From a Nonlinear, Nonconvex Variational Problem to a Linear, Convex Formulation

   Abstract. We propose a general approach to deal with nonlinear, nonconvex variational problems based on a reformulation of the problem resulting in an optimization problem with linear cost functional and convex constraints. As a first step we explicitly explore these ideas to some one-dimensional variational problems and obtain specific conclusions of an analytical and numerical nature.     

From a Nonlinear, Nonconvex Variational Problem to a Linear, Convex Formulation

Abstract. We propose a general approach to deal with nonlinear, nonconvex variational problems based on a reformulation of the problem resulting in an optimization problem with linear cost functional and convex constraints. As a first step we explicitly explore these ideas to some one-dimensional variational problems and obtain specific conclusions of an analytical and numerical nature.     

On Cayley-Transform Methods for the Discretization of Lie-Group Equations

In this paper we develop in a systematic manner the theory of time-stepping methods based on the Cayley transform. Such methods can be applied to discretize differential equations that evolve in some Lie groups, in particular in the orthogonal group and the symplectic group. Unlike many other Lie-group solvers, they do not require the evaluation of matrix exponentials. Similarly to the theory of Magnus expansions in [13], we identify terms in a Cayley expansion with rooted trees, which can be constructed recursively. Each such term is an integral over a polytope but all such integrals can be evaluated to high order by using special quadrature formulas similar to the construction in [13]. Truncated Cayley expansions (with exact integrals) need not be time-symmetric, hence the method does not display the usual advantages associated with time symmetry, e.g., even order of approximation. However, time symmetry (with its attendant benefits) is attained when exact integrals are replaced by certain quadrature formulas. March 7, 2000. Final version received: August 10, 2000. Online publication: January 2, 2001.     

Asymptotic Behavior of Sobolev-Type Orthogonal Polynomials on a Rectifiable Jordan Curve or Arc

Our object of study is the asymptotic behavior of the sequence of polynomials orthogonal with respect to the discrete Sobolev inner product <formula> \langle f, g \rangle = ∈t_{E} f(ξ) \overline{g(ξ)} ρ(ξ) |d ξ|+ f(Z) A g(Z)^H, </formula> where E is a rectifiable Jordan curve or arc in the complex plane f(Z) = (f(z_1), \ldots, f^{(l_1)}(z_1) , \ldots , f(z_m) , \ldots ,f^{(l_m)}(z_m)), A is an M \times M Hermitian matrix, M l ₁ + ⋅s + l _m + m , |d ξ| denotes the arc length measure, ρ is a nonnegative function on E , and z _i ∈Ω, i=1,2,\ldots,m , where Ω is the exterior region to E . July 23, 1999. Dates revised: September 11, 2000 and February 16, 2001. Date accepted: February 26, 2001.     

Explicit Solutions of Nonconvex Variational Problems in Dimension One

We propose a method of finding the generalized solutions of nonconvex variational problems by solving an appropriate differential inclusion that is motivated by necessary conditions of optimality for such generalized minimizers. Accepted 28 September 1998     

Discrete Newton methods and iterated defect corrections

Summary In this paper we present a general theory for discrete Newton methods, iterated defect corrections via neighbouring problems and deferred corrections based on asymptotic expansions of the discretization error.Dedicated to Professor Dr. J. Weisinger on the occasion of his sixty-fifth birthday     

A one-dimensional variational problem with gradient constraint

In this note we give an explicit characterization of the minimum value of one-dimensional integral variational problem with gradient constraint by a positive measurable function.     

On theoretical and numerical aspects of the bang-bang-principle

Summary In the first part of this paper we are dealing with theoretical statements and conditions which finally lead to bang-bang-principles. A careful analysis of these theorems is used for the development of a numerical method. This method consists of two stages: During the first iterations the number and approximate location of the switching points of the optimal control are determined. In the second phase a rapidly convergent algorithm determines the exact location. We apply this method successfully to a parabolic boundary control problem and give an extensive discussion of numerical results.The work of the second author on this paper was partially done during his stay at North Carolina State University, Graduate Program in Operations Research and Department of Mathematics, Raleigh, USA     

Regularization by Functions of Bounded Variation and Applications to Image Enhancement

Optimization problems regularized by bounded variation seminorms are analyzed. The optimality system is obtained and finite-dimensional approximations of bounded variation function spaces as well as of the optimization problems are studied. It is demonstrated that the choice of the vector norm in the definition of the bounded variation seminorm is of special importance for approximating subspaces consisting of piecewise constant functions. Algorithms based on a primal—dual framework that exploit the structure of these nondifferentiable optimization problems are proposed. Numerical examples are given for denoising of blocky images with very high noise. Accepted 29 March 1998     

Uniform Asymptotic Expansions of Symmetric Elliptic Integrals

Symmetric standard elliptic integrals are considered when two or more parameters are larger than the others. The distributional approach is used to derive seven expansions of these integrals in inverse powers of the asymptotic parameters. Some of these expansions also involve logarithmic terms in the asymptotic variables. These expansions are uniformly convergent when the asymptotic parameters are greater than the remaining ones. The coefficients of six of these expansions involve hypergeometric functions with less parameters than the original integrals. The coefficients of the seventh expansion again involve elliptic integrals, but with less parameters than the original integrals. The convergence speed of any of these expansions increases for an increasing difference between the asymptotic variables and the remaining ones. All the expansions are accompanied by an error bound at any order of the approximation. January 31, 2000. Date revised: May 18, 2000. Date accepted: August 4, 2000.     

Approximation of the eigenvalues of a fourth order differential equation with non-smooth coefficients

The eigenvalues of a fourth order, generalized eigenvalue problem in one dimension, with non-smooth coefficients are approximated by a finite element method, introduced in an earlier work by the author and A. Lutoborski, in the context of a similar source problem with non-smooth coefficients. Error estimates for the approximate eigenvalues and eigenvectors are obtained, showing a better performance of this method, when applied to eigenvalue approximation, compared to a standard finite element method with arbitrary mesh.     

Sets of convergence and stability regions

This paper continues the authors' study of the convergence of dynamic iteration methods for large systems of linear initial value problems. We ask for convergence on [0, ) and show how the convergence can be reduced to a graphical test relating the splitting of the matrix to the stability properties of the discretization method.     

On a Conjecture of E. A. Rakhmanov

It is shown that a conjecture of E. A. Rakhmanov is true concerning the zero distribution of orthogonal polynomials with respect to a measure having a discrete real support. We also discuss the case of extremal polynomials with respect to some discrete L _p -norm, 0 < p ≤∈fty , and give an extension to complex supports. Furthermore, we present properties of weighted Fekete points with respect to discrete complex sets, such as the weighted discrete transfinite diameter and a weighted discrete Bernstein—Walsh-like inequality. August 24, 1998. Date revised: March 26, 1999. Date accepted: April 27, 1999.     

The Asymptotic Behavior of Conformal Modules of Quadrilaterals with Applications to the Estimation of Resistance Values

We consider the conformal mapping of ``strip-like' domains and derive a number of asymptotic results for computing the conformal modules of an associated class of quadrilaterals. These results are then used for the following two purposes: (a) to estimate the error of certain engineering formulas for measuring resistance values of integrated circuit networks; and (b) to compute the modules of complicated quadrilaterals of the type that occur frequently in engineering applications. April 17, 1997. Date revised: September 10, 1997.     

Exponents of a class of two-colored digraphs

A two-colored digraph D is primitive if there exist nonnegative integers h and k with h+k>0 such that for each pair (i, j) of vertices there exists an (h, k)-walk in D from i to j. The exponent of the primitive two-colored digraph D is the minimum value of h+k taken over all such h and k. In this article, we consider special primitive two-colored digraphs whose uncolored digraph has n+s vertices and consist of one n-cycle and one (n???2)-cycle. We give the bounds on the exponents, and the characterizations of the extremal two-colored digraphs.     

Nonlinear Smoothing and the EM Algorithm for Positive Integral Equations of the First Kind

We study a modification of the EMS algorithm in which each step of the EMS algorithm is preceded by a nonlinear smoothing step of the form , where S is the smoothing operator of the EMS algorithm. In the context of positive integral equations (à la positron emission tomography) the resulting algorithm is related to a convex minimization problem which always admits a unique smooth solution, in contrast to the unmodified maximum likelihood setup. The new algorithm has slightly stronger monotonicity properties than the original EM algorithm. This suggests that the modified EMS algorithm is actually an EM algorithm for the modified problem. The existence of a smooth solution to the modified maximum likelihood problem and the monotonicity together imply the strong convergence of the new algorithm. We also present some simulation results for the integral equation of stereology, which suggests that the new algorithm behaves roughly like the EMS algorithm. Accepted 1 April 1997     

Nonnegative Linearization for Polynomials Orthogonal with Respect to Discrete Measures

We give conditions for the coefficients in three term recurrence relations implying nonnegative linearization for polynomials orthogonal with respect to measures supported on convergent sequences of points. The previous methods were unable to cover this case. January 14, 1999. Date revised: January 31, 2000. Date accepted: October 24, 2000.     

Asymptotic behaviour for a nonlocal diffusion equation on a lattice

In this paper we study the asymptotic behaviour as t → ∞ of solutions to a nonlocal diffusion problem on a lattice, namely, with t ≥ 0 and . We assume that J is nonnegative and verifies . We find that solutions decay to zero as t → ∞ and prove an optimal decay rate using, as our main tool, the discrete Fourier transform.      

Asymptotic bounds on the errors of one-step methods

Summary Bulirsch and Stoer have shown how to construct asymptotic upper and lower bounds on the true (global) errors resulting from the solution by extrapolation of the initial value problem for a system of ordinary differential equations. It is shown here how to do this for any one-step method endowed with an asymptotically correct local error estimator. The one-step method can be changed at every step.This work performed at Sandia National Laboratories supported by the U.S. Department of Energy under contract number DE-AC04-76DP00789