Collocation Methods for Cauchy Problems of Elliptic Operators via Conditional Stabilities (Communications in Computational Physics, 2019)
Ill-posed Cauchy problems for elliptic partial differential equations appearin many engineering fields. In this paper, we focus on stable reconstruction methods for this kind of inverse problems. Using kernels that reproduce Hilbert spaces $H^m$(Ω), numerical approximations to solutions of elliptic Cauchy problems are formulated as solutions of nonlinear least-squares problems with quadratic inequality constraints (LSQI). A convergence analysis with respect to noise levels and fill distances ofdata points is provided, from which a Tikhonov regularization strategy is obtained. Anonlinear algorithm using generalized singular value decomposition of matrices andLagrange multipliers is proposed to solve the LSQI problem. Numerical experimentsof two-dimensional cases verify our proved convergence results. By comparing withsolutions of MFS and FEM under the discrete Tikhonov regularization by RKHS undersame Cauchy data, we demonstrate that our method can reconstruct stable and highaccuracy solutions for noisy Cauchy data.
Keywords: Cauchy problems; meshfree; Kansa method; error analysis; LSQI problem; Tikhonov regularization.