Fast algorithms for the inverse medium problem. George Biros University of Pennsylvania

Transcription

1 Fast algorithms for the inverse medium problem George Biros University of Pennsylvania

2 Acknowledgments S. Adavani, H. Sundar, S. Rahul (grad students) C. Davatzikos, D. Shen, H. Litt (heart project) Akcelic, Ghattas, B. Van Bloemen Waanders (transport) Department of Energy o DE-FG02-04ER25646 National Science Foundation o CCF/ITR , NSF/IIE PSC, Argonne (PETSc)

3 Overview Inverse medium problems Motivating applications Linear problems o 1D parabolic problem, spatial control o Preconditioning techniques for reduced space approaches Multigrid Numerical results

4 Inverse medium problem

5 General comments on inverse medium problems Difficult as β 0 o Depends on spaces of y, u, z Goal: fast solvers o If y,u are in R N, then Mesh independent O(N) algorithm o Nonlinear robustness

6 Example of a scroll wave Normal propagation Scroll wave

7 Computational work for the forward electrophysiology problem System of reaction-diffusion PDEs Spatial resolution ~ 6*256^3 ~ 100 million Temporal resolution ~ 1000 time steps Inversion parameters o ~50-100K The associated costs suggest an inexact reduced space approach o Several other problems in this category

8 A similar, simpler problem Wind from mesoscale models (MM5) Sparse sensor readings of concentration Akcelic, Biros, Draganescu, Ghattas, Hill, Waanders 05

9 Formulation Given wind w, observations y*, estimate u : Then forward problem can be used to predict transport contaminant

10 Inversion example Inversion Reconstruction Real initial condition

11 Parallel multigrid performance and scalability on PSC EV68 AlphaCluster Fixed size scalability: 257^4 space-time ~9 billion unknowns 3-level MG Isogranular scalability (fixed size/cpu) ~140 billion unknowns for max size, 3 level MG

12 Sensitivity to regularization

13 Need for multilevel solvers for high-resolution inverse medium problems Problems with single level solver o Algorithmic scalability o Globalization (for nonlinear problems) o CG scales well if Hessian is compact perturbation of identity Multigrid to reduce constant o Inverse problems Need solver robustness wrt regularization parameter

14 Consider 1D inverse medium heat equation

15 Optimality conditions KKT optimality conditions KKT matrix

16 Optimality conditions operator form

17 Consider multigrid solvers State-of-the art for positive BVPs o Fast multipole methods and integral equations o Geometric and algebraic multigrid o Multilevel Schur & Schwarz o Wavelets Convexity requires positive BVP for reduced Hessian

18 2-grid multigrid scheme

19 Related work on multigrid Multigrid - elliptic PDEs o Brandt, Braess, Bramble, Hackbusch Multigrid second kind Fredholm o Hackbusch, Hemker & Schippers Multigrid for optimization o Ascher & Haber & Oldenburg, Borzi, Borzi & Kunisch, Borzi & Griesse, Chavent, Dreyer & Maar & Schultz, Draganescu, Hanke & Vogel, Lewis & Nash, Kaltenbacher, King, Kunoth, Ta asan, Tau & Xu, Vogel, Toint Large-scale parallel multigrid/cascade o Akcelic, Biros & Ghattas, Akcelic et al.

20 Challenges in designing fast solvers for reduced Hessians Typically dense Typically only MatVec available Positivity comes in many flavors o I + Compact o Compact o Differential Multigrid available for first and third o Little available in the middle category o Inverse problems when rhs range(hessian) Smoothers (coarse/fine grids work partitioning) Transfer operators (do not contaminate spectrum) Globalization (non-convex problems)

21 Spectrum of reduced Hessian in the unit box with Dirichlet BCs

22 Reduced Hessian spectrum Use CG as a smoother?

23 Using CG for the reduced Hessian - I regularization Fixed β : CG mesh-independent Fixed mesh : CG β-dependent β depends on frequency information that we need to recover o Truncation noise β h 2

24 Using CG for the reduced Hessian - II regularization Fixed β : CG mesh-independent Fixed mesh : CG β-dependent β depends on frequency information that we need to recover o Truncation noise β h 2

25 Motivation for reduced space CG solver Smooth eigenvectors forward operator spectrum Large eigenvectors

26 CG performance Laplacian smoothing factor Reduced Hessian wavenumber

27 Smoothers for optimization-related operators Hackbush smoother(s) for 2 nd -kind Fredholm o Not good in practice, especially for inverse problems Kaltenbacher-King smoothers o Easier to implement but not scalable as regularization 0 Borzi-Griesse pointwise space-time multigrid o Collective, but expensive for general regularization Time domain-decomposition o Lions, Hienkenschloss, Maday

28 Borzi, Griesse, Kunisch o linear elliptic or parabolic, nonlinear term Eliminate decision variables from optimality conditions Diagonally dominant operator for large β

29 Kaltenbacher-King

30 Pointwise smoother

31 Combine with filtering Apply only on high-frequency part Optimal complexity Mesh/ β independent

32 Numerical results Multigrid, with PF smoother Adavani, B., 06

33 Convergence factors

34 Summary Computationally demanding forward problems o Billions of states, millions of opt parameters Need for fast Hessian solvers Special structure of Hessian for inverse problems Multigrid o Standard smoothers won t work o Existing inv.prob smoothers don t scale wrt reg. param. Proposed new smoother Ongoing work o Variable coefficients, non-linear constraints o H1-regularization, TV-regularization

35 A maximum