Mark S. Gockenbach

An Abstract Framework for Elliptic Inverse Problems: Part 1. An Output Least-Squares Approach

The solution of an elliptic boundary value problem is an infinitely differentiable function of the coefficient in the partial differential equation. When the (coefficient-dependent) energy norm is used, the result is a smooth, convex output least-squares functional. Using total variation regularization, it is possible to estimate discontinuous coefficients from interior measurements. The minimization problem is guaranteed to have a solution, which can be obtained in the limit from finite-dimensional discretizations of the problem. These properties hold in an abstract framework that encompasses several interesting problems: the standard (scalar) elliptic BVP in divergence form, the system of isotropic elasticity, and others.

C++ classes for linking optimization with complex simulations

The object-oriented programming paradigm can be used to overcome the incompatibilities between off-the-shelf optimization software and application software. The Hilbert Class Library (HCL) defines the fundamental mathematical objects arising in optimization problems, such as vectors, linear operators, and so forth, as C++ classes, making it possible to write optimization code in a natural fashion, while allowing application software such as simulators to use the most convenient data structures and programming style. In spite of the poor reputation C++ has for runtime performance, the use of mixed-language programming allows performance equal to that achieved by standard Fortran packages, as comparisons with the popular code LBFGS and ARPACK demonstrate.

An Abstract Framework for Elliptic Inverse Problems: Part 2. An Augmented Lagrangian Approach

The coefficient in a linear elliptic partial differential equation can be estimated from interior measurements of the solution. Posing the estimation problem as a constrained optimization problem with the PDE as the constraint allows the use of the augmented Lagrangian method, which is guaranteed to converge. Moreover, the convergence analysis encompasses discretization by finite element methods, so the proposed algorithm can be implemented and will produce a solution to the constrained minimization problem. All of these properties hold in an abstract framework that encompasses several interesting problems: the standard (scalar) elliptic BVP in divergence form, the system of isotropic elasticity, and others. Moreover, the analysis allows for the use of total variation regularization, so rapidly-varying or even discontinuous coefficients can be estimated.

An equation error approach for the elasticity imaging inverse problem for predicting tumor location

The primary objective of this work is to study the elasticity imaging inverse problem of identifying cancerous tumors in the human body. This nonlinear inverse problem not only represents an important and interesting application, it also brings forth noteworthy mathematical challenges since the underlying model is a system of elasticity equations involving incompressibility. Due to the locking effect, classical finite element methods are not effective for incompressible elasticity equations. Therefore, special treatment is necessary for both the direct and inverse problems. To study the inverse problem in an optimization framework, we propose an extension of the equation error approach. We focus on two cases, namely when the material parameter is sufficiently smooth and when it is may be discontinuous. For the latter case, we extend the total variation regularization method to the elasticity imaging inverse problem. We give the existence results for the proposed equation error approach and give the convergence analysis for the discretized problem. We give sufficient details on the discrete formulas as well as on the implementation issues. Numerical examples for smooth and discontinuous coefficients are given.

Optimal Signal Sets for Non-Gaussian Detectors

Identifying a maximally separated set of signals is important in the design of modems. The notion of optimality is dependent on the model chosen to describe noise in the measurements; while some analytic results can be derived under the assumption of Gaussian noise, no such techniques are known for choosing signal sets in the non-Gaussian case. To obtain numerical solutions for non-Gaussian detectors, minimax problems are transformed into nonlinear programs, resulting in a novel formulation yielding problems with relatively few variables and many inequality constraints. Using sequential quadratic programming, optimal signal sets are obtained for a variety of noise distributions.

Recovering Planar Lame Moduli from a Single-Traction Experiment

Under a simple nondegeneracy condition, the displacement and edge traction of a planar, isotropic, linearly elastic solid determine its Lame moduli. When these moduli are constant, they can be recovered exactly; this is demonstrated by a specific traction satisfying the nondegeneracy condition. Spatially varying moduli can be computed numerically by considering the equations of linear elasticity as a hyperbolic system for the unknown moduli. A stable finite difference scheme for solving this system is given; synthetic experiments demonstrate its efficacy.

The dual regularization approach to seismic velocity inversion

Differential semblance optimization (DSO) is a novel way of approaching a class of inverse problems arising in exploration seismology. The promising feature of the DSO method is that it replaces a non-smooth, highly non-convex cost function (the output least-squares (OLS) objective function) with a smooth cost function that is amenable to standard (local) optimization algorithms. The OLS problem can be written abstractly as a partially linear least-squares problem with linear constraints. The DSO objective function is derived from the associated quadratic penalty function. One way to view the DSO objective function is as a regularization of a function that is dual (in a certain sense) to the OLS objective function. Under suitable assumptions, the DSO method defines a parametrized path of minimizers converging to the desired solution and that, for certain values of the parameter, standard optimization techniques can be used to find a point on the path. The results of the theory are illustrated in the plane wave detection problem, a simple model problem for velocity inversion.

Stability and error estimates for an equation error method for elliptic equations

To estimate a parameter in an elliptic boundary value problem, the method of equation error chooses the value that minimizes the error in the PDE and boundary condition (the solution of the BVP having been replaced by a measurement). The estimated parameter converges to the exact value as the measured data converge to the exact value, provided Tikhonov regularization is used to control the instability inherent in the problem. The error in the estimated solution can be bounded in an appropriate quotient norm; estimates can be derived for both the underlying (infinite-dimensional) problem and a finite-element discretization that can be implemented in a practical algorithm. Numerical experiments demonstrate the efficacy and limitations of the method.

A Variational Method for Recovering Planar Lamé Moduli

For a planar, isotropic, linearly elastic square solid, the Lamé moduli are determined by the displacement under a known edge traction, assuming the displacement satisfies a particular non-degeneracy condition. Estimates of the moduli are derived by minimizing a functional and an error bound for the estimated moduli is given.

Efficient and automatic implementation of the adjoint state method

Combination of object-oriented programming with automatic differentiation techniques facilitates the solution of data fitting, control, and design problems driven by explicit time stepping schemes for initial-boundary value problems. The C++ class fdtd takes a complete specification of a single step, along with some associated code, and assembles from it a complete simulator, along with the linearized and adjoint simulations. The result is a (nonlinear) operator in the sense of the Hilbert Class Library (HCL), a C++ software package for optimization. The HCL operator so produced links directly with any of the HCL optimization algorithms. Moreover the performance of simulators constructed in this way is equivalent to that of optimized Fortran implementations.