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

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

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 0427695, NSF/IIE 0431070 PSC, Argonne (PETSc)

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

Inverse medium problem

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

Example of a scroll wave Normal propagation Scroll wave

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

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

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

Inversion example Inversion Reconstruction Real initial condition

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

Sensitivity to regularization

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

Consider 1D inverse medium heat equation

Optimality conditions KKT optimality conditions KKT matrix

Optimality conditions operator form

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

2-grid multigrid scheme

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.

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)

Spectrum of reduced Hessian in the unit box with Dirichlet BCs

Reduced Hessian spectrum Use CG as a smoother?

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

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

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

CG performance Laplacian smoothing factor Reduced Hessian wavenumber

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

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

Kaltenbacher-King

Pointwise smoother

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

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

Convergence factors

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

A maximum