arXiv subjects

Bedřich Sousedík

Publications and source records attributed to Bedřich Sousedík.

At least 19 recordsLinked to original sources

arXiv2026

A Mean-Informed Low-Rank Monolithic Stochastic Galerkin Solver for the Unsteady Navier-Stokes Equations

We study a low-rank solver for the stochastic unsteady incompressible Navier--Stokes equations with uncertain viscosity. The problem is discretized by a stochastic Galerkin method and written in an all-at-once (monolithic) form where the time steps are informed by a sequential solve of the mean problem. The solution vector is represented in Tensor Train (TT) format, while the system matrices are represented in CANDECOMP/PARAFAC (CP) format as needed. We propose preconditioners based on the low-rank CP approximations that retain additional stochastic and nonlinear information compared to mean-based preconditioners. Numerical experiments for a benchmark channel-flow problem are presented to show the effectiveness of the low-rank approximation, the relative tolerance strategy, and the proposed preconditioners.

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arXiv2026

Stochastic Galerkin Method and Hierarchical Preconditioning for PDE-constrained Optimization

We develop efficient hierarchical preconditioners for optimal control problems governed by partial differential equations with uncertain coefficients. Adopting a discretize-then-optimize framework that integrates finite element discretization, stochastic Galerkin projection, and advanced time-discretization schemes, the approach addresses challenges of scaling large and ill-conditioned linear systems arising in uncertainty quantification. By exploiting sparsity of linear systems in stochastic Galerkin method, we formulate hierarchical preconditioners based on truncated stochastic expansion that strike an effective balance between computational cost and preconditioning quality. Numerical experiments demonstrate that the proposed preconditioners significantly accelerate the convergence of iterative solvers compared to existing methods, providing robust and efficient solvers for both steady-state and time-dependent optimal control problems under uncertainty.

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arXiv2022

A stochastic Galerkin method with adaptive time-stepping for the Navier-Stokes equations

We study the time-dependent Navier-Stokes equations in the context of stochastic finite element discretizations. Specifically, we assume that the viscosity is a random field given in the form of a generalized polynomial chaos expansion, and we use the stochastic Galerkin method to extend the methodology from [D. A. Kay et al., \textit{SIAM J. Sci. Comput.} 32(1), pp. 111--128, 2010] into this framework. For the resulting stochastic problem, we explore the properties of the resulting stochastic solutions, and we also compare the results with that of Monte Carlo and stochastic collocation. Since the time-stepping scheme is fully implicit, we also propose strategies for efficient solution of the stochastic Galerkin linear systems using a preconditioned Krylov subspace method. The effectiveness of the stochastic Galerkin method is illustrated by numerical experiments.

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arXiv2022

Inexact and primal multilevel FETI-DP methods: a multilevel extension and interplay with BDDC

We study a framework that allows to solve the coarse problem in the FETI-DP method approximately. It is based on the saddle-point formulation of the FETI-DP system with a block-triangular preconditioner. One of the blocks approximates the coarse problem, for which we use the multilevel BDDC method as the main tool. This strategy then naturally leads to a version of multilevel FETI-DP method, and we show that the spectra of the multilevel FETI-DP and BDDC preconditioned operators are essentially the same. The theory is illustrated by a set of numerical experiments, and we also present a few experiments when the coarse solve is approximated by algebraic multigrid.

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arXiv2022

Stochastic Galerkin finite element method for nonlinear elasticity and application to reinforced concrete members

We develop a stochastic Galerkin finite element method for nonlinear elasticity and apply it to reinforced concrete members with random material properties. The strategy is based on the modified Newton-Raphson method, which consists of an incremental loading process and a linearization scheme applied at each load increment. We consider that the material properties are given by a stochastic expansion in the so-called generalized polynomial chaos (gPC) framework. We search the gPC expansion of the displacement, which is then used to update the gPC expansions of the stress, strain and internal forces. The proposed method is applied to a reinforced concrete beam with uncertain initial concrete modulus of elasticity and a shear wall with uncertain maximum compressive stress of concrete, and the results are compared to those of stochastic collocation and Monte Carlo methods. Since the systems of equations obtained in the linearization scheme using the stochastic Galerkin method are very large, and there are typically many load increments, we also studied iterative solution using preconditioned conjugate gradients. The efficiency of the proposed method is illustrated by a set of numerical experiments.

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arXiv2022

Stochastic Galerkin methods for linear stability analysis of systems with parametric uncertainty

We present a method for linear stability analysis of systems with parametric uncertainty formulated in the stochastic Galerkin framework. Specifically, we assume that for a model partial differential equation, the parameter is given in the form of generalized polynomial chaos expansion. The stability analysis leads to the solution of a stochastic eigenvalue problem, and we wish to characterize the rightmost eigenvalue. We focus, in particular, on problems with nonsymmetric matrix operators, for which the eigenvalue of interest may be a complex conjugate pair, and we develop methods for their efficient solution. These methods are based on inexact, line-search Newton iteration, which entails use of preconditioned GMRES. The method is applied to linear stability analysis of Navier-Stokes equation with stochastic viscosity, its accuracy is compared to that of Monte Carlo and stochastic collocation, and the efficiency is illustrated by numerical experiments.

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arXiv2022

On Surrogate Learning for Linear Stability Assessment of Navier-Stokes Equations with Stochastic Viscosity

We study linear stability of solutions to the Navier\textendash Stokes equations with stochastic viscosity. Specifically, we assume that the viscosity is given in the form of a~stochastic expansion. Stability analysis requires a solution of the steady-state Navier-Stokes equation and then leads to a generalized eigenvalue problem, from which we wish to characterize the real part of the rightmost eigenvalue. While this can be achieved by Monte Carlo simulation, due to its computational cost we study three surrogates based on generalized polynomial chaos, Gaussian process regression and a shallow neural network. The results of linear stability analysis assessment obtained by the surrogates are compared to that of Monte Carlo simulation using a set of numerical experiments.

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arXiv2021

Application of adaptive ANOVA and reduced basis methods to the stochastic Stokes-Brinkman problem

The Stokes-Brinkman equations model fluid flow in highly heterogeneous porous media. In this paper, we consider the numerical solution of the Stokes-Brinkman equations with stochastic permeabilities, where the permeabilities in subdomains are assumed to be independent and uniformly distributed within a known interval. We employ a truncated anchored ANOVA decomposition alongside stochastic collocation to estimate the moments of the velocity and pressure solutions. Through an adaptive procedure selecting only the most important ANOVA directions, we reduce the number of collocation points needed for accurate estimation of the statistical moments. However, for even modest stochastic dimensions, the number of collocation points remains too large to perform high-fidelity solves at each point. We use reduced basis methods to alleviate the computational burden by approximating the expensive high-fidelity solves with inexpensive approximate solutions on a low-dimensional space. We furthermore develop and analyze rigorous a posteriori error estimates for the reduced basis approximation. We apply these methods to 2D problems considering both isotropic and anisotropic permeabilities.

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arXiv2019

A posteriori error estimates and adaptive mesh refinement for the Stokes-Brinkman problem

The Stokes-Brinkman equations model flow in heterogeneous porous media by combining the Stokes and Darcy models of flow into a single system of equations. With suitable parameters, the equations can model either flow without detailed knowledge of the interface between the two regions. Thus, the Stokes-Brinkman equations provide an alternative to coupled Darcy-Stokes models. After a brief review of the Stokes-Brinkman problem and its discretization using Taylor-Hood finite elements, we present a residual-based a posteriori error estimate and use it to drive an adaptive mesh refinement process. We compare several strategies for the mesh refinement, and demonstrate its effectiveness by numerical experiments in both 2D and 3D.

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arXiv2019

On adaptive BDDC for the flow in heterogeneous porous media

We study a method based on Balancing Domain Decomposition by Constraints (BDDC) for a numerical solution of a single-phase flow in heterogenous porous media. The method solves for both flux and pressure variables. The fluxes are resolved in three steps: the coarse solve is followed by subdomain solves and last we look for a divergence-free flux correction and pressures using conjugate gradients with the BDDC preconditioner. Our main contribution is an application of the adaptive algorithm for selection of flux constraints. Performance of the method is illustrated on the benchmark problem from the 10th SPE Comparative Solution Project (SPE 10). Numerical experiments in both 2D and 3D demonstrate that the first two steps of the method exhibit some numerical upscaling properties, and the adaptive preconditioner in the last step allows a significant decrease in number of iterations of conjugate gradients at a small additional cost.

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arXiv2018

Inexact Methods for Symmetric Stochastic Eigenvalue Problems

We study two inexact methods for solutions of random eigenvalue problems in the context of spectral stochastic finite elements. In particular, given a parameter-dependent, symmetric matrix operator, the methods solve for eigenvalues and eigenvectors represented using polynomial chaos expansions. Both methods are based on the stochastic Galerkin formulation of the eigenvalue problem and they exploit its Kronecker-product structure. The first method is an inexact variant of the stochastic inverse subspace iteration [B. Soused\'ık, H. C. Elman, SIAM/ASA Journal on Uncertainty Quantification 4(1), pp. 163--189, 2016]. The second method is based on an inexact variant of Newton iteration. In both cases, the problems are formulated so that the associated stochastic Galerkin matrices are symmetric, and the corresponding linear problems are solved using preconditioned Krylov subspace methods with several novel hierarchical preconditioners. The accuracy of the methods is compared with that of Monte Carlo and stochastic collocation, and the effectiveness of the methods is illustrated by numerical experiments.

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arXiv2017

A Low-rank solver for the Navier--Stokes equations with uncertain viscosity

We study an iterative low-rank approximation method for the solution of the steady-state stochastic Navier--Stokes equations with uncertain viscosity. The method is based on linearization schemes using Picard and Newton iterations and stochastic finite element discretizations of the linearized problems. For computing the low-rank approximate solution, we adapt the nonlinear iterations to an inexact and low-rank variant, where the solution of the linear system at each nonlinear step is approximated by a quantity of low rank. This is achieved by using a tensor variant of the GMRES method as a solver for the linear systems. We explore the inexact low-rank nonlinear iteration with a set of benchmark problems, using a model of flow over an obstacle, under various configurations characterizing the statistical features of the uncertain viscosity, and we demonstrate its effectiveness by extensive numerical experiments.

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arXiv2016

Stochastic Galerkin methods for the steady-state Navier-Stokes equations

We study the steady-state Navier-Stokes equations in the context of stochastic finite element discretizations. Specifically, we assume that the viscosity is a random field given in the form of a generalized polynomial chaos expansion. For the resulting stochastic problem, we formulate the model and linearization schemes using Picard and Newton iterations in the framework of the stochastic Galerkin method, and we explore properties of the resulting stochastic solutions. We also propose a preconditioner for solving the linear systems of equations arising at each step of the stochastic (Galerkin) nonlinear iteration and demonstrate its effectiveness for solving a set of benchmark problems.

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arXiv2015

Inverse subspace iteration for spectral stochastic finite element methods

We study random eigenvalue problems in the context of spectral stochastic finite elements. In particular, given a parameter-dependent, symmetric positive-definite matrix operator, we explore the performance of algorithms for computing its eigenvalues and eigenvectors represented using polynomial chaos expansions. We formulate a version of stochastic inverse subspace iteration, which is based on the stochastic Galerkin finite element method, and we compare its accuracy with that of Monte Carlo and stochastic collocation methods. The coefficients of the eigenvalue expansions are computed from a stochastic Rayleigh quotient. Our approach allows the computation of interior eigenvalues by deflation methods, and we can also compute the coefficients of multiple eigenvectors using a stochastic variant of the modified Gram-Schmidt process. The effectiveness of the methods is illustrated by numerical experiments on benchmark problems arising from vibration analysis.

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arXiv2015

BDDC for Mixed-Hybrid Formulation of Flow in Porous Media with Combined Mesh Dimensions

We extend the Balancing Domain Decomposition by Constraints (BDDC) method to flows in porous media discretised by mixed-hybrid finite elements with combined mesh dimensions. Such discretisations appear when major geological fractures are modelled by 1D or 2D elements inside three-dimensional domains. In this set-up, the global problem as well as the substructure problems have a symmetric saddle-point structure, containing a `penalty' block due to the combination of meshes. We show that the problem can be reduced by means of iterative substructuring to an interface problem, which is symmetric and positive definite. The interface problem can thus be solved by conjugate gradients with the BDDC method as a preconditioner. A parallel implementation of this algorithm is incorporated into an existing software package for subsurface flow simulations. We study the performance of the iterative solver on several academic and real-world problems. Numerical experiments illustrate its efficiency and scalability.

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arXiv2013

Truncated hierarchical preconditioning for the stochastic Galerkin FEM

Stochastic Galerkin finite element discretizations of partial differential equations with coefficients characterized by arbitrary distributions lead, in general, to fully block dense linear systems. We propose two novel strategies for constructing preconditioners for these systems to be used with Krylov subspace iterative solvers. In particular, we present a variation on of the hierarchical Schur complement preconditioner, developed recently by the authors, and an adaptation of the symmetric block Gauss-Seidel method. Both preconditioners take advantage of the hierarchical structure of global stochastic Galerkin matrices, and also, when applicable, of the decay of the norms of the stiffness matrices obtained from the polynomial chaos expansion of the coefficients. This decay allows to truncate the matrix-vector multiplications in the action of the preconditioners. Also, throughout the global matrix hierarchy, we approximate solves with certain submatrices by the associated diagonal block solves. The preconditioners thus require only a limited number of stiffness matrices obtained from the polynomial chaos expansion of the coefficients, and a preconditioner for the diagonal blocks of the global matrix. The performance is illustrated by numerical experiments.

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arXiv2013

Nested BDDC for a saddle-point problem

We propose a Nested BDDC for a class of saddle-point problems. The method solves for both flux and pressure variables. The fluxes are resolved in three-steps: the coarse solve is followed by subdomain solves, and last we look for a divergence-free flux correction and pressure variables using conjugate gradients with a Multilevel BDDC preconditioner. Because the coarse solve in the first step has the same structure as the original problem, we can use this procedure recursively and solve (a hierarchy of) coarse problems only approximately, utilizing the coarse problems known from the BDDC. The resulting algorithm thus first performs several upscaling steps, and then solves a hierarchy of problems that have the same structure but increase in size while sweeping down the levels, using the same components in the first and in the third step on each level, and also reusing the components from the higher levels. Because the coarsening can be quite aggressive, the number of levels can be kept small and the additional computational cost is significantly reduced due to the reuse of the components. We also provide the condition number bound and numerical experiments confirming the theory.

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arXiv2013

Hierarchical Schur complement preconditioner for the stochastic Galerkin finite element methods

Use of the stochastic Galerkin finite element methods leads to large systems of linear equations obtained by the discretization of tensor product solution spaces along their spatial and stochastic dimensions. These systems are typically solved iteratively by a Krylov subspace method. We propose a preconditioner which takes an advantage of the recursive hierarchy in the structure of the global matrices. In particular, the matrices posses a recursive hierarchical two-by-two structure, with one of the submatrices block diagonal. Each one of the diagonal blocks in this submatrix is closely related to the deterministic mean-value problem, and the action of its inverse is in the implementation approximated by inner loops of Krylov iterations. Thus our hierarchical Schur complement preconditioner combines, on each level in the approximation of the hierarchical structure of the global matrix, the idea of Schur complement with loops for a number of mutually independent inner Krylov iterations, and several matrix-vector multiplications for the off-diagonal blocks. Neither the global matrix, nor the matrix of the preconditioner need to be formed explicitly. The ingredients include only the number of stiffness matrices from the truncated Karhunen-Loève expansion and a good preconditioned for the mean-value deterministic problem. We provide a condition number bound for a model elliptic problem and the performance of the method is illustrated by numerical experiments.

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