Stability of the hp-Raviart-Thomas projection operator as a mapping H^1(K) -> H^1(K) on the unit cube K in R^3 has been addressed e.g. in [2], see also [1]. These results are suboptimal with respect to the polynomial degree. In this talk we present quasi-optimal stability estimates for the hp-Raviart-Thomas projection operator on the cube. The analysis involves elements of the polynomial approximation theory on an interval and the real method of Banach space interpolation.

(Joint work with Herbert Egger, TU Darmstadt)

[1] Mark Ainsworth and Katia Pinchedez. hp-approximation theory for BDFM and RT finite elements on quadrilaterals. SIAM J. Numer. Anal., 40(6):2047–2068 (electronic) (2003), 2002.

[2] Dominik Schötzau, Christoph Schwab, and Andrea Toselli. Mixed hp-DGFEM for incompressible flows. SIAM J. Numer. Anal., 40(6):2171–2194 (electronic) (2003), 2002.

# Past Computational Mathematics and Applications Seminar

Nonlinear systems of partial differential equations (PDEs) may permit several distinct solutions. The typical current approach to finding distinct solutions is to start Newton's method with many different initial guesses, hoping to find starting points that lie in different basins of attraction. In this talk, we present an infinite-dimensional deflation algorithm for systematically modifying the residual of a nonlinear PDE problem to eliminate known solutions from consideration. This enables the Newton--Kantorovitch iteration to converge to several different solutions, even starting from the same initial guess. The deflated Jacobian is dense, but an efficient preconditioning strategy is devised, and the number of Krylov iterations is observed not to grow as solutions are deflated. The technique is then applied to computing distinct solutions of nonlinear PDEs, tracing bifurcation diagrams, and to computing multiple local minima of PDE-constrained optimisation problems.

In this talk we will describe the steps towards self-consistent computer simulations of the evolution of the universe beginning soon after the Big Bang and ending with the formation of realistic stellar systems like the Milky Way. This is a multi-scale problem of vast proportions. The first step has been the Millennium Simulation, one of the largest and most successful numerical simulations of the Universe ever carried out. Now we are in the midst of the next step, where this is carried to a much higher level of physical fidelity on the latest supercomputing platforms. This talk will be illustrate how the role of mathematics is essential in this endeavor. Also computer simulations will be shown. This is joint work among others with Volker Springel.

Some problems in scientific computing, like the forward simulation of electromagnetic waves in geophysical prospecting, can be

solved via approximation of f(A)b, the action of a large matrix function f(A) onto a vector b. Iterative methods based on rational Krylov

spaces are powerful tools for these computations, and the choice of parameters in these methods is an active area of research.

We provide an overview of different approaches for obtaining optimal parameters, with an emphasis on the exponential and resolvent function, and the square root. We will discuss applications of the rational Arnoldi method for iteratively generating near-optimal absorbing boundary layers for indefinite Helmholtz problems, and for rational least squares vector fitting.

Gaussian quadrature rules are of theoretical and practical interest because of their role in numerical integration and interpolation. For general weighting functions, their computation can be performed with the Golub-Welsch algorithm or one of its refinements. However, for the specific case of Gauss-Legendre quadrature, computation methods based on asymptotic series representations of the Legendre polynomials have recently been proposed.

For large quadrature rules, these methods provide superior accuracy and speed at the cost of generality. We provide an overview of the progress that was made with these asymptotic methods, focusing on the ideas and asymptotic formulas that led to them.

Finally, the limited generality will be discussed with Gauss-Jacobi quadrature rules as a prominent possibility for extension.

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.