Stochastic Galerkin Method and Hierarchical Preconditioning for PDE-constrained Optimization
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| author | Li, Zhendong Onwunta, Akwum Sousedík, Bedřich |
| author_facet | Li, Zhendong Onwunta, Akwum Sousedík, Bedřich |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_23804 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stochastic Galerkin Method and Hierarchical Preconditioning for PDE-constrained Optimization Li, Zhendong Onwunta, Akwum Sousedík, Bedřich Optimization and Control Numerical Analysis Analysis of PDEs 35R60, 65C20, 65F08, 65F10, 60H35, 65N22 35R60, 35R60, 65C20, 65F08, 65F10, 60H35, 65N22 65C20, 65F08, 65F10, 60H35, 65N22 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. |
| title | Stochastic Galerkin Method and Hierarchical Preconditioning for PDE-constrained Optimization |
| topic | Optimization and Control Numerical Analysis Analysis of PDEs 35R60, 65C20, 65F08, 65F10, 60H35, 65N22 35R60, 35R60, 65C20, 65F08, 65F10, 60H35, 65N22 65C20, 65F08, 65F10, 60H35, 65N22 |
| url | https://arxiv.org/abs/2512.23804 |