Grid refinement and scaling for distributed parameter estimation problems

This paper considers problems of distributed parameter estimation from data measurements on solutions of differential equations. A nonlinear least squares functional is minimized to approximately recover the sought parameter function (i.e. the model). This functional consists of a data fitting term, involving the solution of a finite volume or finite element discretization of the forward differential equation, and a Tikhonov-type regularization term, involving the discretization of a mix of model derivatives. The resulting nonlinear optimization problems can be very large and costly to solve. Thus, we seek ways to solve as much of the problem as possible on coarse grids. We propose to search for the regularization parameter first on a coarse grid. Then, a gradual refinement technique to find both the forward and inverse solutions on finer grids is developed. The grid spacing of the model discretization, as well as the relative weight of the entire regularization term, affect the sort of regularization achieved and the algorithm for gradual grid refinement. We thus investigate a number of questions which arise regarding their relationship, including the correct scaling of the regularization matrix. For nonuniform grids we rigorously associate the practice of using unscaled regularization matrices with approximations of a weighted regularization functional. We also discuss interpolation for grid refinement. Our results are demonstrated numerically using synthetic examples in one and three dimensions.

[1]  P. J. Huber.  Robust Estimation of a Location Parameter , 1964 .

[2]  Susana Gómez,et al.  Multiscale Optimization For Aquifer Parameter Identification With Noisy Data , 1970 .

[3]  Frederick R. Forst,et al.  On robust estimation of the location parameter , 1980 .

[4]  Nariida C. Smith,et al.  Two-Dimensional DC Resistivity Inversion for Dipole-Dipole Data , 1984, IEEE Transactions on Geoscience and Remote Sensing.

[5]  W. Menke.  Geophysical data analysis : discrete inverse theory , 1984 .

[6]  V. A. Morozov,et al.  Methods for Solving Incorrectly Posed Problems , 1984 .

[7]  Jon F. Claerbout,et al.  Imaging the Earth's Interior , 1985 .

[8]  Wolfgang Hackbusch,et al.  Multi-grid methods and applications , 1985, Springer series in computational mathematics.

[9]  L. Watson.  Numerical linear algebra aspects of globally convergent homotopy methods , 1986 .

[10]  E. Allgower,et al.  A mesh-independence principle for operator equations and their discretizations , 1986 .

[11]  Andrew Blake,et al.  Visual Reconstruction , 1987, Deep Learning for EEG-Based Brain–Computer Interfaces.

[12]  C. Kelley.  Iterative Methods for Linear and Nonlinear Equations , 1987 .

[13]  Eugene Isaacson,et al.  Numerical Solution of Boundary Value Problems for Ordinary Differential Equations (Uri M. Ascher, Robert M. M. Mattheij, and Robert D. Russell) , 1989, SIAM Rev..

[14]  A non-linear least-square theory for inverse problems A non-linear least-square theory for inverse p , 1989 .

[15]  C. Chavent,et al.  Multiscale Parametrization for the Estimation of a Diffusion Coefficient in Elliptic and Parabolic Problems , 1989 .

[16]  A. Devaney.  The limited-view problem in diffraction tomography , 1989 .

[17]  G. Wahba.  Spline models for observational data , 1990 .

[18]  K. Kunisch,et al.  The augmented lagrangian method for parameter estimation in elliptic systems , 1990 .

[19]  Grace Wahba,et al.  Spline Models for Observational Data , 1990 .

[20]  P. Wesseling.  An Introduction to Multigrid Methods , 1992 .

[21]  F. Potra,et al.  Asymptotic mesh independence of Newton-Galerkin methods via a refined Mysovskii theorem , 1992 .

[22]  Jun Liu.  A Multiresolution Method for Distributed Parameter Estimation , 1993, SIAM J. Sci. Comput..

[23]  Matthias Heinkenschloss,et al.  Mesh Independence for Nonlinear Least Squares Problems with Norm Constraints , 1993, SIAM J. Optim..

[24]  D. Oldenburg,et al.  Inversion of Induced-Polarization Data , 1993 .

[25]  Per Christian Hansen,et al.  Regularization methods for large-scale problems , 1993 .

[26]  R. Parker.  Geophysical Inverse Theory , 1994 .

[27]  Richard Barrett,et al.  Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods , 1994, Other Titles in Applied Mathematics.

[28]  Robert D. Russell,et al.  Numerical solution of boundary value problems for ordinary differential equations , 1995, Classics in applied mathematics.

[29]  Geophysical Inverse Theory R.L. Parker, Princeton University Press, Princeton, NJ, 1994, 386 pp., ISBN 0-691-03634-9, hardback £30, $39.50 , 1995 .

[30]  L. N. Vicente,et al.  Trust-Region Interior-Point SQP Algorithms for a Class of Nonlinear Programming Problems , 1998 .

[31]  G. Papanicolaou,et al.  High-contrast impedance tomography , 1996 .

[32]  H. Engl,et al.  Regularization of Inverse Problems , 1996 .

[33]  Van Emden Henson,et al.  Multilevel Image Reconstruction with Natural Pixels , 1996, SIAM J. Sci. Comput..

[34]  Volker Schulz,et al.  Reduced SQP Methods for Large-Scale Optimal Control Problems in DAE with Application to Path Plannin , 1996 .

[35]  Eldad Haber,et al.  Numerical strategies for the solution of inverse problems , 1997 .

[36]  M. Hanke.  Regularizing properties of a truncated Newton-CG algorithm for nonlinear inverse problems , 1997 .

[37]  T. Chan,et al.  Augmented Lagrangian and total variation methods for recovering discontinuous coefficients from elliptic equations , 1997 .

[38]  Homer F. Walker,et al.  NITSOL: A Newton Iterative Solver for Nonlinear Systems , 1998, SIAM J. Sci. Comput..

[39]  D. Oldenburg,et al.  NON-LINEAR INVERSION USING GENERAL MEASURES OF DATA MISFIT AND MODEL STRUCTURE , 1998 .

[40]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[41]  Carl Tim Kelley,et al.  Iterative methods for optimization , 1999, Frontiers in applied mathematics.

[42]  Andreas Rieder,et al.  On the regularization of nonlinear ill-posed problems via inexact Newton iterations , 1999 .

[43]  Rainer Helmig,et al.  Conditional statistical inverse modeling in groundwater flow by multigrid methods , 1999 .

[44]  Gene H. Golub,et al.  A Nonlinear Primal-Dual Method for Total Variation-Based Image Restoration , 1999, SIAM J. Sci. Comput..

[45]  E. Haber,et al.  Fast Simulation of 3D Electromagnetic Problems Using Potentials , 2000 .

[46]  G. Newman,et al.  Three-dimensional magnetotelluric inversion using non-linear conjugate gradients , 2000 .

[47]  E. Haber,et al.  On optimization techniques for solving nonlinear inverse problems , 2000 .

[48]  E. Haber,et al.  A GCV based method for nonlinear ill-posed problems , 2000 .

[49]  S. Nash.  A multigrid approach to discretized optimization problems , 2000 .

[50]  Liliana Borcea.  A nonlinear multigrid for imaging electrical conductivity and permittivity at low frequency , 2001 .