报告题目：Adaptive multi-level DCA algorithm for PDE-constrained optimization problems with $L^{1-2}$-control cost

报 告 人：宋晓良（大连理工大学）

报告时间：2025年1月16日  10:30-11:15

报告地点：格物楼528

报告摘要：

In this talk, we investigate the sparse elliptic partial differential equation (PDE)-constrained optimization problem with an L1−2 control cost (L1−2-EOCP). Such problems are inherently nonconvex and nonsmooth, making their discretized versions computationally intensive to solve on a fixed grid. To address this issue, we propose an adaptive multi-level strategy that refines the grid progressively throughout the computation process rather than maintaining a fixed grid level. Our approach begins by employing an inexact difference of convex functions algorithm (iDCA) in function space to transform the nonconvex optimization problem with L1−2 regularization terms into a sequence of convex optimization subproblems, each featuring only L1 regularization terms. The inexactness in the algorithm arises from the discretization of these subproblems using standard piecewise linear finite elements. Crucially, the discretization accuracy is coupled with the iterative process through an adaptive multi-level strategy, resulting in an Adaptive Multilevel Inexact Difference of Convex Functions Algorithm (Am-iDCA). We explore two distinct strategies for implementing Am-iDCA: one based on the number of iterations and the other relying on a posteriori estimates for the residuals of the subproblems. The discretized subproblems are then solved using an inexact alternating direction method of multipliers with a posterior error control (pADMM). Additionally, leveraging the unique structure of the L1−2-EOCP, we provide convergence results for the Am-iDCA. Numerical results demonstrate the efficiency and effectiveness of the Am-iDCA in solving sparse elliptic PDE-constrained optimization problems with L1-2 control costs.

报告人简介：

宋晓良博士，现为大连理工大学数学科学学院副教授。他于2018年在大连理工大学获得博士学位，攻读博士学位期间于2015.09-2017.07在新加坡国立大学进行联合培养；2018.09-2020.09在香港理工大学进行博士后研究工作。他主要从事PDE约束优化问题的数值离散和优化算法的研究。到目前为止，已发表SCI检索论文8篇。