We consider the numerical simulation of an optimal control problem constrained by the unsteady Stokes-Brinkman equation involving random data. More precisely, we treat the state, the control, the target (or the desired state), as well as the viscosity, as analytic functions depending on uncertain parameters. This allows for a simultaneous generalized polynomial chaos approximation of these random functions in the stochastic Galerkin finite element method discretization of the model. The discrete problem yields a prohibitively high dimensional saddle point system with Kronecker product structure. We develop a new alternating iterative tensor method for an efficient reduction of this system by the low-rank Tensor Train representation. Besides, we propose and analyze a robust Schur complement-based preconditioner for the solution of the saddle-point system. The performance of our approach is illustrated with extensive numerical experiments based on twoand three-dimensional examples, where the full problem size exceeds one billion degrees of freedom. The developed Tensor Train scheme reduces the solution storage by two-three orders of magnitude, depending on discretization parameters.
Abstract. We study the solution of linear systems resulting from the discretization of unsteady diffusion equations with stochastic coefficients. In particular, we focus on those linear systems that are obtained using the so-called stochastic Galerkin finite element method (SGFEM). These linear systems are usually very large with Kronecker product structure, and thus solving them can be both time-and computer memory-consuming. Under certain assumptions, we show that the solution of such linear systems can be approximated with a vector of low tensor rank. We then solve the linear systems using low-rank preconditioned iterative solvers. Numerical experiments demonstrate that these lowrank preconditioned solvers are effective, especially when the fluctuations in the random data are not too large relative to their mean values.
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
We consider the simulation of Bayesian statistical inverse problems governed by large-scale linear and nonlinear partial differential equations (PDEs). Markov chain Monte Carlo (MCMC) algorithms are standard techniques to solve such problems. However, MCMC techniques are computationally challenging as they require several thousands of forward PDE solves. The goal of this paper is to introduce a fractional deep neural network based approach for the forward solves within an MCMC routine. Moreover, we discuss some approximation error estimates and illustrate the efficiency of our approach via several numerical examples.
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