TY - JOUR
T1 - Collocation Methods for Cauchy Problems of Elliptic Operators via Conditional Stabilities
AU - Siqing Li & Leevan Ling
JO - Communications in Computational Physics
VL - 3
SP - 785
EP - 808
PY - 2019
DA - 2019/04
SN - 26
DO - http://doi.org/10.4208/cicp.OA-2018-0182
UR - https://global-sci.org/intro/article_detail/cicp/13147.html
KW - Cauchy problems, meshfree, Kansa method, error analysis, LSQI problem, Tikhonov
regularization.
AB - <p style="text-align: justify;">Ill-posed Cauchy problems for elliptic partial differential equations appear
in 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 of
data points is provided, from which a Tikhonov regularization strategy is obtained. A
nonlinear algorithm using generalized singular value decomposition of matrices and
Lagrange multipliers is proposed to solve the LSQI problem. Numerical experiments
of two-dimensional cases verify our proved convergence results. By comparing with
solutions of MFS and FEM under the discrete Tikhonov regularization by RKHS under
same Cauchy data, we demonstrate that our method can reconstruct stable and high
accuracy solutions for noisy Cauchy data.</p>