08:30 to 09:55 Registration 09:55 to 10:00 David Wallace - Welcome INI 1 10:00 to 11:00 Fast high-order high-frequency solvers in computational acoustics and electromagnetics The numerical solution of highly oscillatory wave-propagation and scattering problems presents a variety of significant challenges: these problems require high discretization densities and often give rise to poorly conditioned numerics; realistic engineering configurations, further, usually require consideration of geometries of great complexity and large extent. In this talk we will consider a number of methodologies that were introduced recently to address these difficulties. We will thus discuss algorithms that can solve, with high-order accuracy, problems of scattering for complex three-dimensional geometries---including, possibly, singular elements such as wires, corners, edges and open screens. In particular, we will describe solutions achieved for two realistic three-dimensional problems of very high frequency---surface scattering and atmospheric GPS propagation---which previous three-dimensional solvers could not address adequately. For added efficiency, these solvers, which are based on integral equations, high-order integration and fast Fourier transforms, can be used in conjunction with new regularized combined field equations---which require much smaller numbers of iterations in a iterative linear algebra solver than combined field equations available previously. We will also describe a new class of high-order surface representation methods which, starting from point clouds or CAD data, can produce high-order-accurate surface parametrizations of complex engineering surfaces, as required by high-order solvers. Time permitting, applications of these methodologies to solution of time-domain problems and fast evaluation of fully-nonlocal and convergent computational boundary conditions for time-domain problems will be mentioned. In all cases these algorithms exhibit high-order convergence, they run on low memories and reduced operation counts, and they can produce solutions with a high degree of accuracy. INI 1 11:00 to 11:30 Coffee and Poster session 11:30 to 12:00 Domain decomposition for multiscale elliptic PDEs In the Monte-Carlo simulation of fluid flow in stochastic media one typically has to solve elliptic PDEs with highly oscillatory coefficients. In practical applications in hydrogeology, these (random) oscillatory coefficients can have wavelength of order 10^{-3} or smaller and amplitude of the order of 10^{8}. Accurate finite element approximation in 2D requires of the order of 10^{8} degrees of freedom and the resulting linear systems can have condition number close to 10^{16}. In this talk we will discuss domain decomposition preconditioning for such linear systems. Our overall aim is to solve such problems in a time close to the time required for solving a discretisation of a standard Poisson problem with constant coefficient with the same number of degrees of freedom. The essential step in constructing two- or multi-level preconditioners is to replace the discretisation on the finest grid with a suitable discretisation on a coarser grid or grids. In the present application, standard (piecewise polynomial based) coarsening fails because, even if the fine mesh resolves the oscillations in the coefficient, coarser meshes typically fail to do so. By extending the classical domain decomposition theory to this case, we show that a suitable coarsening strategy for heterogeneous media involves the construction of low energy coarse space basis functions. This naturally suggests that multiscale finite element methods can provide good coarse spaces, and leads to a new class of domain decomposition preconditioners. Recent results of Scheichl and Vainikko used the same theoretical technique to explain the robustness of certain algebraically defined preconditioners. The theoretical results are illustrated by numerical examples on deterministic and random problems. Reference: I.G. Graham, P. Lechner and R. Scheichl, Domain Decomposition for Multiscale PDEs, Numer. Math. DOI 10.1007/s00211-007-0074-1 (2007) . INI 1 12:30 to 13:30 Lunch at Wolfson Court (Residents only) 14:00 to 15:00 E Zuazua ([Madrid])An alternate direction descent method for the control of flows in the presence of shocks We present a new optimization strategy to compute numerical approximations of minimizers for optimal control problems governed by scalar conservation laws in the presence of shocks. We focus on the 1-d inviscid Burgers equation. We first prove the existence of minimizers and, by a Gamma-convergence argument, the convergence of discrete minima obtained by means of numerical approximation schemes satisfying the so called one-sided Lipschitz condition (OSLC). Then we address the problem of developing efficient descent algorithms. We first consider and compare the existing two possible approaches. The first one, the so-called discrete approach, based on a direct computation of gradients in the discrete problem and the so-called continuous one, where the discrete descent direction is obtained as a discrete copy of the continuous one. When optimal solutions have shock discontinuities, both approaches produce highly oscillating minimizing sequences and the effective descent rate is very weak. As a remedy we propose a new method that uses the recent developments of generalized tangent vectors and the linearization around discontinuous solutions. We develop a new descent stratagey, that we shall call "alternating descent method", distinguishing descent directions that move the shock and those that perturb the profile of the solution away of it. As we shall see, a suitable alternating combination of these two classes of descent directions allows building very efficient and fast descent algorithms. INI 1 15:00 to 15:30 Tea and Poster session 15:30 to 16:00 Condition number estimates for oscillatory integral operators We present results obtained during the programme on the conditioning of the standard combined potential boundary integral operators in time-harmonic acoustic scattering (cf. the talk by Oscar Bruno), in particular addressing behaviour as the wavenumber k tends to infinity, when the integral operator becomes increasingly oscillatory. While study of this topic goes back to Kress and Spassov (Numer. Math. 1983), the focus previously has been on the canonical case of a circle/sphere for which spherical harmonics are the eigenfunctions and the singular values are known explicitly. However, even for this case, it is only recently (Dominguez, Graham, Smyshlyaev, Preprint NI07004-HOP and Numer. Math. 2007) that rigorous upper bounds have been obtained on the operator and its inverse as a function of k. For non-spherical scatterers the only result is a recent upper bound on the inverse operator for piecewise smooth starlike domains (Chandler-Wilde & Monk, to appear SIAM J. Math. Anal.). In this talk we derive a range of lower bounds on the operator and its inverse, which show that the behaviour for large k depends subtly on the geometry. The sharpness of these lower bounds is demonstrated by numerical simulation and, in some instances, by provable upper bounds. The main computational message is that, while the condition number grows only mildly (like k^{1/3}) for a circle/sphere, behaviour can be much worse (like k^{5/4}) for non-starlike domains. INI 1 16:00 to 16:30 New Galerkin methods for high-frequency scattering simulations Recently developed integral equation methods for surface scattering simulations ---that combine the advantages of rigorous solvers (error controllability) with those of asymptotic methods (frequency-independent discretizations)--- have displayed the potential of delivering scattering returns within a prescribed error-tolerance in times that do not depend on the wavenumber k. The near-optimal characteristics of these novel schemes, pioneered by Bruno, Geuzaine, Monro and Reitich, have rapidly generated great interest and significant number of work in recent years. In single-scattering configurations, an actual proof that provides a rigorous upper bound for the operation count of O(k^{1/9}) in the case of circular/spherical boundaries was recently established by Dominguez, Graham and Smyshlyaev for a p-version boundary element implementation of a similar approach. As in the original algorithm, they take profound advantage of the exponential decay (with increasing wavenumber k) of the corresponding physical density in the deep shadow region, and approximate this quantity by zero there. In this talk, we present two improved Galerkin schemes for the solution of single-scattering problems, and we show that, within the framework of our first scheme, the error in best approximation of the surface current grows at most at O(k^a) (for any a>0) over the "entire" boundary. Moreover, as we show, our second approach based on a novel change of variables around the "transition regions" reduces this dependency to O(log(k)). INI 1 16:30 to 17:00 Plane wave discontinuous Galerkin methods Standard low order Lagrangian finite element discretization of boundary value problems for the Helmholtz equation $-\Delta u - \omega^{2}u=f$ are afflicted with the so-called pollution phenomenon: though for sufficiently small $h\omega$ an accurate approximation of $u$ is possible, the Galerkin procedure fails to provide it. Attempts to remedy this have focused on incorporating extra information in the form of plane wave functions $\boldsymbol{x}\mapsto \exp(i\omega\boldsymbol{d}\cdot\boldsymbol{x})$, $|\boldsymbol{d}|=1$, into the trial spaces. Prominent examples of such methods are the plane wave partition of unity finite element method of Babuska and Melenk, and the ultra-weak Galerkin discretization due to Cessenat and Despres. Both perform well in computations. It turns out that the latter method can be recast as a special so-called discontinuous Galerkin (DG) method employing local trial spaces spanned by a few plane waves. This perspective paves the way for marrying plane wave approximation with many of the various DG methods developed for 2nd-order elliptic boundary value problems. We have pursued this for a generic mixed DG method and a primal DG method which generalized the ultra-weak scheme. For these methods we have developed a convergence analysis for the $h$-version, which achieves convergence through mesh refinement. Key elements are approximation estimates for plane waves and sophisticated duality techniques. The latter entail estimating how well local plane waves can approximate the solution of a dual problem. Unfortunately, we could not help invoking general polynomial estimates in Sobolev spaces for this purpose. This incurs unsatisfactory pollution-affected final error bounds $O(\omega^{2}h)$. On the other hand, a more detailed 1D analysis confirms that the plane wave DG method does not suffer from pollution. To bridge the gap is an open theoretical challenge. INI 1 17:00 to 18:00 Welcome Wine Reception and Poster session 18:45 to 19:30 Dinner at Wolfson Court (Residents only)
 09:00 to 10:00 S Vandewalle ([Leuven])Numerical integration of highly oscillatory functions based on analytic continuation We consider the integration of highly oscillatory functions. Based on analytic continuation, rapidly converging quadrature rules are derived for a fairly general class of oscillatory integrals with an analytic integrand. The accuracy of the quadrature increases both for the case of a fixed number of points and increasing frequency, and for the case of an increasing number of points and fixed frequency. These results are then used to obtain quadrature rules for more general oscillatory integrals, i.e., for functions that exhibit some smoothness but that are not analytic. The approach described in this paper is related to the steepest descent or saddle point method from complex analysis. However, it does not employ asymptotic expansions. It can be used for small or moderate frequencies as well as for very high frequencies. We consider both the one-dimensional case, and the multi-dimensional case. Finally, we briefly elaborate on the use of the new integration rules in the context of solving highly oscillatory integral equations. INI 1 10:00 to 10:30 H Brunner ([Newfoundland])Open problems in the computational solution of volterra functional equations with highly oscillatory solutions The approximation of solutions to Volterra integral equations by collocation or discontinuous Galerkin methods leads to a set of semi-discretised' equations that in general are not amenable to numerical computation: an additional discretisation process that is able efficiently and accurately to cope with the highly oscillatory nature of the kernel of the given Volterra integral operator is needed. Here, the use of Filon-type quadrature is an obvious possibility; however, it is not yet clear how best to do this when the kernel is weakly singular. In this talk I will describe current work related to the above problem, and especially to collocation methods for various types of Volterra functional equations, including equations with variable (and possibly vanishing) delay arguments. It will also be shown that in the case of Volterra integral equations of the first kind, the choice of the quadrature scheme in discontinuous Galerkin methods can have a major effect on the convergence properties of the approximate solution. INI 1 10:30 to 11:00 Coffee and Poster session 11:00 to 11:30 Asymptotic least squares approximation for highly oscillatory differential equations This talk presents a new approach for approximating highly oscillatory ordinary differential equations. By using the asymptotic expansion in a least squares system, we are able to obtain a result that preserves the asymptotic accuracy of the expansion, while converging rapidly to the exact solution. We are thus able to accurately approximate such differential equations by solving a very small linear system. We apply this method to the computation of highly oscillatory integrals, as well as second order oscillatory differential equations. INI 1 11:30 to 12:00 A Iserles ([Cambridge])Modified fourier expansions and spectral problems for highly oscillatory Fredholm operators Although highly oscillatory Fredholm operators are compact and have point spectrum, their calculation by standard means, e.g. the finite section method, is notoriously difficult. As an alternative, we propose expanding the underlying eigenfunctions in modified Fourier series. This leads to infinite-dimensional algebraic eigenvalue problems that exhibit intriguing structure and rapid decay of coefficients. This is exploited in an effective numerical algorithm. INI 1 12:30 to 13:30 Lunch at Wolfson Court (Residents only) 14:00 to 15:00 Approximation of non-adiabatic quantum dynamics by surface hopping The time-dependent Schroedinger equation provides the fundamental description of quantum mechanical molecular dynamics. Its multiscale character suggests a splitting in two coupled subproblems, the so called Born-Oppenheimer approximation: One solves a family of stationary Schroedinger equations in the electronic degrees of freedom (one equation for each nucelonic configuration) and subsequently a time-dependent Schroedinger equation in the nucleonic degrees of freedom, whose potential has been determined by the electronic problem. This splitting fails to provide an approximation, if different electronic eigenvalues are not uniformly separated for all nucleonic configurations. The talk explains a microlocal point of view on this non-adiabatic coupling between electronic and nucleonic degrees of freedom and derives an associated deterministic surface hopping algorithm. Its numerical realization crucially relies on the sampling of highly oscillatory initial data on high-dimensional configuration spaces, which can be tackled by a Monte Carlo approach. The presented results are joint work with C. Fermanian, S. Kube, and M. Weber. INI 1 15:00 to 15:30 Tea and Poster session 15:30 to 16:00 C Sparber ([Vienna])Asymptotics for linear and nonlinear Schr\"odinger equations with periodic potentials We consider semiclassically scaled linear and nonlinear Schrödinger equation with highly oscillatory periodic potentials. We will discuss rigorous asymptotic descriptions for such problems as well as a recently developed numerical approach based on Bloch decomposition. INI 1 16:00 to 16:30 Computational quantum dynamics using Hagedorn wavepackets This talk, which is based on joint work with Erwan Faou and Vasile Gradinaru, reports on work in progress on a newly developed numerical approach to many-body quantum dynamics in the semi-classical regime. We present a symmetric splitting integrator for the propagation of multidimensional extensions of Gauss-Hermite wavepackets appearing in analytical work by Hagedorn. The integrator evolves positions and momenta of the wavepackets according to the Stoermer-Verlet integrator of classical mechanics, and gains its computational feasibility and efficiency for many particles by the possibility of thinning out the moving basis sets according to a hyperbolic cross approximation or a Hartree-type approximation in a moving frame. The algorithm reduces to the Strang splitting of the Schroedinger equation in the limit of the full set of orthonormal basis functions, and it is robust in the semiclassical limit. INI 1 16:45 to 17:15 On macroscale variables for multiple time scale problems INI 1 17:15 to 17:45 A multiscale method for computing highly oscillatory ODEs We prsent a multiscale method for computing the effective behavior of a class of stiff and highly oscillatory ordinary differential equations. The oscillations may be in resonance with one another and thereby generate hidden slow dynamics. The proposed method relies on correctly tracking a set of slow variables whose dynamics is closed up to $\epsilon$ perturbation, and is sufficient to approximate any variable and functional that are slow under the dynamics of the ODE. This set of variables is detected numerically as a preprocessing step in the numerical methods. Error and complexity estimates are obtained. The advantages of the method is demonstrated with a few examples, including a commonly studied problem of Fermi, Pasta, and Ulam. INI 1 19:30 to 23:00 Conference Garden Party at Clare Hall