@Article{rael14,
  Title                    = {Solving an Ill-Posed Cauchy Problem for a Two-Dimensional Parabolic PDE with Variable Coefficients Using a Preconditioned GMRES Method},
  Author                   = {Ranjbar, Z. and Eld\'en, L.},
  Journal                  = {SIAM Journal on Scientific Computing},
  Year                     = {2014},
  Number                   = {5},
  Pages                    = {B868-B886},
  Volume                   = {36},

  Abstract                 = {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 L2 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.},
  Doi                      = {10.1137/130951166},
  Eprint                   = {http://dx.doi.org/10.1137/130951166},
  File                     = {rael14.pdf:rael14.pdf:PDF},
  Url                      = {http://dx.doi.org/10.1137/130951166}
}