Wave Equation Numerical Resolution: a Comprehensive Mechanized Proof of a C Program

We formally prove correct a C program that implements a numerical scheme for the resolution of the one-dimensional acoustic wave equation. Such an implementation introduces errors at several levels: the numerical scheme introduces method errors, and floating-point computations lead to round-off errors. We annotate this C program to specify both method error and round-off error. We use Frama-C to generate theorems that guarantee the soundness of the code. We discharge these theorems using SMT solvers, Gappa, and Coq. This involves a large Coq development to prove the adequacy of the C program to the numerical scheme and to bound errors. To our knowledge, this is the first time such a numerical analysis program is fully machine-checked.Comment: No. RR-7826 (2011

Nonlinear partial differential equations: sequential and weak solutions

Nonlinear partial differential equation

Generalized solutions of nonlinear partial differential equations

During the last few years, several fairly systematic nonlinear theories of generalized solutions of rather arbitrary nonlinear partial differential equations have emerged. The aim of this volume is to offer the reader a sufficiently detailed introduction to two of these recent nonlinear theories which have so far contributed most to the study of generalized solutions of nonlinear partial differential equations, bringing the reader to the level of ongoing research.The essence of the two nonlinear theories presented in this volume is the observation that much of the mathematics concerni

Abstract differential geometry, differential algebras of generalized functions, and de Rham cohomology

Abstr. differential geometry is a recent extension of classical differential geometry on smooth manifolds which, however, does no longer use any notion of Calculus. Instead of smooth functions, one starts with a sheaf of algebras, i.e., the structure sheaf, considered on an arbitrary topological space, which is the base space of all the sheaves subsequently involved. Further, one deals with a sequence of sheaves of modules, interrelated with appropriate ‘differentials’, i.e., suitable ‘Leibniz’ sheaf morphisms, which will constitute the ‘differential complex’. This abstract approach captures much of the essence of classical differential geometry, since it places a powerful apparatus at our disposal which can reproduce and, therefore, extend fundamental classical results. The aim of this paper is to give an indication of the extent to which this apparatus can go beyond the classical framework by including the lamest class of singularities dealt with so far. Thus, it is shown that, instead of the classical structure sheaf of algebras of smooth functions, one can start with a significantly larger, and nonsmooth, sheaf of so-called nowhere dense differential algebras of generalized functions. These latter algebras, which contain the Schwartz distributions, also provide global solutions for arbitrary analytic nonlinear PDEs. Moreover, unlike the distributions, and as a matter of physical interest, these algebras can deal with the vastly larger class of singularities which are concentrated on arbitrary closed, nowhere dense subsets and, hence, can have an arbitrary large positive Lebesgue measure. Within the abstract differential geometric context, it is shown that, starting with these nowhere dense differential algebras as a structure sheaf, one can recapture the exactness of the corresponding de Rham complex, and also obtain the short exponential sequence. These results are the two fundamental ingredients in developing differential geometry along classical, as well as abstract lines. Although the commutative framework is used here, one can easily deal with a class of singularities which is far larger than any other one dealt with so far, including in noncommutative theories

Solution of continuous nonlinear PDEs through order completion

This work inaugurates a new and general solution method for arbitrary continuous nonlinear PDEs. The solution method is based on Dedekind order completion of usual spaces of smooth functions defined on domains in Euclidean spaces. However, the nonlinear PDEs dealt with need not satisfy any kind of monotonicity properties. Moreover, the solution method is completely type independent. In other words, it does not assume anything about the nonlinear PDEs, except for the continuity of their left hand term, which includes the unkown function. Furthermore the right hand term of such nonlinear PDEs can in fact be given any discontinuous and measurable function

Nonlinear SPDEs: Colombeau solutions and pathwise limits

This paper is devoted to the semilinear stochastic wave equatio

A branch and bound algorithm for Choquet optimization in multicriteria problems

Summary. This paper is devoted to the search for Choquet-optimal solutions in multicriteria combinatorial optimization with application to spanning tree problems and knapsack problems. After recalling basic notions concerning the use of Choquet integrals for preference aggregation, we present a condition (named preference for interior points) that characterizes preferences favouring well-balanced solutions, a natural attitude in multicriteria optimization. When using a Choquet integral as preference model, this condition amounts to choosing a submodular (resp. supermodular) capacity when criteria have to be minimized (resp. maximized). Under this assumption, we investigate the determination of Choquet-optimal solutions in the multicriteria spanning tree problem and the multicriteria 0-1 knapsack problem. For both problems, we introduce a linear bound for the Choquet integral, computable in polynomial time, and propose a branch and bound procedure using this bound. We provide numerical experiments that show the actual efficiency of the algorithms on various instances of different sizes