The usual variational formulations of the Helmholtz equation are sign-indefinite (i.e. not coercive). In this talk, I will argue that this indefiniteness is not an inherent feature of the Helmholtz equation itself, only of its standard formulations. I will do this by presenting new sign-definite formulations of several Helmholtz boundary value problems.

This is joint work with Andrea Moiola (Reading).
 

Incomplete Cholesky factorizations are commonly used as black-box preconditioners for the iterative solution of large sparse symmetric positive definite linear systems. Traditionally, incomplete 
factorizations are obtained by dropping (i.e., replacing by zero) some entries of the factors during the factorization process. Here we consider a less common way to approximate the factors : through low-rank approximations of some off-diagonal blocks. We focus more specifically on approximation schemes that satisfy the orthogonality condition: the approximation should be orthogonal to the corresponding approximation error.

The resulting incomplete Cholesky factorizations have attractive theoretical properties. First, the underlying factorization process can be shown breakdown-free. Further, the condition number of the 
preconditioned system, that characterizes the convergence rate of standard iterative schemes, can be shown bounded as a function of the accuracy of individual approximations. Hence, such a bound can benefit from better approximations, but also from some algorithmic peculiarities. Eventually, the above results can be shown to hold for any symmetric positive definite system matrix.

On the practical side, we consider a particular variant of the preconditioner. It relies on a nested dissection ordering of unknowns to  insure an attractive memory usage and operations count. Further, it exploits in an algebraic way the low-rank structure present in system matrices that arise from PDE discretizations. A preliminary implementation of the method is compared with similar Cholesky and 
incomplete Cholesky factorizations based on dropping of individual entries.

The Dynamic Dictionary of Mathematical Functions (or DDMF, http://ddmf.msr-inria.inria.fr/) is an interactive website on special functions inspired by reference books such as the NIST Handbook of Special Functions. The originality of the DDMF is that each of its “chapters” is automatically generated from a short mathematical description of the corresponding function.

To make this possible, the DDMF focuses on so-called D-finite (or holonomic) functions, i.e., complex analytic solutions of linear ODEs with polynomial coefficients. D-finite functions include in particular most standard elementary functions (exp, log, sin, sinh, arctan...) as well as many of the classical special functions of mathematical physics (Airy functions, Bessel functions, hypergeometric functions...). A function of this class can be represented by a finite amount of data (a differential equation along with sufficiently many initial values), 
and this representation makes it possible to develop a computer algebra framework that deals with the whole class in a unified way, instead of ad hoc algorithms and code for each particular function. The DDMF attempts to put this idea into practice.

In this talk, I will present the DDMF, some of the algorithms and software libraries behind it, and ongoing projects based on similar ideas, with an emphasis on symbolic-numeric algorithms.

The coefficients in mathematical models of physical processes are often impossible to determine fully or accurately, and are hence subject to uncertainty. It is of great importance to quantify the uncertainty in the model outputs based on the (uncertain) information that is available on the model inputs. This invariably leads to very high dimensional quadrature problems associated with the computation of statistics of quantities of interest, such as the time it takes a pollutant plume in an uncertain subsurface flow problem to reach the boundary of a safety region or the buckling load of an airplane wing. Higher order methods, such as stochastic Galerkin or polynomial chaos methods, suffer from the curse of dimensionality and when the physical models themselves are complex and computationally costly, they become prohibitively expensive in higher dimensions. Instead, some of the most promising approaches to quantify uncertainties in continuum models are based on Monte Carlo sampling and the “multigrid philosophy”. Multilevel Monte Carlo (MLMC) Methods have been introduced recently and successfully applied to many model problems, producing significant gains. In this talk I want to recall the classical MLMC method and then show how the gains can be improved further (significantly) by using quasi-Monte Carlo (QMC) sampling rules. More importantly the dimension independence and the improved gains can be justified rigorously for an important model problem in subsurface flow. To achieve uniform bounds, independent of the dimension, it is necessary to work in infinite dimensions and to study quadrature in sequence spaces. I will present the elements of this new theory for the case of lognormal random coefficients in a diffusion problem and support the theory with numerical experiments.

For many tomographic imaging problems there are explicit inversion formulas, and depending on the completeness of the data these are unstable to differing degrees. Increasingly we are solving tomographic problems as though they were any other linear inverse problem using numerical linear algebra. I will illustrate the use of numerical singular value decomposition to explore the (in)stability for various problems. I will also show how standard techniques from numerical linear algebra, such as conjugate gradient least squares, can be employed with systematic regularization compared with the ad hoc use of slowly convergent iterative methods more traditionally used in computed tomography. I will mainly illustrate the talk with examples from three dimensional x-ray tomography but I will also touch on tensor tomography problems.