Computer Assisted Proofs in Solving Linear Parametric Problems
Consider a linear system A(p)x b(p) whose input data depend on a number of uncertain parameters p (p1,,pk) varying within given intervals [p] The objective is to verify by numerical computations monotonic (and convexity/concavity) dependence of a solution component xi(p) with respect to a parameter pj over the interval box [p] or more general, to prove if some boundary inf sup xi(p) for all p isin [p] is attained at the end points of [p] Such knowledge is useful in many applications in order to facilitate the solution of some underlying linear parametric problem involving uncertainties. In this paper we present a technique, for proving the desired properties of the parametric solution, which is alternative to the approaches based on extreme point computations. The proposed computer aided proof is based on guaranteed interval enclosures for the partial derivatives of the parametric solution for all p isin [p] The availability of self validated methods providing guaranteed enclosure of a parametric solution set by floating point computations is a key for the efficiency and the expanded scope of applicability of the proposed approach. Linear systems involving nonlinear parameter dependencies, and dependencies between A(p) and b(p) as well as non square linear parametric systems can be handled successfully. Presented are details of the algorithm design and mathematica tools implementing the proposed approach. Numerical examples from structural mechanics illustrate its application.
12th GAMM IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN 2006)