TY - JOUR

T1 - A matching Schur complement preconditioning technique for inverse source problems

AU - Lin, Xuelei

AU - Ng, Michael K.

N1 - The work of Xue-lei Lin was partially supported by research grants: 12301480 from NSFC, HA45001143 from Harbin Institute of Technology, Shenzhen, HA11409084 from Shenzhen. The work of Michael K. Ng was partially supported by research grants: HKRGC GRF 17201020, 17300021, C7004-21GF and Joint NSFC-RGC N-HKU76921.
Publisher Copyright:
© 2024 IMACS

PY - 2024/7

Y1 - 2024/7

N2 - Numerical discretization of a regularized inverse source problem leads to a non-symmetric saddle point linear system. Interestingly, the Schur complement of the non-symmetric saddle point system is Hermitian positive definite (HPD). Then, we propose a preconditioner matching the Schur complement (MSC). Theoretically, we show that the preconditioned conjugate gradient (PCG) method for a linear system with the preconditioned Schur complement as coefficient has a linear convergence rate independent of the matrix size and value of the regularization parameter involved in the inverse problem. Fast implementations are proposed for the matrix-vector multiplication of the preconditioned Schur complement so that the PCG solver requires only quasi-linear operations. To the best of our knowledge, this is the first solver with guarantee of linear convergence for the inversion of Schur complement arising from the discrete inverse problem. Combining the PCG solver for inversion of the Schur complement and the fast solvers for the forward problem in the literature, the discrete inverse problem (the saddle point system) is solved within a quasi-linear complexity. Numerical results are reported to show the performance of the proposed matching Schur complement (MSC) preconditioning technique.

AB - Numerical discretization of a regularized inverse source problem leads to a non-symmetric saddle point linear system. Interestingly, the Schur complement of the non-symmetric saddle point system is Hermitian positive definite (HPD). Then, we propose a preconditioner matching the Schur complement (MSC). Theoretically, we show that the preconditioned conjugate gradient (PCG) method for a linear system with the preconditioned Schur complement as coefficient has a linear convergence rate independent of the matrix size and value of the regularization parameter involved in the inverse problem. Fast implementations are proposed for the matrix-vector multiplication of the preconditioned Schur complement so that the PCG solver requires only quasi-linear operations. To the best of our knowledge, this is the first solver with guarantee of linear convergence for the inversion of Schur complement arising from the discrete inverse problem. Combining the PCG solver for inversion of the Schur complement and the fast solvers for the forward problem in the literature, the discrete inverse problem (the saddle point system) is solved within a quasi-linear complexity. Numerical results are reported to show the performance of the proposed matching Schur complement (MSC) preconditioning technique.

KW - Inverse space-dependent source problem

KW - Preconditioning technique for Schur complement

KW - Saddle point system

KW - Size-independent convergence of iterative solver

UR - http://www.scopus.com/inward/record.url?scp=85189516453&partnerID=8YFLogxK

UR - https://www.sciencedirect.com/science/article/abs/pii/S0168927424000795?via%3Dihub

U2 - 10.1016/j.apnum.2024.03.018

DO - 10.1016/j.apnum.2024.03.018

M3 - Journal article

AN - SCOPUS:85189516453

SN - 0168-9274

VL - 201

SP - 404

EP - 418

JO - Applied Numerical Mathematics

JF - Applied Numerical Mathematics

ER -