Name
Title | ||
|---|---|---|
A Data Scalable Augmented Lagrangian KKT Preconditioner for Large-Scale Inverse Problems. |
Abstract | ||
|---|---|---|
Current state-of-the-art preconditioners for the reduced Hessian and the Karush-Kuhn Tucker (KKT) operator for large-scale inverse problems are typically based on approximating the reduced Hessian with the regularization operator. However, the quality of this approximation degrades with increasingly informative observations or data. Thus the best case scenario from a scientific standpoint (fully informative data) is the worse case scenario from a computational perspective. In this paper we present an augmented Lagrangian-type preconditioner based on a block diagonal approximation of the augmented upper left block of the KKT operator. The preconditioner requires solvers for two linear subproblems that arise in the augmented KKT operator, which we expect to be much easier to precondition than the reduced Hessian. Analysis of the spectrum of the preconditioned KKT operator indicates that the preconditioner is effective when the regularization is chosen appropriately. In particular, it is effective when the regularization does not overpenalize highly informed parameter modes and does not underpenalize uninformed modes. Finally, we present a numerical study for a large data/low noise Poisson source inversion problem, demonstrating the effectiveness of the preconditioner. In this example, three MINRES iterations on the KKT system with our preconditioner results in a reconstruction with better accuracy than 50 iterations of CG on the reduced Hessian system with regularization preconditioning. |
Year | DOI | Venue |
|---|---|---|
2017 | 10.1137/16M1084365 | SIAM JOURNAL ON SCIENTIFIC COMPUTING |
Keywords | Field | DocType |
PDE constrained inverse problems,data scalability,augmented Lagrangian,preconditioning,KKT matrix,Krylov subspace methods | Mathematical optimization,Preconditioner,Mathematical analysis,Hessian matrix,Regularization (mathematics),Augmented Lagrangian method,Inverse problem,Operator (computer programming),Karush–Kuhn–Tucker conditions,Mathematics,Block matrix | Journal |
Volume | Issue | ISSN |
39 | 5 | 1064-8275 |
Citations | PageRank | References |
1 | 0.35 | 27 |
Authors | ||
4 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Nick Alger | 1 | 1 | 0.69 |
| Umberto Villa | 2 | 30 | 6.64 |
| Tan Bui-Thanh | 3 | 102 | 12.50 |
| Omar Ghattas | 4 | 697 | 61.43 |