University of Reading

Standard numerical schemes for acoustic scattering problems suffer from the restriction that the number of degrees of freedom required to achieve a prescribed level of accuracy must grow at least linearly with respect to frequency in order to maintain accuracy as frequency increases. In this talk, we review recent progress on the development and analysis of hybrid numerical-asymptotic boundary integral equation methods for these problems. The key idea of this approach is to form an ansatz for the solution based on knowledge of the high frequency asymptotics, allowing one to achieve any required accuracy via the approximation of only (in many cases provably) non-oscillatory functions. In particular, we discuss very recent work extending these ideas for the first time to non-convex scatterers.

Data assimilation aims to correct a forecast of a physical system, such as the atmosphere or ocean, using observations of that system, in order to provide a best estimate of the current system state. Since it is not possible to observe the whole state it is important to know how errors in different variables of the forecast are related to each other, so that all fields may be corrected consistently. In the first part of this talk we consider how we may impose constraints between different physical variables in data assimilation. We examine how we can use our knowledge of atmospheric physics to pose the assimilation problem in variables that are assumed to be uncorrelated. Using a shallow-water model we demonstrate that this is best achieved by using potential vorticity rather than vorticity to capture the balanced part of the flow. The second part of the talk will consider a further transformation of variables to represent spatial correlations. We show how the accuracy and efficiency with which we can solve the transformed assimilation problem (as measured by the condition number) is affected by the observation distribution and accuracy and by the assumed correlation lengthscales. Theoretical results will be illustrated using the Met Office variational data assimilation scheme.

Feedback design for a second order control system leads to an

eigenstructure assignment problem for a quadratic matrix polynomial. It is

desirable that the feedback controller not only assigns specified

eigenvalues to the second order closed loop system, but also that the

system is robust, or insensitive to perturbations. We derive here new

sensitivity measures, or condition numbers, for the eigenvalues of the

quadratic matrix polynomial and define a measure of robustness of the

corresponding system. We then show that the robustness of the quadratic

inverse eigenvalue problem can be achieved by solving a generalized linear

eigenvalue assignment problem subject to structured perturbations.

Numerically reliable methods for solving the structured generalized linear

problem are developed that take advantage of the special properties of the

system in order to minimize the computational work required.

We consider the problem of recovering the position of a scattering surface

from measurements of the scattered field on a finite line above the surface.

A point source algorithm is proposed, based on earlier work by Potthast,

which reconstructs, in the first instance, the scattered field in the whole

region above the scattering surface. This information is used in a second

stage to locate the scatterer. We summarise the theoretical results that can

be obtained (error bounds on the reconstructed field as a function of the

noise level in the original measurements). For the case of a point source of

the incident field we present numerical experiments for both a steady source

(time harmonic excitation) and a pulse source typical of an antenna in

ground penetrating radar applications.

\\

This is joint work with Claire Lines (Brunel University).

Standard finite element or boundary element methods for high frequency scattering problems, with piecewise polynomial approximation spaces, suffer from the limitation that the number of degrees of freedom required to achieve a prescribed level of accuracy grows at least linearly with respect to the frequency. Here we present a new boundary element method for which, by including in the approximation space the products of plane wave basis functions with piecewise polynomials supported on a graded mesh, we can demonstrate a computational cost that grows only logarithmically with respect to the frequency.

Variational data assimilation techniques for optimal state estimation in very large environmental systems currently use approximate Gauss-Newton (GN) methods. The GN method solves a sequence of linear least squares problems subject to linearized system constraints. For very large systems, low resolution linear approximations to the model dynamics are used to improve the efficiency of the algorithm. We propose a new approach for deriving low order system approximations based on model reduction techniques from control theory which can be applied to unstable stochastic systems. We show how this technique can be combined with the GN method to retain the response of the dynamical system more accurately and improve the performance of the approximate GN method.

TBA

Starting from the classical Saltzman two-dimensional convection equations, we derive via a severe spectral truncation a minimal 10 ODE system which includes the thermal effect of viscous dissipation. Neglecting this process leads to a dynamical system which includes a decoupled generalized Lorenz system. The consideration of this process breaks an important symmetry and couples the dynamics of fast and slow variables, with the ensuing modifications to the structural properties of the attractor and of the spectral features. When the relevant nondimensional number (Eckert number) Ec is different from zero, an additional time scale of O(Ec^(?1)) is introduced in the system, as shown with standard multiscale analysis and made clear by several numerical evidences. Moreover, the system is ergodic and hyperbolic, the slow variables feature long-term memory with 1/ f^(3/2) power spectra, and the fast variables feature amplitude modulation. Increasing the strength of the thermal-viscous feedback has a stabilizing effect, as both the metric entropy and the Kaplan-Yorke attractor dimension decrease monotonically with Ec. The analyzed system features very rich dynamics: it overcomes some of the limitations of the Lorenz system and might have prototypical value in relevant processes in complex systems dynamics, such as the interaction between slow and fast variables, the presence of long-term memory, and the associated extreme value statistics. This analysis shows how neglecting the coupling of slow and fast variables only on the basis of scale analysis can be catastrophic. In fact, this leads to spurious invariances that affect essential dynamical properties (ergodicity, hyperbolicity) and that cause the model losing ability in describing intrinsically multiscale processes.

Invariant subspaces are a well-established tool in the theory of linear eigenvalue problems. They are also computationally more stable objects than single eigenvectors if one is interested in a group of closely clustered eigenvalues. A generalization of invariant subspaces to matrix polynomials can be given by using invariant pairs.

We investigate some basic properties of invariant pairs and give perturbation results, which show that invariant pairs have similarly favorable properties for matrix polynomials than do invariant subspaces have for linear eigenvalue problems. In the second part of the talk we discuss computational aspects, namely how to extract invariant pairs from linearizations of matrix polynomials and how to do efficient iterative refinement on them. Numerical examples are shown using the NLEVP collection of nonlinear eigenvalue test problems.

This talk is joint work with Daniel Kressner from ETH Zuerich.

Work with N.K. Nichols (Reading), C. Boess & A. Bunse-Gerstner (Bremen)

The Gauss-Newton (GN) method is a well known iterative technique for solving nonlinear least squares problems subject to dynamical system constraints. Such problems arise commonly from applications in optimal control and state estimation. Variational data assimilation systems for weather, ocean and climate prediction currently use approximate GN methods. The GN method solves a sequence of linear least squares problems subject to linearized system constraints. For very large systems, low resolution linear approximations to the model dynamics are used to improve the efficiency of the algorithm. We propose a new method for deriving low order system approximations based on model reduction techniques from control theory. We show how this technique can be combined with the GN method to give a state estimation technique that retains more of the dynamical information of the full system. Numerical experiments using a shallow-water model illustrate the superior performance of model reduction to standard truncation techniques.