Low-Rank Solution of Unsteady Diffusion Equations with Stochastic Coefficients

Benner, Peter; Onwunta, Akwum; Stoll, Martin

doi:10.1137/130937251

Cited by 35 publications

(64 citation statements)

References 26 publications

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“…An effective approach to approximate blocks (1, 1) and (2, 2) is the application of Chebyshev semi-iteration to the mass matrices in each of the two blocks [27]. Approximating the Schur complement S, that is, block (3,3), poses more difficulty, however. One possibility [22] is to approximate S by dropping the term An alternative and more robust approach, which we adopt here and in the rest of this paper, was proposed in [20] (see also [8,Chapter 5]) in the context of deterministic optimal control problems.…”

Section: 1mentioning

confidence: 99%

“…. , N, one could approximate Z using, for example, the block-diagonal mean-based preconditioner [3,21]:…”

Section: It Turns Out Then That (38) Holds If and Only Ifmentioning

confidence: 99%

“…Note that an important feature of low-rank MINRES is that the iterates of the solution matrices Y , U , and F in the algorithm are truncated by a truncation operator T with a prescribed tolerance . This is accomplished via QR decomposition as in [14] or truncated singular value decomposition (SVD) as in [3,25]. …”

Section: 2mentioning

confidence: 99%

“…This class of problems often leads to prohibitively high dimensional linear systems with Kronecker product structure, especially when discretized with the stochastic Galerkin finite element method (SGFEM). Moreover, a typical model for an optimal control problem with stochastic inputs (SOCP) will usually be used for the quantification of the statistics of the system response-a task that could in turn result in additional enormous computational expense.Stochastic finite element-based solvers for a large range of PDEs with random data have been studied extensively [1,3,12,21,24]. However, optimization problems constrained by PDEs with random inputs have, in our opinion, not yet received adequate attention.…”

mentioning

confidence: 99%

“…Stochastic finite element-based solvers for a large range of PDEs with random data have been studied extensively [1,3,12,21,24]. However, optimization problems constrained by PDEs with random inputs have, in our opinion, not yet received adequate attention.…”

mentioning

confidence: 99%

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Block-Diagonal Preconditioning for Optimal Control Problems Constrained by PDEs with Uncertain Inputs

Benner¹,

Onwunta²,

Stoll³

2016

SIAM J. Matrix Anal. & Appl.

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Abstract. The goal of this paper is the efficient numerical simulation of optimization problems governed by either steady-state or unsteady partial differential equations involving random coefficients. This class of problems often leads to prohibitively high dimensional saddle-point systems with tensor product structure, especially when discretized with the stochastic Galerkin finite element method. Here, we derive and analyze robust Schur complement-based block-diagonal preconditioners for solving the resulting stochastic optimality systems with all-at-once low-rank iterative solvers. Moreover, we illustrate the effectiveness of our solvers with numerical experiments.Key words. stochastic Galerkin system, iterative methods, PDE-constrained optimization, saddle-point system, low-rank solution, preconditioning, Schur complement AMS subject classifications. 35R60, 60H15, 60H35, 65N22, 65F10, 65F50 DOI. 10.1137/15M10185021. Introduction. Optimization problems constrained by deterministic steadystate partial differential equations (PDEs) are computationally challenging. This is even more so if the constraints are deterministic unsteady PDEs since one would then need to solve a system of PDEs coupled globally in time and space, and time-stepping methods quickly reach their limitations due to the enormous demand for storage [25]. Yet, more challenging than the aforementioned are problems constrained by unsteady PDEs involving (countably many) parametric or uncertain inputs. This class of problems often leads to prohibitively high dimensional linear systems with Kronecker product structure, especially when discretized with the stochastic Galerkin finite element method (SGFEM). Moreover, a typical model for an optimal control problem with stochastic inputs (SOCP) will usually be used for the quantification of the statistics of the system response-a task that could in turn result in additional enormous computational expense.Stochastic finite element-based solvers for a large range of PDEs with random data have been studied extensively [1,3,12,21,24]. However, optimization problems constrained by PDEs with random inputs have, in our opinion, not yet received adequate attention. Hence, this study is aimed at pushing the research frontier with respect to the numerical simulation of the latter class of stochastic problems (that is, SOCPs) toward larger and more challenging problems. Some of the papers on SOCPs include [12,13,24]. While [12] studies the existence and the uniqueness of solutions to control problems constrained by elliptic PDEs with random inputs, the emphasis in

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Section: 1mentioning

confidence: 99%

“…. , N, one could approximate Z using, for example, the block-diagonal mean-based preconditioner [3,21]:…”

Section: It Turns Out Then That (38) Holds If and Only Ifmentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Block-Diagonal Preconditioning for Optimal Control Problems Constrained by PDEs with Uncertain Inputs

Benner¹,

Onwunta²,

Stoll³

2016

SIAM J. Matrix Anal. & Appl.

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Low‐rank solution of an optimal control problem constrained by random Navier‐Stokes equations

Benner

Dolgov

Onwunta

et al. 2020

Numerical Methods in Fluids

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Many problems in computational science and engineering are simultaneously characterized by the following challenging issues: uncertainty, nonlinearity, nonstationarity and high dimensionality. Existing numerical techniques for such models would typically require considerable computational and storage resources. This is the case, for instance, for an optimization problem governed by time-dependent Navier-Stokes equations with uncertain inputs. In particular, the stochastic Galerkin finite element method often leads to a prohibitively high dimensional saddle-point system with tensor product structure. In this paper, we approximate the solution by the low-rank Tensor Train decomposition, and present a numerically efficient algorithm to solve the optimality equations directly in the low-rank representation. We show that the solution of the vorticity minimization problem with a distributed control admits a representation with ranks that depend modestly on model and discretization parameters even for high Reynolds numbers. For lower Reynolds numbers this is also the case for a boundary control. This opens the way for a reduced-order modeling of the stochastic optimal flow control with a moderate cost at all stages.

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Tensor method for optimal control problems constrained by fractional three‐dimensional elliptic operator with variable coefficients

Schmitt

Khoromskij

Khoromskaia

et al. 2021

Numerical Linear Algebra App

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We introduce the tensor numerical method for solving optimal control problems that are constrained by fractional two-(2D) and three-dimensional (3D) elliptic operators with variable coefficients. We solve the governing equation for the control function which includes a sum of the fractional operator and its inverse, both discretized over large 3D n × n × n spacial grids. Using the diagonalization of the arising matrix-valued functions in the eigenbasis of the one-dimensional Sturm-Liouville operators, we construct the rank-structured tensor approximation with controllable precision for the discretized fractional elliptic operators and the respective preconditioner. The right-hand side in the constraining equation (the optimal design function) is supposed to be represented in a form of a low-rank canonical tensor. Then the equation for the control function is solved in a tensor structured format by using preconditioned CG iteration with the adaptive rank truncation procedure that also ensures the accuracy of calculations, given an 𝜀-threshold. This method reduces the numerical cost for solving the control problem to O(n log n) (plus the quadratic term O(n 2 ) with a small weight), which outperforms traditional approaches with O(n 3 log n) complexity in the 3D case. The storage for the representation of all 3D nonlocal operators and functions involved is also estimated by O(n log n).This essentially outperforms the traditional methods operating with fully populated n 3 × n 3 matrices and vectors in R n 3 . Numerical tests for 2D/3D control problems indicate the almost linear complexity scaling of the rank truncated preconditioned conjugate gradient iteration in the univariate grid size n.

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Low-Rank Solution of Unsteady Diffusion Equations with Stochastic Coefficients

Cited by 35 publications

References 26 publications

Block-Diagonal Preconditioning for Optimal Control Problems Constrained by PDEs with Uncertain Inputs

Block-Diagonal Preconditioning for Optimal Control Problems Constrained by PDEs with Uncertain Inputs

Low‐rank solution of an optimal control problem constrained by random Navier‐Stokes equations

Tensor method for optimal control problems constrained by fractional three‐dimensional elliptic operator with variable coefficients

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