Volume 26, Issue 3
Collocation Methods for Cauchy Problems of Elliptic Operators via Conditional Stabilities

Siqing Li & Leevan Ling

Commun. Comput. Phys., 26 (2019), pp. 785-808.

Published online: 2019-04

Preview Full PDF 15 1325
Export citation
  • Abstract

. 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 Hm(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.

  • Keywords

Cauchy problems, meshfree, Kansa method, error analysis, LSQI problem, Tikhonov regularization.

  • AMS Subject Headings

65D15, 65N35, 65N21

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address
  • References
  • Hide All
    View All

  • BibTex
  • RIS
  • TXT
@Article{CiCP-26-785, author = {}, title = {Collocation Methods for Cauchy Problems of Elliptic Operators via Conditional Stabilities}, journal = {Communications in Computational Physics}, year = {2019}, volume = {26}, number = {3}, pages = {785--808}, abstract = {

. 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 Hm(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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2018-0182}, url = {http://global-sci.org/intro/article_detail/cicp/13147.html} }
TY - JOUR T1 - Collocation Methods for Cauchy Problems of Elliptic Operators via Conditional Stabilities JO - Communications in Computational Physics VL - 3 SP - 785 EP - 808 PY - 2019 DA - 2019/04 SN - 26 DO - http://dor.org/10.4208/cicp.OA-2018-0182 UR - https://global-sci.org/intro/cicp/13147.html KW - Cauchy problems, meshfree, Kansa method, error analysis, LSQI problem, Tikhonov regularization. AB -

. 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 Hm(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.

Siqing Li & Leevan Ling. (2019). Collocation Methods for Cauchy Problems of Elliptic Operators via Conditional Stabilities. Communications in Computational Physics. 26 (3). 785-808. doi:10.4208/cicp.OA-2018-0182
Copy to clipboard
The citation has been copied to your clipboard