Low-rank solvers for unsteady Stokes–Brinkman optimal control problem with random data

Benner, Peter; Dolgov, Sergey; Onwunta, Akwum; Stoll, Martin

doi:10.1016/j.cma.2016.02.004

Cited by 46 publications

(76 citation statements)

References 57 publications

(122 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…which is consistent with the Kronecker product A = A (1) ⊗ A (2) in the case d = 2 and R 1 = 1, and allows a natural multiplication with (68) returning the result in the same format.…”

Section: Solving the Optimality Systems From The Unsteady Problem Fisupporting

confidence: 55%

“…Recasting the above SOCP given by (2) and (13) into a saddle-point formulation, Chen and Quarteroni in [5] prove the existence and uniqueness of its solution. More precisely, the following result holds.…”

Section: Representation Of Random Inputsmentioning

confidence: 99%

“…Each of the basis functions is associated with a multi-index ν = (ν 1 , ν 2 ), where the components represent the degrees of the polynomials in ξ 1 and ξ 2 . Since the total degree of the polynomial is three, we have the possibilities ν = (0,0), (1,0), (2,0), (3,0), (0,1), (1,1), (2,1), (0,2), (1,2), and (0, 3). Since the univariate Legendre polynomials of degrees…”

Section: Representation Of Random Inputsmentioning

confidence: 99%

“…Again, U 2 can be reshaped to a three-dimensional tensor y (2) α1,α2 (i 2 ) = U 2 (α 1 i 2 , α 2 ) of moderate size, and the decomposition also applied to Σ 2 V T 2 . Proceeding in this manner, one eventually obtains the TT-format:…”

Section: Solving the Optimality Systems From The Unsteady Problem Fimentioning

confidence: 99%

See 3 more Smart Citations

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

Benner¹,

Onwunta²,

Stoll³

2016

SIAM J. Matrix Anal. & Appl.

Self Cite

View full text Add to dashboard Cite

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

show abstract

“…which is consistent with the Kronecker product A = A (1) ⊗ A (2) in the case d = 2 and R 1 = 1, and allows a natural multiplication with (68) returning the result in the same format.…”

Section: Solving the Optimality Systems From The Unsteady Problem Fisupporting

confidence: 55%

Section: Representation Of Random Inputsmentioning

confidence: 99%

Section: Representation Of Random Inputsmentioning

confidence: 99%

Section: Solving the Optimality Systems From The Unsteady Problem Fimentioning

confidence: 99%

See 2 more Smart Citations

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

Benner¹,

Onwunta²,

Stoll³

2016

SIAM J. Matrix Anal. & Appl.

Self Cite

View full text Add to dashboard Cite

show abstract

“…discretization via finite elements on possibly irregular domains. We aim to base our approaches for these applications on recent results presented in [9,7]. …”

Section: Discussionmentioning

confidence: 99%

Fast tensor product solvers for optimization problems with fractional differential equations as constraints

Dolgov

Pearson

Savostyanov

et al. 2016

Applied Mathematics and Computation

Self Cite

View full text Add to dashboard Cite

Fractional differential equations have recently received much attention within computational mathematics and applied science, and their numerical treatment is an important research area as such equations pose substantial challenges to existing algorithms. An optimization problem with constraints given by fractional differential equations is considered, which in its discretized form leads to a high-dimensional tensor equation. To reduce the computation time and storage, the solution is sought in the tensor-train format. We compare three types of solution strategies that employ sophisticated iterative techniques using either preconditioned Krylov solvers or tailored alternating schemes. The competitiveness of these approaches is presented using several examples with constant and variable coefficients.

show abstract

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

Benner

Dolgov

Onwunta

et al. 2020

Numerical Methods in Fluids

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Low-rank solvers for unsteady Stokes–Brinkman optimal control problem with random data

Cited by 46 publications

References 57 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

Fast tensor product solvers for optimization problems with fractional differential equations as constraints

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

Contact Info

Product

Resources

About