Name
Abstract | ||
|---|---|---|
We discuss the efficient solution of a long sequence of slowly varying linear systems arising in computations for diffuse optical tomographic imaging.The reconstruction of three-dimensional absorption and scattering information by matching computed solutions from a parameterized model to measured data leads to a nonlinear least squares problem that we solve using the Gauss--Newton method with a line search. This algorithm requires the solution of a long sequence of linear systems. Each choice of parameters in the nonlinear least squares algorithm results in a different matrix describing the optical properties of the medium. These matrices change slowly from one step to the next, but may change significantly over many steps. For each matrix we must solve a set of linear systems with multiple shifts and multiple right-hand sides.For this problem, we derive strategies for recycling Krylov subspace information that exploit properties of the application and the nonlinear optimization algorithm to significantly reduce the total number of iterations over all linear systems. Furthermore, we introduce variants of GCRO that exploit symmetry and that allow simultaneous solution of multiple shifted systems using a single Krylov subspace in combination with recycling. Although we focus on a particular application and optimization algorithm, our approach is applicable generally to problems where sequences of linear systems must be solved. This may guide other researchers to exploit the opportunities of tunable solvers.We provide results for two sets of numerical experiments to demonstrate the effectiveness of the resulting method. |
Year | DOI | Venue |
|---|---|---|
2006 | 10.1137/040610271 | SIAM J. Scientific Computing |
Keywords | Field | DocType |
gcro,multiple shift,krylov subspace,optimization algorithm,efficient solution,minres,diffuse optical tomography,long sequence,eigenvalue,nonlinear optimization algorithm,simultaneous solution,computed solution,recycle,invariant subspace,multiple right-hand side,recycling subspace information,squares algorithm result,linear system,nonlinear least squares,three dimensional,line search,eigenvalues,nonlinear optimization | Krylov subspace,Mathematical optimization,Linear system,Matrix (mathematics),Nonlinear programming,Invariant subspace,Non-linear least squares,Eigenvalues and eigenvectors,Numerical linear algebra,Mathematics | Journal |
Volume | Issue | ISSN |
27 | 6 | 1064-8275 |
Citations | PageRank | References |
26 | 1.46 | 21 |
Authors | ||
2 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Misha E. Kilmer | 1 | 320 | 39.27 |
| Eric de Sturler | 2 | 398 | 27.32 |