Mathematical Methods in Inverse Problems for Partial Differential Equations.

William Rundell, Texas A&M University, Chair

Contents

Introduction

Inverse problems in mathematics are as old as the discipline itself; from earliest times questions have been posed that require the determination of an underlying structure from auxiliary information. However, inverse problems in differential equations have a much more recent history. While the concept of a differential equation dates from the time of Newton and the main analytic techniques for solving them are at least a hundred years old, the study of inverse problems in differential equations is, in reality, less than fifty years old with a real explosion only in the last ten or twenty years. From a mathematical standpoint the development of nonlinear functional analysis and of modern, sophisticated numerical analysis was a key factor. The other driving force was an increase in the number of important applications of inverse problems; these applications now span the entire range of applied science.

Conference main themes

Perhaps the central mathematical issue in any inverse problem is the question of uniqueness - is the additional data provided sufficient to uniquely determine the unknown coefficient or parameter? Of course, the optimal situation would be to determine the exact amount of overposed data to answer the uniqueness question in the affirmative. For all but the simplest problems, this is extremely difficult. There have been many important breakthroughs in recent years and we now have an expanded set of tools. It is important that techniques that were effective in one application are brought to the attention of researchers working in others.

Algorithms for effective reconstruction of the coefficient are also clearly important. Here consideration must always be given to the fact that the unknown coefficient to data map is usually both highly nonlinear and compact; standard optimization schemes generally fail due to the presence of many local minima and Newton-type iteration schemes have the problem that the derivative of the map is not invertible in the desired topologies. This has been an area of much recent research and several new ideas have shown considerable promise. ``Steepest descent'' methods based on Landweber-Fridman iteration have been shown to be surprisingly effective in certain problems; one is trading convergence speed for provable progress towards a solution. New regularization ideas have been developed - for example the use of the total variation penalty function has proved remarkably effective in several problems where the coefficient to be reconstructed had an ''almost piecewise constant'' form. The use of innovative and adaptive basis sets together with corresponding fast inversion algorithms have revolutionized the art of inverting first kind integral equations; these are the building blocks of general inverse problems as well as image reconstruction. The extension of these techniques, which have allowed enormous advances in dealing with the effects of ill-conditioning, to nonlinear models offers exciting mathematical challenges.

Finally, to obtain an actual solution to many inverse problem requires a statistical analysis of the data and numerical computation: an algorithm, and its implementation. Many problems of scientific interest, in particular those arising in astronomy and geophysics, of necessity generate extremely large data sets. The building of a ``correct'' mathematical model for the underlying problem and the issues of data reduction and statistical consistency are critical. A second emerging computational area is the development of software libraries of object-oriented code to solve many of the ``standard features'' of inverse problems. This is a rich area for computational mathematicians with the potential for considerable impact towards a wide variety of applications.

Preliminary Schedule

SUNDAY
8:30 - 8:40 Welcoming Remarks
8:40 - 9:20 Heinz Engl
(Universität Linz)		 "Iterative Regularization of Nonlinear Inverse Probems"
9:20 - 10:00 Brian Sleeman
(University of Leeds)		 "Acoustic Scattering by Irregular Obstacles"
10:00 - 10:30 Coffee
10:30 - 11:10 Zuhair Nashed
(University of Delaware)		 "Kernel Methods in Numerical Conformal Mapping and Inverse Problems"
11:10 - 11:50 Andreas Kirsch
(Universität Karlsruhe)		 "The Regularized Spectral Method in Inverse Scattering"
12:00 - 1:00 Lunch
1:00 - 1:40 Frank Hettlich
(Universität Erlangen-Nürnberg)		 "A regularized second degree method for inverse problems"
1:40 - 2:20 Philip Stark
(University of Berkeley)		 "Misfit Measures and Statistical Inconsistency in Linear Inverse Problems"
2:20 - 3:00 Bob Anderssen
(University of Canberra)		 "Non-Linear Industrial Inverse Problems: Asymetric Flow-FFF"
3:00 - 3:30 Coffee
3:30 - 4:10 David Dobson
(Texas A&M University)		 "Maximizing band gaps in periodic dielectric structures"
4:10 - 4:50 Graham Gladwell
(University of Waterloo)		 "Title pending"
4:50 - 5:30 Lassi Päivärinta
(University of Oulu)		 "On inverting singularities from back-scattering data"
6:00 Dinner
MONDAY
8:40 - 9:20 Alfred Louis
(Universität des Saarlandes)		 "The approximate inverse for linear and some nonlinear problems"
9:20 - 10:00 Steve Cox
(Rice University)		 "Extracting the Cable Properties of Dendritic Neurons from Input Impedance Data"
10:00 - 10:30 Coffee
10:30 - 11:10 George Papanicolaou
(Stanford University)		 "High contrast impedance tomography"
11:10 - 11:50 Patricia Lamm
(Michigan State University)		 "Variable local regularization of ill-posed inverse problems"
12:00 - 1:00 Lunch
1:00 - 1:40 Rainer Kress
(Universität Göttingen)		 "Approximation of inverse media problems by inverse transmission problems"
1:40 - 2:20 Paul Sacks
(Iowa State University)		 "The inverse spectral problem for a radially symmetric potential"
2:20 - 3:00 Victor Isakov
(Wichita State University)		 "On inverse problems for linear and nonlinear parabolic equations"
3:00 - 3:30 Coffee
3:30 - 4:10 Martin Hanke
(Universität Karlsruhe)		 "A numerical solution of an inverse nonlinear heat diffusion problem"
4:10 - 4:50 Lester Caudill
(University of Richmond)		 "Recent Progress on Inverse Problems in Thermal Imaging"
4:50 - 5:30 Lars Elden
(Linköping University)		 "Numerical solution of some ill-posed Cauchy problems for PDE"
6:00 Dinner
TUESDAY
8:10 - 8:50 John Sylvester
(University of Washington)		 "LayerStripping via the Riesz Transform"
8:50 - 9:30 Curt Vogel
(Montana State)		 "Phase diversity-based deconvolution and phase retrieval"
9:30 - 10:10 Peter Maass
(Universität Potsdam)		 "A bilinear Landweber--algorithm with applications in emission tomography"
10:10 - 10:40 Coffee
10:40 - 11:20 Slava Kurylev
(University of Loughborough)		 "Gromov's compactness and stability of the Gel'fand inverse problem"
11:20 - 12:00 Joyce McLaughlin
(Rensselaer Polytechnic Institute)		 "Natural Frequencies and Mode Shapes: Rich Data Sets"
12:00 - 1:00 Lunch
1:00 - 5:30 Excursion
6:00 Dinner
WEDNESDAY
8:40 - 9:20 Gunther Uhlmann
(University of Washington)		 "Inverse scattering in anisotropic media"
9:20 - 10:00 Peter Monk
(University of Delaware)		 "The Regularized Sampling Method in Inverse Scattering"
10:00 - 10:30 Coffee
10:30 - 11:10 Pierre Sabatier
(Université de Montpellier)		 "Side Aspects of Inverse Problem Analyses"
11:10 - 11:50 Otmar Scherzer
(Universität Linz)		 "A posteriori error estimates for nonlinear ill-posed problems"
12:00 - 1:00 Lunch
1:00 - 1:40 Michael Klibanov
(NC Charlotte)		 "Elliptic Systems Numerical Method in Optical Tomography"
1:40 - 2:20 Gang Bao
(University of Florida)		 " Determination of locations of epileptic foci inthe living human brain"
2:20 - 3:00 Alfredo Lorenzi
(Universitá Milan)		 "Identification problems for linear evolution integrodifferential equations with operator coefficients"
3:00 - 3:30 Coffee
3:30 - 4:10 Ziqi Sun
(Wichita State University)		 Canceled
4:10 - 4:50 Margaret Cheney
(Rensselaer Polytechnic Institute)		 "Uniqueness for a half-space inverse problem"
4:50 - 5:30 Karl Kunisch
(Universität Graz)		 "Canceled"
6:00 Dinner
THURSDAY
8:40 - 9:20 Michael Vogelius
(Rutgers University)		 "Determination of conductivity imperfections of small area (volume) by boundary measurements"
9:20 - 10:00 Tom Seidman
(University of Maryland)		 "Sideways heat equation with spatially variable coefficients"
10:00 - 10:30 Coffee
10:30 - 11:10 Andreas Rieder
(Universität des Saarlandes)		 "Inexact Newton Techniques for the Regularization of Nonlinear Ill-Posed Problems"
11:10 - 11:50 Yu Chen
(Courant Institute)		 "Stable Layer Stripping in Two Dimensions"
12:00 - 1:00 Lunch
1:00 End of Conference


Colloquium Abstracts

Speaker: Bob Anderssen (Canberra)

Abstract: Initially, the talk will review some of the mathematical challenges posed by a variety of industrial inverse problems including electromagnetic and acoustic non-destructive testing, aquifer parameter identification, infiltration, polymer dynamics and food processing. Among other things, these applications will be used to show how simple linear forward problems can spawn a wide variety of quite challenging and complex non-linear inverse problems. Even for the more complex non-linear inverse problems which arise in industry, the nature of the question being examined and the available data often act as a guide to simplification which can be invoked to allow the underlying problem to be assessed mathematically as well as solved computationally and efficiently. In order to illustrate this process, the talk will focus on the asymetric flow field-flow fractionation (flow-FFF) procedure for determining the molecular weight distributions of proteins and starches.

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Speaker: Steven Cox (Rice University)

Abstract: We contribute to the long-standing question of the degree to which input impedance measurements determine the cable properties of biological cells. The cable model of the cell amounts to viewing the cell as a uniaxial conductor with axial (or cytoplasm) resistance R, lateral (or membrane) conductance G, and lateral (or membrane) capacitance C.
The objective is to determine R, G, and C from measurements taken at one end of the cell, for example at the soma in the case of nerve cell. More precisely, one supposes the distal end of the cell to be insulated, injects a known current pulse, i(t), into the near end and simultaneously measures the associated potential, v(t), at this end. The ratio of the respective Laplace transforms of v and i is written z(s) and referred to as the input impedance of the cell. The literature on the measurement of this impedance and its use in the characterization of R, G, and C is vast. The best general reference is probably Impedance Measurements in Biological Cells, written by O.F. Schanne and E.R. P.-Ceretti, and published by John Wiley in 1978. While the experimental technique appears to have reached a reasonable level of maturity the companion problem of fitting the data to a cable model has remained ad hoc. In particular, there has yet to appear any confirmation that the input impedance indeed uniquely determines R, G, and C.
Our solution to this problem performs well on synthetic data and leads to a highly practical algorithm. More precisely, we show that evaluating z and its first two derivatives at s=0 suffices to uniquely determine R, G, and C. Recalling basic properties of the Laplace transform it follows that our data, z(0), z'(0), and z''(0), may be expressed as rational combinations of the first 3 moments of i and v. As our data arises from integrals of v and i it tends to smooth noisy transients, a characteristic that bodes well for its application to experimental data.

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Speaker: David Dobson (Texas A&M)

Abstract: "Photonic crystals" are optical microstructures composed of periodic arrangements of dielectric materials. Such structures can exhibit interesting and useful spectral behavior, including the existence of band gaps. As fabrication issues are resolved, these structures are expected to have a dramatic impact on many applications in optics and electronics.

We consider the problem of determining material arrangements which result in maximal band gaps. The approach is to first formulate an appropriate optimization problem, which turns out to have a nonsmooth but Lipschitz continuous cost functional. The problem formulation allows for generalization to an inverse problem setting. A generalized gradient is calculated, and a simple generalized gradient ascent minimization algorithm is proposed. Numerical results are presented in which new structures with large band gaps are obtained.

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Speaker: Lars Elden (Linköping University)

Abstract: We consider the following Cauchy problems for partial differential equations: the sideways heat equation, the heat equation backwards in time, the Cauchy problem for an elliptic equation. In simple geometries these problems can be regularized by replacing an unbounded differential operator by a bounded one, e.g. by using spectral or wavelet approximations. The resulting well-posed problem can be considered as an initial value problem for an ODE, and can be solved by standard methods, e.g. a Runge-Kutta method. Numerical aspects of these approximations are discussed: We analyze and compare Fourier and wavelet methods for solving the such problems, with respect to efficiency and robustness to perturbations of the data. Error estimates are given, as well as recipes for choosing the parameter that controls stability. As test problems we take model equations and a problem from an industrial application.

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Speaker: Heinz Engl (Linz)

Abstract: We first give a short introduction into the (by now classical) convergence theory for Tikhonov regularization of nonlinear ill-posed problems. Convergence rates always need "source conditions" involving an adjoint to the linearized operator. Based on recent joint work with J.Zou (Hongkong), we present a variant using an adjoint involving the weak formulation of the underlying equation that gives more flexibility for applying this theory e.g. to parameter identification problems.
We then present a new approach, based on invariance principles, to prove convergence rates for iterative regularization methods for solving nonlinear ill-posed problems, and illustrate this method for Landweber iteration and iteratively regularized Gauss-Newton methods. (Joint work with P.Deuflhard,Berlin, and O.Scherzer,Linz). Finally, we apply Landweber iteration to a parameter identification problem in a mathematical model for growth of polymers. (Joint work with M.Burger,Linz, and V.Capasso,Milano).

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Speaker: Martin Hanke (Karlsruhe)

Abstract: We consider the inverse problem of reconstructing the diffusion coefficient in a quasilinear parabolic differential equation in divergence form from measurements of the solution at a finite number of points in the interior of the domain. An equation error method is developed which transforms the inverse problem into a system of linear operator equations for the diffusion coefficient, and which can be solved by the conjugate gradient method in a very efficient and stable manner. A detailed error analysis relates the required number of measurements with their accuracy. Numerical results illustrate the performance of the method. This is joint work with Otmar Scherzer (Linz).

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Speaker: Frank Hettlich (Erlangen-Nürnberg)

Abstract: Based on recent results on regularized Newton type methods a general iterative scheme approximating solutions of nonlinear ill-posed equations which uses the second derivative is suggested. A predictor-corrector approach to the nonlinear equation allows the application of the linear Tikhonov regularization in any iteration step. Combined with a certain stopping criterion local convergence of the method is shown under similar conditions as known for the regularized Newton method or the Landweber iteration.

As a test example an inverse scattering problem is considered which is known to be highly ill-posed. From a representation of the second derivative of the domain to far field operator an efficient numerical implementation of the second degree method can be obtained. The performance of the method illustrates the advantage which can be gained from using the second derivative information.

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Speaker: Victor Isakov (Wichita State)

Abstract: In this talk we discuss uniqueness and stability of identification of (smooth and discontinuous) diffusion coefficients and of nonlinear terms of parabolic equations and of some systems. We consider single and many lateral boundary measurerements. We give applications to recovery of constitutive law of diffusion and to option pricing as well as some numerical algorithms.

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Speaker: Andreas Kirsch (Karlsruhe)

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Speaker: Rainer Kress (Universität Göttingen)

Abstract: For the inverse electrical impedance tomography problem, in some medical applications it is known a priori that the conductivity is almost piecewise constant corresponding to different conductivities of different organs. In these cases rather than determining the value of the impedances in the internal organs it is of interest to determine their boundaries. A typical example for this situation occurs in the monitoring of the lung activities during a breath cycle.

This observation leads to considering an inverse transmission problem for the Laplace equation as an approximation for the inverse impedance tomography problem. We present a regularized Newton type method for the numerical solution of the inverse transmission problem via boundary integral equation techniques. Through a comparison of the numerical results with those obtained via a standard finite element method, both for synthetic data and for real data from clinical applications, we illustrate the feasibility of this approach to inverse impedance tomography.

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Speaker: Michael Klibanov (NC Charlotte)

Abstract: The authors have recently developed a novel approach to the numerical solution of a rather broad class of multidimensional inverse problems with single measurement. They call this approach the Elliptic Systems Method (ESM) [1]. This algorithm has roots in earlier works of Klibanov devoted to applications of Carleman estimates to inverse problems. The ESM has been tested successfully for a broad variety of conditions for the case of Optical Tomography (OT) with the time dependent data. OT amounts to the solution of an inverse problem for the parabolic equation. OT has intriguing applications in optical imaging of human organs. In this talk, the ESM will be presented and its numerical performance for both simulated and experimental data will be discussed.

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Speaker: Patricia Lamm (Michigan State University)

Abstract: We will give an overview of local regularization methods for general ill-posed integral equations, stating recent results that we have obtained in the theory for linear Volterra integral equations. In particular, these results appear to allow the extension of this theory to problems of non-Volterra (i.e., Fredholm) type.

In addition, we present some numerical results for the local Tikhonov regularization of Volterra problems and show how a local, variable Tikhonov parameter $\alpha = \alpha(t)$ may be selected using a sequence of local discrepancy principles. As will be seen in various examples, the numerical process determines $\alpha$ to be relatively large in flat areas of the solution and small in steep or sharp areas of the solution.

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Speaker: Alfred Louis (Universität des Saarlandes)

Abstract: For solving a linear euqation we precompute a reconstruction kernel by solving an auxiliary problem for the adjoint operator where the right-hand side is a 'mathematical lense' that is chosen according to the desired information on the solution. A parallel processing of the solution process is possible. Using invariance properties of the operator the evaluation can be considerably accelerated.

It is shown that this approximate inverse is a general form of a regularization method which can be interpreted as a combination of the generalized inverse and a smoothing in any order.

Applications are given for different problems as x-ray computerized tomography and for the solution of Volterra convolution equations where marching schemes result.

Generalizations to nonlinear problems are discussed and an example for identification of a vibrating string is presented.

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Speaker: Peter Maass (Berlin)

Abstract:

We propose an iterative algorithm, which is based on the well--known Landweber--Iteration for the solution of operator equations of the form $A(f)+B(f,\mu)=y$. Here, $A$ is a linear and compact operator and $B$ is a bilinear, completely continous operator. We show that the algorithm converges in a weak sense to a solution of the equation.

For the inversion of data from single--photon emission computed tomography (SPECT), we approximate the Attenuated Radon Transform by a bilinear operator equation and present some results from the reconstructions.

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Speaker: Joyce McLaughlin (Rensselaer Polytechnic Institute)

Abstract: We give an overview of how these sets can be used, what mathematical properties are established to make their efffective use and how the data sets are measured.

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Speaker: Andreas Rieder (Saarbrücken)

Abstract: We present inexact Newton methods for the stable solution of nonlinear ill-posed problems. The corresponding inner scheme can be chosen to be any linear regularization technique with a sufficient modulus of convergence.

The regularization property of these Newton-type algorithms will be verified, that is, the iterates converge to a solution of the nonlinear problem with exact data when the noise level tends to zero. We also give convergence rates.

For an implementation of our inexact Newton iteration the issue of choosing the forcing terms is vital. We suggest a selection strategy based on our convergence analysis.

Finally, we report on numerical experiments with respect to a parameter identification problem for an elliptic PDE. The results reproduce nicely theoretical predictions and show the efficiency of the proposed scheme.

Key words: Nonlinear ill-posed problems, regularization, inexact Newton iteration
Subject classification: AMS(MOS) 65J15, 65J20

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Speaker: Pierre Sabatier (Montpellier)

Abstract: Inverse Problem analyses generally appear in attempts for recovering the parameters of a physical system from data, what can be called data inversion. However, they have also been used in studies of integrable systems, resolution improvements, measurements design, nonlinear signal processing, etc,. Two new examples in this set of side aspects are presented here, both because they are of current concern for the author and because several features of the "inverse analyses" which they contain are not quite similar to those which appear in data inversion.

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Speaker: Gunther Uhlmann (Washington)

Abstract: In inverse scattering one attempts to determine the properties of an inhomogeneous medium by measuring the response of the medium to incident plane waves. There has been considerably progress in understanding this problem in the case that the medium parameters are isotropic, i.e. they does not depend on direction. However there are several important physical examples of medium parameters exhibiting anisotropy. These include the electromagnetic parameters of crystals, the elastic parameters of the Earth and the conductivity of muscle tissue.

In this talk we consider one of the basic inverse scattering problems in anisotropic media. This involves the determination of a Riemannian metric which is euclidian outside a ball from scattering information. This problem is closely related to the question of whether one can determine a Riemannian metric on a bounded domain by measuring the travel times of geodesics through the domain. We will discuss the relationship between these two problems and some recent results.

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Speaker: Curt Vogel (Montana State)

Abstract: In conventional deconvolution, one is given data

d(x)=\int s(x-y)f(x)dx.

In optical imaging s is called the Point Spread Function (PSF), $f$ is the object, and $d$ is the (blurred) image. It is often the case that both $s$ and $f$ are unknown. This problem is called blind deconvolution, and is highly underdetermined. Fortunately, one can sometimes make use of physics to avoid blind deconvoluton.

In atmospheric optics an important variable is the phase $\phi$, which quantifies the deviation from planarity of a wave front which has propagated through the atmosphere. The dependence of the PSF of the phase is given by

s[\phi] = |{\cal F}(pe^{\imath\phi})|^2,

where ${\cal F}$ denotes the Fourier transform, and $p$ is the pupil, or aperture, function for the optics. By deliberately perturbing the unknown phase by a known amount, one can generate additional images. This procedure is called phase diversity.

In this talk, we will discuss various aspects of the problem of estimating both the phase and the object form phase diversity data. Included will be discussion of ill-posedness and regularization, numerical optimization, and stochastic modeling of (both temporally and spatially varying) phase.

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Organizing Committee

Last Updated: 12 Jun 98