Strictly feasible equation-based methods for mixed complementarity problems

Summary.   We introduce a new algorithm for the solution of the mixed complementarity problem (MCP) which has stronger properties than most existing methods. In fact, typical solution methods for the MCP either generate feasible iterates but have to solve relatively complicated subproblems (like quadratic programs or linear complementarity problems), or they have relatively simple subproblems (like linear systems of equations) but generate not necessarily feasible iterates. The method to be presented here combines the nice features of these two classes of methods: It has to solve only one linear system of equations (of reduced dimension) at each iteration, and it generates feasible (more precisely: strictly feasible) iterates. The new method has some nice global and local convergence properties. Some preliminary numerical results will also be given. Received August 26, 1999 / Revised version recived April 11, 2000 / Published online February 5, 2001     

An iterative two-step algorithm for linear complementarity problems

Summary. We propose an algorithm for the numerical solution of large-scale symmetric positive-definite linear complementarity problems. Each step of the algorithm combines an application of the successive overrelaxation method with projection (to determine an approximation of the optimal active set) with the preconditioned conjugate gradient method (to solve the reduced residual systems of linear equations). Convergence of the iterates to the solution is proved. In the experimental part we compare the efficiency of the algorithm with several other methods. As test example we consider the obstacle problem with different obstacles. For problems of dimension up to 24\,000 variables, the algorithm finds the solution in less then 7 iterations, where each iteration requires about 10 matrix-vector multiplications. Received July 14, 1993 / Revised version received February 1994     

Numerical validation of solutions of complementarity The nonlinear case

Summary. This paper proposes a validation method for solutions of nonlinear complementarity problems. The validation procedure performs a computational test. If the result of the test is positive, then it is guaranteed that a given multi-dimensional interval either includes a solution or excludes all solutions of the nonlinear complementarity problem. Received September 22, 2000 / Revised version received April 11, 2001 / Published online October 17, 2001     

A Cholesky dual method for proximal piecewise linear programming

Summary. A quadratic programming method is given for minimizing a sum of piecewise linear functions and a proximal quadratic term, subject to simple bounds on variables. It may be used for search direction finding in nondifferentiable optimization algorithms. An efficient implementation is described that updates a Cholesky factorization of active constraints and provides good accuracy via iterative refinement. Numerical experience is reported for some large problems. Received March 29, 1993 / Revised version received December 18, 1993     

An operator splitting method for nonlinear convection-diffusion equations

Summary. We present a semi-discrete method for constructing approximate solutions to the initial value problem for the -dimensional convection-diffusion equation . The method is based on the use of operator splitting to isolate the convection part and the diffusion part of the equation. In the case , dimensional splitting is used to reduce the -dimensional convection problem to a series of one-dimensional problems. We show that the method produces a compact sequence of approximate solutions which converges to the exact solution. Finally, a fully discrete method is analyzed, and demonstrated in the case of one and two space dimensions. ReceivedFebruary 1, 1996 / Revised version received June 24, 1996     

Numerical validation of solutions of linear complementarity problems

Summary. This paper proposes a validation method for solutions of linear complementarity problems. The validation procedure consists of two sufficient conditions that can be tested on a digital computer. If the first condition is satisfied then a given multidimensional interval centered at an approximate solution of the problem is guaranteed to contain an exact solution. If the second condition is satisfied then the multidimensional interval is guaranteed to contain no exact solution. This study is based on the mean value theorem for absolutely continuous functions and the reformulation of linear complementarity problems as nonsmooth nonlinear systems of equations. Received August 21, 1997 / Revised version July 2, 1998     

Improved smoothing-type methods for the solution of linear programs

Summary. We consider a smoothing-type method for the solution of linear programs. Its main idea is to reformulate the primal-dual optimality conditions as a nonlinear and nonsmooth system of equations, and to apply a Newton-type method to a smooth approximation of this nonsmooth system. The method presented here is a predictor-corrector method, and is closely related to some methods recently proposed by Burke and Xu on the one hand, and by the authors on the other hand. However, here we state stronger global and/or local convergence properties. Moreover, we present quite promising numerical results for the whole netlib test problem collection. Received August 9, 2000 / Revised version received September 28, 2000 / Published online June 7, 2001     

A new line-search step based on the Weierstrass \wp-function for minimizing a class of logarithmic barrier functions

Summary. We define the notion of self-concordance of order two for the restriction of a logarithmic barrier function to a given line. Based on this notion we prove an inner approximation of the domain of , as well as a lower bound of the distance from a point $t$ to the minimum of . These results provide the theoretical tools to develop a simple and efficient search step for finding the minimum of the barrier function along a given line. The new bound on the size of the line-search step is better than the optimal bound known for the case of a self-concordant function (of order one). We conclude with some numerical examples that illustrate the promise of the new line-search step. Received May 24, 1993 / Revised version received February 1994     

Semi-implicit finite volume scheme for solving nonlinear diffusion equations in image processing

Summary. We propose and prove a convergence of the semi-implicit finite volume approximation scheme for the numerical solution of the modified (in the sense of Catté, Lions, Morel and Coll) Perona–Malik nonlinear image selective smoothing equation (called anisotropic diffusion in the image processing). The proof is based on a-priori estimates and Kolmogorov's compactness theorem. The implementation aspects and computational results are discussed. Received January 7, 1999 / Revised version received May 31, 2000 / Published online March 20, 2001     

An acceleration theorem for the ρ-algorithm

Summary. The ρ-algorithm of Wynn is an excellent device for accelerating the convergence of some logarithmically convergent sequences. Until now a convergence theorem and an acceleration theorem for the ρ-algorithm have not been obtained. The purpose of this paper is to give an acceleration theorem for the ρ-algorithm. Moreover, it is proved that the ρ-algorithm cannot accelerate linear convergence. Numerical examples are given. Received October 20, 1994 / Revised version received July 2, 1995     

Analysis of the quasi-Laguerre method

The quasi-Laguerre's iteration formula, using first order logarithmic derivatives at two points, is derived for finding roots of polynomials. Three different derivations are presented, each revealing some different properties of the method. For polynomials with only real roots, the method is shown to be optimal, and the global and monotone convergence, as well as the non-overshooting property, of the method is justified. Different ways of forming quasi-Laguerre's iteration sequence are addressed. Local convergence of the method is proved for general polynomials that may have complex roots and the order of convergence is . Received June 30, 1996 / Revised version received August 12, 1996     

On the numerical analysis of non-convex variational problems

Summary. We discuss a numerical method for finding Young-measure-valued minimizers of non-convex variational problems. To have any hope of a convergence theorem, one must work in a setting where the minimizer is unique and minimizing sequences converge strongly. This paper has two main goals: (i) we specify a method for producing strongly-convergent minimizing sequences, despite the failure of strict convexity; and (ii) we show how uniqueness of the Young measure can be parlayed into a numerical convergence theorem. The treatment of (ii) is done in the setting of two model problems, one involving scalar valued functions and a multiwell energy, the other from micromagnetics. Received July 29, 1995     

Numerical solution of constant coefficient linear delay differential equations as abstract Cauchy problems

Summary. In this paper we present an approach for the numerical solution of delay differential equations where , and , different from the classical step-by-step method. We restate (1) as an abstract Cauchy problem and then we discretize it in a system of ordinary differential equations. The scheme of discretization is proved to be convergent. Moreover the asymptotic stability is investigated for two significant classes of asymptotically stable problems (1). Received May 4, 1998 / Revised version received January 25, 1999 / Published online November 17, 1999     

On an a posteriori parameter choice strategy for Tikhonov regularization of nonlinear ill-posed problems

Summary. In the study of the choice of the regularization parameter for Tikhonov regularization of nonlinear ill-posed problems, Scherzer, Engl and Kunisch proposed an a posteriori strategy in 1993. To prove the optimality of the strategy, they imposed many very restrictive conditions on the problem under consideration. Their results are difficult to apply to concrete problems since one can not make sure whether their assumptions are valid. In this paper we give a further study on this strategy, and show that Tikhonov regularization is order optimal for each with the regularization parameter chosen according to this strategy under some simple and easy-checking assumptions. This paper weakens the conditions needed in the existing results, and provides a theoretical guidance to numerical experiments. Received August 8, 1997 / Revised version received January 26, 1998     

A numerical approach to variational problems subject to convexity constraint

Summary. We describe an algorithm to approximate the minimizer of an elliptic functional in the form on the set of convex functions u in an appropriate functional space X. Such problems arise for instance in mathematical economics [4]. A special case gives the convex envelope of a given function . Let be any quasiuniform sequence of meshes whose diameter goes to zero, and the corresponding affine interpolation operators. We prove that the minimizer over is the limit of the sequence , where minimizes the functional over . We give an implementable characterization of . Then the finite dimensional problem turns out to be a minimization problem with linear constraints. Received November 24, 1999 / Published online October 16, 2000     

Analysis of a coupled finite-infinite element method for exterior Helmholtz problems

Summary. This analysis of convergence of a coupled FEM-IEM is based on our previous work on the FEM and the IEM for exterior Helmholtz problems. The key idea is to represent both the exact and the numerical solution by the Dirichlet-to-Neumann operators that they induce on the coupling hypersurface in the exterior of an obstacle. The investigation of convergence can then be related to a spectral analysis of these DtN operators. We give a general outline of our method and then proceed to a detailed investigation of the case that the coupling surface is a sphere. Our main goal is to explore the convergence mechanism. In this context, we show well-posedness of both the continuous and the discrete models. We further show that the discrete inf-sup constants have a positive lower bound that does not depend on the number of DOF of the IEM. The proofs are based on lemmas on the spectra of the continuous and the discrete DtN operators, where the spectral characterization of the discrete DtN operator is given as a conjecture from numerical experiments. In our convergence analysis, we show algebraic (in terms of N) convergence of arbitrary order and generalize this result to exponential convergence. Received April 10, 1999 / Revised version received November 10, 1999 / Published online October 16, 2000     

The appearance of microstructures in problems with incompatible wells and their numerical approach

Summary. The goal of the paper is to analyze the creation of microstructure in problems of Calculus of Variations with wells. More precisely we consider a case with strong incompatibility between the wells. This forces the minimizing sequences to use other gradients than the wells in a puzzling way. Using a method we are then able to single out discrete minimizing sequences and to give energy estimates in terms of the mesh size. Received December 23, 1997 / Revised version received September 7, 1998 / Published online: June 29, 1999     

The rate of convergence of Toeplitz based PCG methods for second order nonlinear boundary value problems

Summary. In previous works [21–23] we proposed the use of [5] and band Toeplitz based preconditioners for the solution of 1D and 2D boundary value problems (BVP) by means of the preconditioned conjugate gradient (PCG) methods. As and band Toeplitz linear systems can be solved [4] by using fast sine transforms [8], these methods become especially attractive in a parallel environment of computation. In this paper we extend this technique to the nonlinear, nonsymmetric case and, in addition, we prove some clustering properties for the spectra of the preconditioned matrices showing why these methods exhibit a convergence speed which results to be more than linear. Therefore these methods work much finer than those based on separable preconditioners [18,45], on incomplete LU factorizations [36,13,27], and on circulant preconditioners [9,30,35] since the latter two techniques do not assure a linear rate of convergence. On the other hand, the proposed technique has a wider range of application since it can be naturally used for nonlinear, nonsymmetric problems and for BVP in which the coefficients of the differential operator are not strictly positive and only piecewise smooth. Finally the several numerical experiments performed here and in [22,23] confirm the effectiveness of the theoretical analysis. Received December 19, 1995 / Revised version received September 15, 1997     

New properties of a nonlinear conjugate gradient method

Summary. This paper provides several new properties of the nonlinear conjugate gradient method in [5]. Firstly, the method is proved to have a certain self-adjusting property that is independent of the line search and the function convexity. Secondly, under mild assumptions on the objective function, the method is shown to be globally convergent with a variety of line searches. Thirdly, we find that instead of the negative gradient direction, the search direction defined by the nonlinear conjugate gradient method in [5] can be used to restart any optimization method while guaranteeing the global convergence of the method. Some numerical results are also presented. Received March 12, 1999 / Revised version received April 25, 2000 / Published online February 5, 2001     

Shape optimization by the homogenization method

Summary. In the context of shape optimization, we seek minimizers of the sum of the elastic compliance and of the weight of a solid structure under specified loading. This problem is known not to be well-posed, and a relaxed formulation is introduced. Its effect is to allow for microperforated composites as admissible designs. In a two-dimensional setting the relaxed formulation was obtained in [6] with the help of the theory of homogenization and optimal bounds for composite materials. We generalize the result to the three dimensional case. Our contribution is twofold; first, we prove a relaxation theorem, valid in any dimensions; secondly, we introduce a new numerical algorithm for computing optimal designs, complemented with a penalization technique which permits to remove composite designs in the final shape. Since it places no assumption on the number of holes cut within the domain, it can be seen as a topology optimization algorithm. Numerical results are presented for various two and three dimensional problems. Received July 4, 1995