|
Case Western Reserve University DEPARTMENT OF MATHEMATICS |
|
HOME | APPLY | GIVE | VISIT | DIRECTORIES | REQUEST INFO | SEARCH |
DEPARTMENT OF MATHEMATICS COLLOQUIUMS |
|
ColloquiumProfessor Clyde F. MARTIN
Professor Daniel SZYLD ABSTRACT We study the use of inexact and truncated Krylov subspace methods for the solution of the linear systems arising in the discretized solution of the optimal control of a parabolic partial differential equation. An all-at-once temporal discretization and a reduction approach are used to obtain a symmetric positive definite system for the control variables only, where a Conjugate Gradient (CG) method can be used at the cost of the solution of two very large linear systems in each iteration. We propose to use inexact Krylov subspace methods, in which the solution of the two large linear systems are not solved exactly, and their approximate solutions can be progressively less exact. The option we propose is the use of the parareal-in-time algorithm for approximating the solution of these two linear systems. The use of less parareal iterations makes it possible to reduce the time integration costs and to improve the time parallel scalability, and therefore, making it possible to really consider otimization in real time. We also show that truncated methods could be used without much delay in convergence, but with important savings in storage. Spectral bounds are provided and numerical experiments with inexact versions of CG, the full orthogonalization method (FOM), and of truncated FOM, are presented, illustrating the potential of the proposed methods.
Professor Matthias BECK ABSTRACT A common theme of enumerative combinatorics is formed by counting functions that are polynomials. For example, one proves in any introductory graph theory course that the number of proper k-colorings of a given graph G is a polynomial in k, the chromatic polynomial of G. Combinatorics is abundant with polynomials that count something when evaluated at positive integers, and many of these polynomials have a (completely different) interpretation when evaluated at negativeintegers: these instances go by the name of combinatorial reciprocity theorems. For example, when we evaluate the chromatic polynomial of G at -1, weobtain (up to a sign) the number of acyclic orientations of G, that is, those orientations of G that do not contain a coherently oriented cycle. Combinatorial reciprocity theorems appear all over combinatorics. This talk will attempt to show some of the charm (and usefulness!) these theorems exhibit. Our goal is to weave a unifying thread through various combinatorial reciprocity theorems, by looking at them through
Professor Manish PATNAIK
Professor Tom LEINSTER
Professor AlessandoVENEZIANI
Case Western Reserve University |
|
|