SOLVING AN ILL-POSED CAUCHY PROBLEM FOR A TWO-DIMENSIONAL PARABOLIC PDE WITH VARIABLE COEFFICIENTS USING A PRECONDITIONED GMRES METHOD

作者：Ranjbar Zohreh^*; Eldent Lars

来源：SIAM Journal on Scientific Computing, 2014, 36(5): B868-B886.

DOI：10.1137/130951166

• Cauchy problem
• inverse problem
• ill-posed
• iterative methods
• GMRES preconditioning
• singular preconditioner
• parabolic PDE
• FFT

The sideways parabolic equation (SPE) is a model of the problem of determining the temperature on the surface of a body from the interior measurements. Mathematically it can be formulated as a noncharacteristic Cauchy problem for a parabolic partial differential equation. This problem is severely ill-posed in an L-2 setting. We use a preconditioned generalized minimum residual method (GMRES) to solve a two-dimensional SPE with variable coefficients. The preconditioner is singular and chosen in a way that allows efficient implementation using the FFT. The preconditioner is a stabilized solver for a nearby problem with constant coefficients, and it reduces the number of iterations in the GMRES algorithm significantly. Numerical experiments are performed that demonstrate the performance of the proposed method.

• 出版日期2014