23ℓγn0 · r0) MLMC approximates E [gL] ≈ E [g] using the following telescopic sum: E [gL] ≈ m−1 0 m0 X i=1 g (0,i) 0 + L X ℓ=1 m−1 ℓ mℓ X i=1 (g (ℓ,i) ℓ − g (ℓ,i) ℓ−1) . Minimize F(m0, . . . , mL) := PL ℓ=0 mℓsℓ + µ2 Vℓ mℓ , obtain mℓ = ε−2 q Vℓ sℓ PL i=0 √ Visi. 2. Numerics Figure 2: The mean value E [cm(t, x)] at t = {7τ, 19τ, 40τ, 94τ}. In all cases, E [cm] (t, x) ∈ [0, 1]. Figure 3: The variance V [cm(t, x)] at t = {7τ, 19τ, 40τ, 94τ}. Maximal values (dark red colour) of V [cm] are 1.9·10−3, 3.4 · 10−3, 2.9 · 10−3, 2.4 · 10−3 respectively. The dark blue colour corresponds to a zero value. g 0 g 1 - g 0 g 2 - g 1 g 3 - g 2 g 4 - g 3 -6 -4 -2 0 g 0 g 1 - g 0 g 2 - g 1 g 3 - g 2 g 4 - g 3 -16 -14 -12 -10 -8 -6 -4 0 1 2 3 4 10 -3 10 -2 0 1 2 3 4 10 -8 10 -7 10 -6 10 -5 10 -4 Figure 4: (1-2)The weak and the strong convergences, QoI is g := cm(t15, x1), α = 1.07, ζ1 = −1.1, β = 1.97, ζ2 = −8. (3-4) Decay comparison of (left) E [gℓ − gℓ−1] and E [gℓ] vs. ℓ; (right) V [gℓ − gℓ−1] and V [gℓ]. QoI g = cm(t18, x1), x1 = (1.1, −0.8). Level ℓ nℓ, ( nℓ nℓ−1 ) rℓ, ( rℓ rℓ−1 ) τℓ = 6016/rℓ Computing times (sℓ), ( sℓ sℓ−1 ) average min. max. 0 608 188 32 3 2.4 3.4 1 2368 (3.9) 376 (2) 16 22 (7.3) 15.5 27.8 2 9344 (3.9) 752 (2) 8 189 (8.6) 115 237 3 37120 (4) 1504 (2) 4 1831 (10) 882 2363 4 147968 (4) 3008 (2) 2 18580 (10) 7865 25418 Table 1: # ndofs nℓ, number of time steps rℓ, step size in time τℓ, average, minimal, and maxi- mal comp. times. Estimated the weak and strong convergence rates α and β, and constants C1 and C2 for different QoIs. QoI are so- lution in a point xi and an integral Ii(t, ω) := R x∈∆i cm(t, x, ω)ρ(cm(t, x, ω))dx, ∆i := [xi−0.1, xi+0.1]×[yi−0.1, yi+0.1], i = 1, . . . , 6. QoI α c1 β C2 cm(x1, 15τ) -1.07 0.47 -1.97 4 · 10−3 I2(cm) -1.5 8.3 -2.6 0.4 ε m0 m1 m2 m3 m4 0.2 28 0 0 0 0 0.1 10 1 0 0 0 0.05 57 4 1 0 0 0.025 342 22 4 1 0 0.01 3278 205 35 6 1 0 10 20 30 40 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 10 20 30 40 0 1 2 3 4 5 10 -4 0 10 20 30 40 -8 -6 -4 -2 0 10 -3 0 10 20 30 40 0 0.2 0.4 0.6 0.8 1 10 -6 Figure 5: (1) The coefficient of variance CVℓ := CV (gℓ), (2) the variance V [gℓ], (3) the mean E [gℓ − gℓ−1], (4) the variance V [gℓ − gℓ−1]. The QoI is g = cm(t, x1). The small oscillations in the two lower pictures are due to the dependence of the recharge q̂x on the time, cf. (7). 10 -2 10 -1 10 0 10 5 10 10 Complexity comparison of ML and MLMC vs. accuracy ε (horizontal axis) in log-log scale. Acknowledgements: Alexander von Humboldt foundation and KAUST HPC. 1. D. Logashenko, A. Litvinenko, R. Tempone, E. Vasilyeva, G. Wittum, Uncertainty quantification in the Henry problem using the multilevel Monte Carlo method, Journal of Computational Physics, Vol. 503, 2024, 112854, https://doi.org/10.1016/j.jcp.2024.112854 2. D. Logashenko, A. Litvinenko, R. Tempone, G. Wittum, Estimation of uncertainties in the density driven flow in fractured porous media using MLMC, arXiv:2404.18003, 2024