Numerical analysis of elliptic inverse problems with interior data

A number of algorithms have been proposed and analyzed for estimating a coefficient in an elliptic boundary value problem when interior data is available. Most of the analysis has been done for the simple scalar BVP -Δ a Δ u = f in Ω, a (∂ u / ∂ n) g on ∂ Ω However, some methods and the associated analysis have been extended to the problem of estimating the Lamé moduli in the system of linear, isotropic elasticity. Under certain idealized conditions, convergence of estimates to the exact Lame moduli has been proved for two techniques, the output least-squares method and a variational method similar to the equation error approach

Generalizing the GSVD

The generalized singular value decomposition (GSVD) of a pair of matrices is the natural tool for certain problems defined on Euclidean space, such as certain weighted least-squares problems, the result of applying Tikhonov regularization to such problems (sometimes called regularization with seminorms), and equality-constrained least-squares problems. There is an extension of the GSVD to pairs of bounded linear operators defined on Hilbert space that turns out to be a natural representation for analyzing the same problems in the infinite-dimensional setting

On the convergence of a heuristic parameter choice rule for Tikhonov regularization

Multiplicative regularization solves a linear inverse problem by minimizing the product of the norm of the data misfit and the norm of the solution. This technique is related to Tikhonov regularization with the parameter chosen to make the data misfit and regularization terms (of the Tikhonov objective function) equal. This suggests a heuristic parameter choice method, equivalent to the rule previously proposed by Reginska. Reginska\u27s rule is well defined provided the data is sufficiently close to exact data and does not lie in the range of the operator. If a sufficiently large portion of the data error lies outside the range of the operator, then the solution defined by Reginska\u27s rule converges weakly to the exact solution as the data error converges to zero. The regularization parameter converges to zero like the square of the norm of the data noise, leading to under-regularization for small noise levels. Nevertheless, the method performs well on a suite of test problems, as shown by comparison with the L-curve, generalized cross-validation, quasi-optimality, and Hanke--Raus parameter choice methods. A modification of the approach yields a heuristic parameter choice rule that is provably convergent (in the norm topology) under the restrictions on the data error described above, as long as the exact solution has a small amount of additional smoothness. On the test problems considered here, the modified rule outperforms all of the above heuristic methods, although it is only slightly better than the quasi-optimality rule

Approximating the generalized singular value expansion

The generalized singular value expansion (GSVE) simultaneously diagonalizes a pair of operators on Hilbert space. From a theoretical point of view, the GSVE enables a straightforward analysis of, for example, weighted least-squares problems and the method of Tikhonov regularization with seminorms. When the operators are discretized, an approximate GSVE can be computed from the generalized singular value decomposition of a pair of Galerkin matrices. Unless the discretization is carefully chosen, spurious modes can appear, but a natural condition on the discretization guarantees convergence of the approximate GSVE to the exact one. Numerical examples illustrate the pitfalls of a poor discretization and efficacy of the convergence conditions

Article the singular value expansion for arbitrary bounded linear operators

The singular value decomposition (SVD) is a basic tool for analyzing matrices. Regarding a general matrix as defining a linear operator and choosing appropriate orthonormal bases for the domain and co-domain allows the operator to be represented as multiplication by a diagonal matrix. It is well known that the SVD extends naturally to a compact linear operator mapping one Hilbert space to another; the resulting representation is known as the singular value expansion (SVE). It is less well known that a general bounded linear operator defined on Hilbert spaces also has a singular value expansion. This SVE allows a simple analysis of a variety of questions about the operator, such as whether it defines a well-posed linear operator equation and how to regularize the equation when it is not well posed

An Infeasible Point Method for Minimizing the Lennard-Jones Potential

Minimizing the Lennard-Jones potential, the most-studied modelproblem for molecular conformation, is an unconstrained globaloptimization problem with a large number of local minima. In thispaper, the problem is reformulated as an equality constrainednonlinear programming problem with only linear constraints. Thisformulation allows the solution to approached through infeasibleconfigurations, increasing the basin of attraction of the globalsolution. In this way the likelihood of finding a global minimizeris increased. An algorithm for solving this nonlinear program isdiscussed, and results of numerical tests are presented.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/44788/1/10589_2004_Article_140555.pd

Finite-dimensional linear algebra

Linear algebra forms the basis for much of modern mathematics-theoretical, applied, and computational. Finite-Dimensional Linear Algebra provides a solid foundation for the study of advanced mathematics and discusses applications of linear algebra to such diverse areas as combinatorics, differential equations, optimization, and approximation

The output least-squares approach to estimating Lamé moduli

The Lamé moduli of a heterogeneous, isotropic, planar membrane can be estimated by observing the displacement of the membrane under a known edge traction, and choosing estimates of the moduli that best predict the observed displacement under a finite-element simulation. This algorithm converges to the exact moduli given pointwise measurements of the displacement on an increasingly fine mesh. The error estimates that prove this convergence also show the instability of the inverse problem

Partial differential equations: Analytical and numerical methods, Second edition

Partial differential equations (PDEs) are essential for modeling many physical phenomena. This undergraduate textbook introduces students to the topic with a unique approach that emphasizes the modern finite element method alongside the classical method of Fourier analysis.Additional features of this new edition include broader coverage of PDE methods and applications, with new chapters on the method of characteristics, Sturm–Liouville problems, and Green\u27s functions, and a new section on the finite difference method for the wave equation. The author continues to emphasize Fourier series and finite element methods, which were the primary scope of the first edition.The book also features emphasis on linear algebra, particularly the idea of best approximation; realistic physical parameters and meaningful experiments for many of the examples and exercises; and tutorials for the most popular software (MATLAB™, Mathematica™, and Maple™) that can be used to reproduce the examples and solve the exercises

Understanding and implementing the finite element method

The finite element method is the most powerful general-purpose technique for computing accurate solutions to partial differential equations. Understanding and Implementing the Finite Element Method is essential reading for those interested in understanding both the theory and the implementation of the finite element method for equilibrium problems. This book contains a thorough derivation of the finite element equations as well as sections on programming the necessary calculations, solving the finite element equations, and using a posteriori error estimates to produce validated solutions. Accessible introductions to advanced topics, such as multigrid solvers, the hierarchical basis conjugate gradient method, and adaptive mesh generation, are provided. Each chapter ends with exercises to help readers master these topics. Understanding and Implementing the Finite Element Method includes a carefully documented collection of MATLAB programs implementing the ideas presented in the book. Readers will benefit from a careful explanation of data structures and specific coding strategies and will learn how to write a finite element code from scratch. Students can use the MATLAB codes to experiment with the method and extend them in various ways to learn more about programming finite elements