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.
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,
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.
Propagation of Uncertainties in Density Driven Groundwater FlowAlexander Litvinenko
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.
In many countries, groundwater is the strategic reserve, which is used as drinking water and as an irrigation resource. Therefore, accurate modeling of the pollution of the soil and groundwater aquifer is highly important. As a model, we consider a density-driven groundwater flow problem with uncertain porosity and permeability. This problem may arise in geothermal reservoir simulation, natural saline-disposal basins, modeling of contaminant plumes and subsurface flow.
This strongly non-linear problem describes how salt or polluted water streams down building ''fingers". The solving process requires a very fine unstructured mesh and, therefore, high computational resources. Consequently, we run the parallel multigrid solver UG4 (https://github.com/UG4/ughub.wiki.git) on Shaheen II supercomputer.
The parallelization is done in both - the physical space and the stochastic space. The novelty of this work is the estimation of risks that the pollution will achieve a specific critical concentration. Additionally, we demonstrate how the multigrid UG4 solver can be run in a black-box fashion for testing different scenarios in the density-driven flow.
We solve Elder's problem in 2D and 3D domains, where unknown porosity and permeability are modeled by random fields. For approximations in the stochastic space, we use the generalized polynomial chaos expansion. We compute different quantities of interest such as the mean, variance and exceedance probabilities of the concentration. As a reference solution, we use the solution, obtained from the quasi-Monte Carlo method.
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.
Simulation of propagation of uncertainties in density-driven groundwater flowAlexander Litvinenko
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.
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,
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.
Propagation of Uncertainties in Density Driven Groundwater FlowAlexander Litvinenko
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.
In many countries, groundwater is the strategic reserve, which is used as drinking water and as an irrigation resource. Therefore, accurate modeling of the pollution of the soil and groundwater aquifer is highly important. As a model, we consider a density-driven groundwater flow problem with uncertain porosity and permeability. This problem may arise in geothermal reservoir simulation, natural saline-disposal basins, modeling of contaminant plumes and subsurface flow.
This strongly non-linear problem describes how salt or polluted water streams down building ''fingers". The solving process requires a very fine unstructured mesh and, therefore, high computational resources. Consequently, we run the parallel multigrid solver UG4 (https://github.com/UG4/ughub.wiki.git) on Shaheen II supercomputer.
The parallelization is done in both - the physical space and the stochastic space. The novelty of this work is the estimation of risks that the pollution will achieve a specific critical concentration. Additionally, we demonstrate how the multigrid UG4 solver can be run in a black-box fashion for testing different scenarios in the density-driven flow.
We solve Elder's problem in 2D and 3D domains, where unknown porosity and permeability are modeled by random fields. For approximations in the stochastic space, we use the generalized polynomial chaos expansion. We compute different quantities of interest such as the mean, variance and exceedance probabilities of the concentration. As a reference solution, we use the solution, obtained from the quasi-Monte Carlo method.
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.
Simulation of propagation of uncertainties in density-driven groundwater flowAlexander Litvinenko
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.
In this tutorial I will provide a survey of recent research efforts on the application of QMC methods to PDEs with random coefficients. Such PDE problems occur in the area of uncertainty quantification. A prime example is the flow of water through a disordered porous medium. There is a huge body of literature on this topic using a variety of methods. QMC methods are relatively new to this application area. The aim of this tutorial is to provide an entry point for QMC experts wanting to start research in this direction, for PDE analysts and practitioners wanting to tap into contemporary QMC theory and methods, and for anyone else who sees how to cross-fertilize the ideas to other application areas.
A fundamental numerical problem in many sciences is to compute integrals. These integrals can often be expressed as expectations and then approximated by sampling methods. Monte Carlo sampling is very competitive in high dimensions, but has a slow rate of convergence. One reason for this slowness is that the MC points form clusters and gaps. Quasi-Monte Carlo methods greatly reduce such clusters and gaps, and under modest smoothness demands on the integrand they can greatly improve accuracy. This can even take place in problems of surprisingly high dimension. This talk will introduce the basics of QMC and randomized QMC. It will include discrepancy and the Koksma-Hlawka inequality, some digital constructions and some randomized QMC methods that allow error estimation and sometimes bring improved accuracy.
Dependence of Charge Carriers Mobility in the P-N-Heterojunctions on Composit...msejjournal
In this paper we consider manufacturing a p-n-junctions by dopant diffusion or ion implantation into a multilayer structure. We introduce an approach to increase sharpness of these p-n-junctions and at the same time to increase homogeneity of distributions of dopants in enriched by these dopants areas. We consider influence of the above changing of distribution of dopant on charge carrier mobility. We also consider an approach to decrease value of mismatch-induced stress in the considered multilayer structure by using a buffer layer. The decreasing gives a possibility to increase value of charge carrier mobility.
We research behavior and sharp bounds for the zeros of infinite sequences of polynomials orthogonal with respect to a Geronimus perturbation of a positive Borel measure on the real line.
Density Driven Groundwater Flow with Uncertain Porosity and PermeabilityAlexander Litvinenko
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.
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.
In this tutorial I will provide a survey of recent research efforts on the application of QMC methods to PDEs with random coefficients. Such PDE problems occur in the area of uncertainty quantification. A prime example is the flow of water through a disordered porous medium. There is a huge body of literature on this topic using a variety of methods. QMC methods are relatively new to this application area. The aim of this tutorial is to provide an entry point for QMC experts wanting to start research in this direction, for PDE analysts and practitioners wanting to tap into contemporary QMC theory and methods, and for anyone else who sees how to cross-fertilize the ideas to other application areas.
A fundamental numerical problem in many sciences is to compute integrals. These integrals can often be expressed as expectations and then approximated by sampling methods. Monte Carlo sampling is very competitive in high dimensions, but has a slow rate of convergence. One reason for this slowness is that the MC points form clusters and gaps. Quasi-Monte Carlo methods greatly reduce such clusters and gaps, and under modest smoothness demands on the integrand they can greatly improve accuracy. This can even take place in problems of surprisingly high dimension. This talk will introduce the basics of QMC and randomized QMC. It will include discrepancy and the Koksma-Hlawka inequality, some digital constructions and some randomized QMC methods that allow error estimation and sometimes bring improved accuracy.
Dependence of Charge Carriers Mobility in the P-N-Heterojunctions on Composit...msejjournal
In this paper we consider manufacturing a p-n-junctions by dopant diffusion or ion implantation into a multilayer structure. We introduce an approach to increase sharpness of these p-n-junctions and at the same time to increase homogeneity of distributions of dopants in enriched by these dopants areas. We consider influence of the above changing of distribution of dopant on charge carrier mobility. We also consider an approach to decrease value of mismatch-induced stress in the considered multilayer structure by using a buffer layer. The decreasing gives a possibility to increase value of charge carrier mobility.
We research behavior and sharp bounds for the zeros of infinite sequences of polynomials orthogonal with respect to a Geronimus perturbation of a positive Borel measure on the real line.
Density Driven Groundwater Flow with Uncertain Porosity and PermeabilityAlexander Litvinenko
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.
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.
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.
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`.
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.
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.
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.
Response Surface in Tensor Train format for Uncertainty QuantificationAlexander Litvinenko
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.
ON INCREASING OF DENSITY OF ELEMENTS IN A MULTIVIBRATOR ON BIPOLAR TRANSISTORSijcsitcejournal
In this paper we consider an approach to increase density of elements of a multivibrator on bipolar transistors.
The considered approach based on manufacturing a heterostructure with necessity configuration,
doping by diffusion or ion implantation of required areas to manufacture the required type of conductivity
(p or n) in the areas and optimization of annealing of dopant and/or radiation defects to manufacture more
compact distributions of concentrations of dopants. We also introduce an analytical approach to prognosis
technological process.
Quantum Annealing for Dirichlet Process Mixture Models with Applications to N...Shu Tanaka
Our paper entitled “Quantum Annealing for Dirichlet Process Mixture Models with Applications to Network Clustering" was published in Neurocomputing. This work was done in collaboration with Dr. Issei Sato (Univ. of Tokyo), Dr. Kenichi Kurihara (Google), Professor Seiji Miyashita (Univ. of Tokyo), and Prof. Hiroshi Nakagawa (Univ. of Tokyo).
http://www.sciencedirect.com/science/article/pii/S0925231213005535
The preprint version is available:
http://arxiv.org/abs/1305.4325
佐藤一誠さん(東京大学)、栗原賢一さん(Google)、宮下精二教授(東京大学)、中川裕志教授(東京大学)との共同研究論文 “Quantum Annealing for Dirichlet Process Mixture Models with Applications to Network Clustering" が Neurocomputing に掲載されました。
http://www.sciencedirect.com/science/article/pii/S0925231213005535
プレプリントバージョンは
http://arxiv.org/abs/1305.4325
からご覧いただけます。
Anomalous Diffusion Through Homopolar Membrane: One-Dimensional Model by Guilherme Garcia Gimenez and Adélcio C Oliveira* in Evolutions in Mechanical Engineering
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.
Determination of the Probability Size Distribution
of Solid Particles in a Technical Water by Tomantschger Kurt W*, Petrović Dragan V and Radojević Rade L in Evolutions in Mechanical Engineering
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 presented on this workshop "Workshop: Imaging With Uncertainty Quantification (IUQ), September 2022",
https://people.compute.dtu.dk/pcha/CUQI/IUQworkshop.html
We consider a weakly supervised classification problem. It
is a classification problem where the target variable can be unknown
or uncertain for some subset of samples. This problem appears when
the labeling is impossible, time-consuming, or expensive. Noisy measurements
and lack of data may prevent accurate labeling. Our task
is to build an optimal classification function. For this, we construct and
minimize a specific objective function, which includes the fitting error on
labeled data and a smoothness term. Next, we use covariance and radial AQ1
basis functions to define the degree of similarity between points. The further
process involves the repeated solution of an extensive linear system
with the graph Laplacian operator. To speed up this solution process,
we introduce low-rank approximation techniques. We call the resulting
algorithm WSC-LR. Then we use the WSC-LR algorithm for analysis
CT brain scans to recognize ischemic stroke disease. We also compare
WSC-LR with other well-known machine learning algorithms.
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.
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...Alexander Litvinenko
Very often one has to deal with high-dimensional random variables (RVs). A high-dimensional RV can be described by its probability density (\pdf) and/or by the corresponding probability characteristic functions (\pcf), or by a function representation. Here the interest is mainly to compute characterisations like the entropy, or
relations between two distributions, like their Kullback-Leibler divergence, or more general measures such as $f$-divergences,
among others. 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 represent the density by a high order tensor product, and approximate this in a low-rank format.
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...Alexander Litvinenko
We develop fast and efficient stochastic methods for characterizing scattering
from objects of uncertain shapes. This is highly needed in the
fields of electromagnetics, optics, and photonics.
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
work. Consequently, the total execution time is significantly
reduced, in comparison to the standard MC scheme.
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.
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...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.
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.
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.
In this paper, we solve a semi-supervised regression
problem. Due to the luck of knowledge about the
data structure and the presence of random noise, the considered data model is uncertain. We propose a method which combines graph Laplacian regularization and cluster ensemble methodologies. The co-association matrix of the ensemble is calculated on both labeled and unlabeled data; this matrix is used as a similarity matrix in the regularization framework to derive the predicted outputs. We use the low-rank decomposition of the co-association matrix to significantly speedup calculations and reduce memory. Two clustering problem examples are presented.
Full version is here https://arxiv.org/abs/1901.03919
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.
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 Alexander Litvinenko
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
Willie Nelson Net Worth: A Journey Through Music, Movies, and Business Venturesgreendigital
Willie Nelson is a name that resonates within the world of music and entertainment. Known for his unique voice, and masterful guitar skills. and an extraordinary career spanning several decades. Nelson has become a legend in the country music scene. But, his influence extends far beyond the realm of music. with ventures in acting, writing, activism, and business. This comprehensive article delves into Willie Nelson net worth. exploring the various facets of his career that have contributed to his large fortune.
Follow us on: Pinterest
Introduction
Willie Nelson net worth is a testament to his enduring influence and success in many fields. Born on April 29, 1933, in Abbott, Texas. Nelson's journey from a humble beginning to becoming one of the most iconic figures in American music is nothing short of inspirational. His net worth, which estimated to be around $25 million as of 2024. reflects a career that is as diverse as it is prolific.
Early Life and Musical Beginnings
Humble Origins
Willie Hugh Nelson was born during the Great Depression. a time of significant economic hardship in the United States. Raised by his grandparents. Nelson found solace and inspiration in music from an early age. His grandmother taught him to play the guitar. setting the stage for what would become an illustrious career.
First Steps in Music
Nelson's initial foray into the music industry was fraught with challenges. He moved to Nashville, Tennessee, to pursue his dreams, but success did not come . Working as a songwriter, Nelson penned hits for other artists. which helped him gain a foothold in the competitive music scene. His songwriting skills contributed to his early earnings. laying the foundation for his net worth.
Rise to Stardom
Breakthrough Albums
The 1970s marked a turning point in Willie Nelson's career. His albums "Shotgun Willie" (1973), "Red Headed Stranger" (1975). and "Stardust" (1978) received critical acclaim and commercial success. These albums not only solidified his position in the country music genre. but also introduced his music to a broader audience. The success of these albums played a crucial role in boosting Willie Nelson net worth.
Iconic Songs
Willie Nelson net worth is also attributed to his extensive catalog of hit songs. Tracks like "Blue Eyes Crying in the Rain," "On the Road Again," and "Always on My Mind" have become timeless classics. These songs have not only earned Nelson large royalties but have also ensured his continued relevance in the music industry.
Acting and Film Career
Hollywood Ventures
In addition to his music career, Willie Nelson has also made a mark in Hollywood. His distinctive personality and on-screen presence have landed him roles in several films and television shows. Notable appearances include roles in "The Electric Horseman" (1979), "Honeysuckle Rose" (1980), and "Barbarosa" (1982). These acting gigs have added a significant amount to Willie Nelson net worth.
Television Appearances
Nelson's char
"Understanding the Carbon Cycle: Processes, Human Impacts, and Strategies for...MMariSelvam4
The carbon cycle is a critical component of Earth's environmental system, governing the movement and transformation of carbon through various reservoirs, including the atmosphere, oceans, soil, and living organisms. This complex cycle involves several key processes such as photosynthesis, respiration, decomposition, and carbon sequestration, each contributing to the regulation of carbon levels on the planet.
Human activities, particularly fossil fuel combustion and deforestation, have significantly altered the natural carbon cycle, leading to increased atmospheric carbon dioxide concentrations and driving climate change. Understanding the intricacies of the carbon cycle is essential for assessing the impacts of these changes and developing effective mitigation strategies.
By studying the carbon cycle, scientists can identify carbon sources and sinks, measure carbon fluxes, and predict future trends. This knowledge is crucial for crafting policies aimed at reducing carbon emissions, enhancing carbon storage, and promoting sustainable practices. The carbon cycle's interplay with climate systems, ecosystems, and human activities underscores its importance in maintaining a stable and healthy planet.
In-depth exploration of the carbon cycle reveals the delicate balance required to sustain life and the urgent need to address anthropogenic influences. Through research, education, and policy, we can work towards restoring equilibrium in the carbon cycle and ensuring a sustainable future for generations to come.
Artificial Reefs by Kuddle Life Foundation - May 2024punit537210
Situated in Pondicherry, India, Kuddle Life Foundation is a charitable, non-profit and non-governmental organization (NGO) dedicated to improving the living standards of coastal communities and simultaneously placing a strong emphasis on the protection of marine ecosystems.
One of the key areas we work in is Artificial Reefs. This presentation captures our journey so far and our learnings. We hope you get as excited about marine conservation and artificial reefs as we are.
Please visit our website: https://kuddlelife.org
Our Instagram channel:
@kuddlelifefoundation
Our Linkedin Page:
https://www.linkedin.com/company/kuddlelifefoundation/
and write to us if you have any questions:
info@kuddlelife.org
WRI’s brand new “Food Service Playbook for Promoting Sustainable Food Choices” gives food service operators the very latest strategies for creating dining environments that empower consumers to choose sustainable, plant-rich dishes. This research builds off our first guide for food service, now with industry experience and insights from nearly 350 academic trials.
UNDERSTANDING WHAT GREEN WASHING IS!.pdfJulietMogola
Many companies today use green washing to lure the public into thinking they are conserving the environment but in real sense they are doing more harm. There have been such several cases from very big companies here in Kenya and also globally. This ranges from various sectors from manufacturing and goes to consumer products. Educating people on greenwashing will enable people to make better choices based on their analysis and not on what they see on marketing sites.
Talk Alexander Litvinenko on SIAM GS Conference in Houston
1. Solution of Density Driven Groundwater Flow
with Uncertain Porosity and Permeability
Coefficients
Alexander Litvinenko,
joint work with
Dmitry Logashenko, Raul Tempone, Gabriel Wittum and David Keyes
RWTH Aachen and KAUST
SIAM GS 2019 Conference in Houston, USA
2. The structure of the talk
Major Goal: estimate risks of the pollution in a subsurface flow
How?: we solve density-driven groundwater flow with uncertain
porosity and permeability
Plan:
1. Set up density-driven groundwater flow problem
2. Review stochastic modeling and stochastic methods
3. Modeling of uncertainty in porosity and permeability
4. Numerical methods to solve deterministic problem
5. 2D numerical experiments
6. 3D numerical experiments
7. Conclusion
2
3. Motivation
As groundwater is an essential nutrition and irrigation resource, its
pollution may lead to catastrophic consequences.
seawater intrusion into coastal aquifers,
https://water.usgs.gov/ogw/gwrp/saltwater/salt.html
Saudi Arabia agriculture:
Every dark point is a green
field or farm with diameter
few hundreds meters
Possible applications: seawater intrusion into coastal aquifers,
radioactive waste disposal, contaminant plumes etc.
3
4. What do we compute?
Accurate modeling of the pollution of the soil and groundwater
aquifer is difficult due to presence of uncertainties in geological
parameters.
Use random variables to model input and output uncertainty.
Compute
The mean, variance, and pdf of QoIs
Forecast the pollution map in 1-5-10 years
Estimate the risk that the pollutant concentration exceeds a
certain level
4
5. Governing equations
Use well-known model, see [1,2,3]:
Domain D is filled with two phases: solid porous matrix and
solution of salt in water (brine)
∂t(φρ) + · (ρq) = 0, (1)
∂t(φρc) + · (ρcq − ρD c) = 0, (2)
where c is the mass fraction of the salt, the tensor field D
represents the molecular diffusion and the mechanical dispersion of
the salt. We assume the Darcy’s law for q:
q = −
K
µ
( p − ρg), (3)
where p(t, x) is the hydrostatic pressure and g the gravity,
ρ = ρ(c) density.
5
6. Governing equations
Assume permeability
K = KI, where K = K(φ) ∈ R, and I ∈ Rd×d
the identity matrix.
Use the linear dependence for the density:
ρ(c) = ρ0 + (ρ1 − ρ0)c,
where ρ0 and ρ1 denote the densities of pure water and the brine,
respectively.
Thus, c ∈ [0, 1] with c = 0 corresponding to the pure water and
c = 1 to the saturated solution.
Assume
D = φDmI,
Dm the coefficient of the molecular diffusion. We neglect the
dispersion.
7. Computational domains
c = 1c = 0
c = 0
c = 0
600 m
300 m
150 m
Schema of 3 layers 2D reservoir D = (0, 600) × (0, 150) and 3
realisations of the porosity.
φ(x) ∈ [0.05, 0.09], φ(x) ∈ [0.077, 0.11], and φ(x) ∈ [0.097, 0.115].
7
8. Two 3D reservoirs
(left) D = (0, 600) × (0, 600) × (0, 150) m3.
BC: Zero-flux for the entire fluid phase; concentration: c = 1 in
the red spot, c = 0 otherwise on the top and Neumann-0 at the
other boundaries.
8
9. Mathematical equation
We introduce uncertain porosity φ = φ(x, θ), x ∈ D,
θ = (θ1, ..., θM, ...), θi is a random variable.
Assume that permeability (Kozeny-Carman-like equation) is
K(φ) = κKC ·
φ3
1 − φ2
, (4)
where κKC is a scaling factor.
9
10. Statistics
Can apply sampling methods, such as (quasi-)Monte Carlo,
collocations, sparse grids to evaluate:
The empirical mean
c(t, x) ≈
Nq
i=1
wi c(t, x, θi )
def
=
Nq
i=1
wi ci ,
where Nq - number of quadrature points, wi quadrature weights, ci
are “scenarios”.
The empirical variance
Var[c](t, x) ≈
Nq
i=1
wi (c(t, x) − c(t, x, θi ))2
.
Exceedance probability (risks)
P(c > c∗
) ≈
#{c(θi ) : c(θi ) > c∗, i = 1, . . . , Ns}
Ns
.
10
11. gPCE based surrogate
An alternative to sampling is a functional approximation:
We approximate unknown QoI by a surrogate (e.g., gPCE)
c(t, x, θ) =
β∈J
cβ(t, x)Ψβ(θ) ≈ c(t, x, θ) =
β∈JM,p
cβ(t, x)Ψβ(θ),
where {Ψβ} is a multivariate Legendre basis, β = (β1, ..., βj , ...) a
multiindex and J a multiindex set.
Ψβ(θ) :=
∞
j=1
ψβj
(θj ); ∀θ ∈ RN
,
ψβj
(·) are Legendre monomials, and coefficients
cβ(t, x) ≈
1
Ψβ, Ψβ
Nq
i=1
Ψβ(t, θi )c(t, x, θi )wi ,
11
12. Truncation and approximation errors
By introducing gPCE surrogate, we also introduce truncation and
approximation errors:
Et = c(t, x, θ)−c(t, x, θ)| =
β∈Jc
cβ(t, x)Ψβ(θ), JM,p∪Jc = J .
Additionally, cβ(t, x) ≈ cβ(t, x):
Ea =
β∈JM,p
cβ(t, x)Ψβ(θ) −
β∈JM,p
cβ(t, x)Ψβ(θ)
Et+Ea =
β∈Jc
cβ(t, x)Ψβ(θ)
truncation error
+
β∈JM,p
(cβ(t, x) − cβ(t, x))Ψβ(θ)
approximation error
.
See [Constantine’12, Sinsbeck’15,Conrad’13].
12
13. Utilised numerical methods to solve one realization
UG4 is a flexible software system for simulating PDE based models
on high performance parallel clusters (G. Wittum and his group).
Computation of one scenario:
1. Spatial discretization on unstructured grids.
2. Implicit Euler schema in time.
3. Newton method with line search.
4. Solution of linearized systems by BiCGStab with multigrid
preconditioning (V-cycle, ILU-smoothers).
5. Parallelisation is based on the distribution of the domain
between cores.
M scenarios are computed in parallel.
Run on 4-8 spatial grid levels with n = 0.5 . . . 8 Mio grid points.
Used 1 . . . 800 Shaheen nodes, computation time is 2-24 hours,
1000-1800 time steps, modeling time interval 5 − 8 years.
13
14. Ex.1: 2D example with 1 RV and small variance
φ(x, ω) = 0.09 + 0.005ξ(cos(x/300) + sin(y/150)),
where ξ ∼ U[−1, 1], in 5.5 years.
1st row: c(x) ∈ (0, 1) computed via qMC (200 simulations) and
via gPCE4 (m = 1, p = 4);
2nd row: Var[c]qMC ∈ (0, 0.021), Var[c]gPCE4 ∈ (0, 0.023).
Observed: 9 GL points gives almost the same result as 200 qMC.
14
15. Ex.2: 2D example with 2 RVs and larger variance
φ(x, ω) = 0.1 + 0.01(ξ1 cos(x/1200) + ξ2 sin(y/300)),
where ξ1, ξ2 ∼ U[−1, 1], in 1.5 years.
1st row: c(x), computed via qMC (1500 simulations) and via
gPCE, c(x) ∈ (0, 1), Var[c]qMC ∈ (0, 0.076),
2nd row: Var[c]gPCE5 ∈ (0, 0.068), Var[c]gPCE7 ∈ (0, 0.0714),
Var[c]gPCE9 ∈ (0, 0.0847).
Observed: Our surrogate and qMC give similar c.
Var[c] computed by surrogate of order 7 is most close to the qMC
variance. 15
16. Ex.3: Difficulties caused by uncertain porosity
Relative small variations in the porosity may result in 3 different
realizations of the mass fraction: with (a) 5 fingers; (b) 4 fingers;
(c) 4.5 fingers.
(a) (b) (c)
Non-linearity may result in several stationary solutions.
16
17. Ex.4: Evolution of variance in time
Below we plot Var[c] after 2.75, 5.5 and 8.25 years.
The variance is accumulated and growing.
(a) 2.75year,
Var[c](x) ∈ (0, 0.023)
(b) 5.5 year,
Var[c](x) ∈ (0, 0.055)
(c) 8.25year,
Var[c](x) ∈ (0, 0.07)
Results are obtained with 700 quasi qMC samples.
17
18. Ex.5. Elder’s problem with 5 RVs and three layers
φ(x, y, ω) =
0.08 + 0.01 5
i=1 θi sin(ixπ/600) sin(iyπ/150), 120 ≤ y ≤ 150.
0.06 + 0.01 5
i=1 θi sin(ixπ/600) sin(iyπ/150), 50 ≤ y < 120
0.09 + 0.01 5
i=1 θi sin(ixπ/600) sin(iyπ/150), 0 ≤ y < 50
(a) cgPCE; (b) cqMC in t = 5.5 years.
(c) Var[c]gPC (x, t) ∈ [0, 0.0466]; (d) Var[c]qMC (x, t) ∈ [0, 0.0556].
(e) porosity φ ∈ [0.0514, 0.09]; (f) c(x, t, 0) − c(x, t) (difference
between deterministic solution (corresponding to θ = 0) and the
mean value in t = 5.5 years,
18
19. Ex.6: 3D reservoir
φ(x, θ) = 0.1 + exp(θ1 sin(πx/600) + θ2 sin(πy/600) + θ3 sin(πz/150) + θ1 sin(πx/600)+
+ θ1 sin(πx/600) sin(πy/600) + θ2 sin(πx/600) sin(πz/150) + θ3 sin(πy/600) sin(πz/150)).
(a) (b)
Five isosurfaces of c after 9.6 years
19
20. Ex.7: Isosurfaces of Var[c] in 3D reservoir
(a) (b)
Var[c] after 4.8 years, N ≈ 8 · 106 grid points.
20
21. Ex.8: Propagation of the contamination
Evolution of the mean concentration in time after a) 0, b) 0.55, c)
1.1, d) 2.2 years. The cutting plane is (150, y, z). 21
22. Ex.9: Evolution of probability density function in a point
PDFs at aquifer point (100, 0, −25) after (a) 0.6, (b) 1.2, (c) 1.8,
and (d) 2.4 years.
22
23. Ex.10: Isosurfaces of Var[c] in 3D reservoir
(a) (b)
Isosurfaces of the variance of the mass fraction after 3 years;
(a) surface where Var[c] = 0.05; (b) surface where Var[c] = 0.15.
23
24. Ex.11: 3D reservoir with three layers
1st row: three layers of the porosity; two profiles of c;
2nd row: isosurface |cdet − c|0.25; isosurfaces Var[c]0.05 and
Var[c]0.12.
24
25. Conclusion
Solved time-dependent density driven flow problem with
uncertain porosity and permeability in 2D and 3D
Computed propagation of uncertainties from porosity and
permeability into the mass fraction. Computed the mean,
variance, exceedance probabilities, quantiles and risks
Such QoIs as the number of fingers, their size, shape,
propagation time can be unstable
For moderate perturbations, our gPCE surrogate results are
similar to qMC results
Used highly scalable parallel solver on up to 800 computing
nodes
25
26. Acknowledgement
1. Alexander von Humboldt foundation
2. Elmar Zander (TU Braunschweig) for the sglib library.
3. KAUST, Shaheen project k1051, 2.7 Mio hours.
4. KAUST Supercomputing Lab.
5. KAUST Visualization Lab.
THANK YOU FOR YOUR ATTENTION !
26
27. Literature
1. S. Reiter, A. Vogel, I. Heppner, M. Rupp, and G. Wittum, A massively parallel geometric multigrid solver
on hierarchically distributed grids, Computing and visualization in science 16, 4 (2013), pp 151-164, DOI:
10.1007/s00791-014-0231-x
2. A. Vogel, S. Reiter, M. Rupp, A. N¨agel, and G. Wittum, UG4 – a novel flexible software system for
simulating PDE based models on high performance computers. Computing and visualization in science 16,
4 (2013), pp 165-179, DOI: 10.1007/s00791-014-0232-9
3. A. Schneider, H. Zhao, J. Wolf, D. Logashenko, S. Reiter, M. Howahr, M. Eley, M. Gelleszun, H.
Wiederhold, Modeling saltwater intrusion scenarios for a coastal aquifer at the German North Sea, E3S
Web of Conferences 54, 00031 (2018), DOI:10.1051/e3sconf/20185400031
4. P. Waehnert, W.Hackbusch, M. Espig, A. Litvinenko, H. Matthies: Efficient low-rank approximation of the
stochastic Galerkin matrix in the tensor format, Computers & Mathematics with Applications, 67 (4), pp
818-829, 2014
5. A. Litvinenko, D. Keyes, V. Khoromskaia, B. N. Khoromskij, H. G. Matthies, Tucker Tensor Analysis of
Matern Functions in Spatial Statistics, DOI: 10.1515/cmam-2018-0022, Computational Methods in
Applied Mathematics , 2018.
6. A. Litvinenko, Application of hierarchical matrices for solving multiscale problems, Doctoral Dissertation,
Leipzig University, Germany, http://www.wire.tu-bs.de/mitarbeiter/litvinen/diss.pdf
27
28. Literature
7. S. Dolgov, B.N. Khoromskij, A. Litvinenko, H.G. Matthies, Computation of the Response Surface in the
Tensor Train data format arXiv preprint arXiv:1406.2816, 2014
8. S. Dolgov, B.N. Khoromskij, A. Litvinenko, H.G. Matthies, Polynomial Chaos Expansion of Random
Coefficients and the Solution of Stochastic Partial Differential Equations in the Tensor Train Format,
IAM/ASA J. Uncertainty Quantification 3 (1), pp 1109-1135, 2015
9. A. Litvinenko, H.G. Matthies, T.A. El-Moselhy, Sampling and low-rank tensor approximation of the
response surface, Monte Carlo and Quasi-Monte Carlo Methods 2012, pp 535-551, Springer, 2013.
10. A. Litvinenko, H.G. Matthies, Inverse problems and uncertainty quantification, arXiv:1312.5048, 2013.
11. A. Litvinenko, Y. Sun, M. G. Genton, D. E. Keyes, Likelihood approximation with hierarchical matrices for
large spatial datasets, Computational Statistics & Data Analysis, Volume 137, 2019, pp 115-132, ISSN
0167-9473, https://doi.org/10.1016/j.csda.2019.02.002.
12. A. Litvinenko, A. C. Yucel, H. Bagci, J. Oppelstrup, E. Michielssen and R. Tempone, Computation of
Electromagnetic Fields Scattered From Objects With Uncertain Shapes Using Multilevel Monte Carlo
Method, in IEEE Journal on Multiscale and Multiphysics Computational Techniques, vol. 4, pp. 37-50,
2019. doi: 10.1109/JMMCT.2019.2897490
13. V. Berikov, A. Litvinenko, Semi-Supervised Regression using Cluster Ensemble and Low-Rank
Co-Association Matrix Decomposition under Uncertainties, arXiv preprint
https://arxiv.org/abs/1901.03919, 2019
14. E. Bernholdt, David, R. Ciancosa, Mark, L. Green, David, J.H. Law, Kody, Litvinenko, Alexander, and M.
Park, Jin. Comparing theory based and higher-order reduced models for fusion simulation data. United
States: N. p., 2018. Web. doi:10.3934/BigDIA.2018.2.41. (2018)
28