Efficient numerical method for shape optimization problem constrained by stochastic elliptic interface equation

Qingguang Guan; Xu Guo; Wenju Zhao

doi:10.3934/cac.2023016

Article Contents

2023, Volume 1, Issue 4: 321-346. Doi: 10.3934/cac.2023016

This issue Previous Article Next Article Reconstruction of extended regions in EIT with a generalized Robin transmission condition

Efficient numerical method for shape optimization problem constrained by stochastic elliptic interface equation

1.
School of Mathematics and Natural Sciences, University of Southern Mississippi, Hattiesburg, MS 39406, USA
2.
Geotechnical and Structural Engineering Research Center, Shandong University, Jinan, Shandong 250061, China
3.
School of Mathematics, Shandong University, Jinan, Shandong 250100, China

^*Corresponding author: Wenju Zhao
^*Corresponding author: Wenju Zhao

Received: June 2023

Revised: October 2023

Early access: October 26, 2023

Published online: October 26, 2023

The second author is funded in part by the National Natural Science Foundation of China under Grants 12271303, and by Taishan Scholars Program of Shandong Province of China (tsqn201909044).
The third author is partially supported by the National Natural Science Foundation of China (Nos. 12001325)..

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Abstract

This paper provides a computational framework for stochastic shape optimization problems. The primary control objective involves minimizing the expected value of a tracking cost function while adhering to constraints posed by a stochastic elliptic interface equation. This study incorporates stochastic shape variation and shape derivatives, enabling the establishment of a diminishing sequence of admissible interfaces. The finite element method is employed for discretizing both state and adjoint systems, yielding mesh displacement directions. Notably, the mesh manipulation technique utilized within the optimization process is subject to regularization through cubic spline interpolation employing a centripetal scheme. To ensure the fidelity of the interface curve and prevent undesirable twisting, the convex hull technique is incorporated, thus upholding mesh displacement consistency. To mitigate the computational demands associated with uncertainty quantification, the sparse grid collocation method is enlisted. This technique facilitates the matching of probability distributions, particularly in scenarios involving relatively high-dimensional problems. Moreover, for extensive-scale sampling optimization, a stochastic sampling-based descent method is integrated. The effectiveness and efficiency of the proposed algorithms are underscored through a series of numerical experiments, which serve to validate their performance.

Keywords:

Stochastic shape optimization,
shape variation,
sparse grid,
stochastic sampling-based descent method.

Mathematics Subject Classification: Primary: 35R60, 49Q10, 93E20.

Citation:

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Figure 1. In domian $ \mathcal{D} $, $ \Gamma $ moves to $ \Gamma' $ by a distance $ \kappa( {\bf{x}}) $ along the outer normal vector $ {\bf{n}}( {\bf{x}}) $

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Figure 2. The numerical solutions on mesh Level 4. Left: numerical solution with the circular interface; Right: Numerical solution with concave-convex interface

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Figure 3. Circular and concave-convex interfaces

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Figure 4. Evolution of deterministic cost functional

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Figure 5. Evolution of deterministic state variable $ u_{iter}( {\bf{x}}) $

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Figure 6. The evolution of deterministic state variable $ u_{iter}( {\bf{x}}) $

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Figure 7. Evolution of cost functional with sparse grid collocation

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Figure 8. The evolution of $ \mathbb E [u_{iter}( {\bf{x}})] $

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Figure 9. The evolution of $ \mathbb E[u_{iter}( {\bf{x}})] $

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Figure 10. Evolution of sampling-based cost functional

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Figure 11. The evolution of sampling-based state variable $ u_{iter}( {\bf{x}}, \omega_t) $

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Figure 12. The evolution of sampling-based state variable $ u_{iter}( {\bf{x}}, \omega_t) $

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Figure 13. Mesh moving with projection

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Figure 14. Mesh moving without projection

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Table 1. Numerical convergence results with the circular interface

Mesh	#Element	$ L^2 $ Error	$ L^2 $ Order	$ H^1 $ Error	$ H^1 $ Order
Level 1	2716	1.3371E-04	-	4.6025E-03	-
Level 2	10864	1.6407E-05	3.0267	1.1588E-03	1.9897
Level 3	43456	2.0530E-06	2.9985	2.9124E-04	1.9924
Level 4	173824	2.5745E-07	2.9953	7.3030E-05	1.9956

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Table 2. Numerical convergence results with the concave-convex interface

Mesh	#Element	$ L^2 $ Error	$ L^2 $ Order	$ H^1 $ Error	$ H^1 $ Order
Level 1	1940	1.5739E-04	-	5.3861E-03	-
Level 2	7760	1.9463E-05	3.0155	1.3578E-03	1.9879
Level 3	31040	2.4334E-06	2.9996	3.4123E-04	1.9924
Level 4	124160	3.0472E-07	2.9974	8.5553E-05	1.9958

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Table 3. Comparison between Experiment 6.3 and 6.4

	sample number	iteration number	total number
Experiment 6.3	385	850	327250
Experiment 6.4	1	10000	10000

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