Can we get rigorous answers when computing with real and complex numbers? There are now many applications where this is possible thanks to a combination of tools from computer algebra and traditional numerical computing. I will give an overview of such methods in the context of two projects I'm developing. The first project, Arb, is a library for arbitrary-precision ball arithmetic, a form of interval arithmetic enabling numerical computations with rigorous error bounds. The second project, Fungrim, is a database of knowledge about mathematical functions represented in symbolic form. It is intended to function both as a traditional reference work and as a software library to support symbolic-numeric methods for problems involving transcendental functions. I will explain a few central algorithmic ideas and explain the research goals of these projects.

# Computational Mathematics and Applications Seminar

Please note that the list below only shows forthcoming events, which may not include regular events that have not yet been entered for the forthcoming term. Please see the past events page for a list of all seminar series that the department has on offer.

The development of effective solvers for high frequency wave propagation problems, such as those described by the Helmholtz equation, presents significant challenges. One promising class of solvers for such problems are parallel domain decomposition methods, however, an appropriate coarse space is typically required in order to obtain robust behaviour (scalable with respect to the number of domains, weakly dependant on the wave number but also on the heterogeneity of the physical parameters). In this talk we introduce a coarse space based on generalised eigenproblems in the overlap (GenEO) for the Helmholtz equation. Numerical results within FreeFEM demonstrate convergence that is effectively independent of the wave number and contrast in the heterogeneous coefficient as well as good performance for minimal overlap.

Domain decomposition methods are widely employed for the numerical solution of partial differential equations on parallel computers. We develop an adjoint-based a posteriori error analysis for overlapping multiplicative Schwarz domain decomposition and for overlapping additive Schwarz. In both cases the numerical error in a user-specified functional of the solution (quantity of interest), is decomposed into a component that arises due to the spatial discretization and a component that results from of the finite iteration between the subdomains. The spatial discretization error can be further decomposed in to the errors arising on each subdomain. This decomposition of the total error can then be used as part of a two-stage approach to construct a solution strategy that efficiently reduces the error in the quantity of interest.

The fully coupled numerical simulation of different physical processes, which can typically occur

at variable time and space scales, is often a very challenging task. A common feature of such models is that

their discretization gives rise to systems of linearized equations with an inherent block structure, which

reflects the properties of the set of governing PDEs. The efficient solution of a sequence of systems with

matrices in a block form is usually one of the most time- and memory-demanding issue in a coupled simulation.

This effort can be carried out by using either iteratively coupled schemes or monolithic approaches, which

tackle the problem of the system solution as a whole.

This talk aims to discuss recent advances in the monolithic solution of coupled multi-physics problems, with

application to poromechanical simulations in fractured porous media. The problem is addressed either by proper

sparse approximations of the Schur complements or special splittings that can partially uncouple the variables

related to different physical processes. The selected approaches can be included in a more general preconditioning

framework that can help accelerate the convergence of Krylov subspace solvers. The generalized preconditioner

relies on approximately decoupling the different processes, so as to address each single-physics problem

independently of the others. The objective is to provide an algebraic framework that can be employed as a

general ``black-box'' tool and can be regarded as a common starting point to be later specialized for the

particular multi-physics problem at hand.

Numerical experiments, taken from real-world examples of poromechanical problems and fractured media, are used to

investigate the behaviour and the performance of the proposed strategies.

Everybody is familiar with the concept of eigenvalues of a matrix. In this talk, we consider the nonlinear eigenvalue problem. These are problems for which the eigenvalue parameter appears in a nonlinear way in the equation. In physics, the Schroedinger equation for determining the bound states in a semiconductor device, introduces terms with square roots of different shifts of the eigenvalue. In mechanical and civil engineering, new materials often have nonlinear damping properties. For the vibration analysis of such materials, this leads to nonlinear functions of the eigenvalue in the system matrix.

One particular example is the sandwhich beam problem, where a layer of damping material is sandwhiched between two layers of steel. Another example is the stability analysis of the Helmholtz equation with a noise excitation produced by burners in a combustion chamber. The burners lead to a boundary condition with delay terms (exponentials of the eigenvalue).

We often receive the question: “How can we solve a nonlinear eigenvalue problem?” This talk explains the different steps to be taken for using Krylov methods. The general approach works as follows: 1) approximate the nonlinearity by a rational function; 2) rewrite this rational eigenvalue problem as a linear eigenvalue problem and then 3) solve this by a Krylov method. We explain each of the three steps.

I will present an analysis of a continuous version of the compressed sensing problem, where the l^1 norm is replaced by the total variation of measures, and one aims to recover the positions and amplitudes of Dirac masses. We show that provided that the Diracs are sufficiently separated under a Fisher metric (which accounts for the geometry of the problem), stable recovery can be achieved when the number of random samples scales linearly with sparsity (up to log factors). This is joint work with Nicolas Keriven and Gabriel Peyre.