The mathematical analysis and numerical simulation of acoustic and electromagnetic wave scattering by planar screens is a classical topic. The standard technique involves reformulating the problem as a boundary integral equation on the screen, which can be solved numerically using a boundary element method. Theory and computation are both well-developed for the case where the screen is an open subset of the plane with smooth (e.g. Lipschitz or smoother) boundary. In this talk I will explore the case where the screen is an arbitrary subset of the plane; in particular, the screen could have fractal boundary, or itself be a fractal. Such problems are of interest in the study of fractal antennas in electrical engineering, light scattering by snowflakes/ice crystals in atmospheric physics, and in certain diffraction problems in laser optics. The roughness of the screen presents challenging questions concerning how boundary conditions should be enforced, and the appropriate function space setting. But progress is possible and there is interesting behaviour to be discovered: for example, a sound-soft screen with zero area (planar measure zero) can scatter waves provided the fractal dimension of the set is large enough. Accurate computations are also challenging because of the need to adapt the mesh to the fine structure of the fractal. As well as presenting numerical results, I will outline some of the outstanding open questions from the point of view of numerical analysis. This is joint work with Simon Chandler-Wilde (Reading) and Andrea Moiola (Pavia).

# Past Computational Mathematics and Applications Seminar

In this talk, I will present a novel solution strategy to efficiently and accurately compute approximate solutions to semilinear optimal control problems, focusing on the optimal control of phase field formulations of geometric evolution laws.

The optimal control of geometric evolution laws arises in a number of applications in fields including material science, image processing, tumour growth and cell motility.

Despite this, many open problems remain in the analysis and approximation of such problems.

In the current work we focus on a phase field formulation of the optimal control problem, hence exploiting the well developed mathematical theory for the optimal control of semilinear parabolic partial differential equations.

Approximation of the resulting optimal control problem is computationally challenging, requiring massive amounts of computational time and memory storage.

The main focus of this work is to propose, derive, implement and test an efficient solution method for such problems. The solver for the discretised partial differential equations is based upon a geometric multigrid method incorporating advanced techniques to deal with the nonlinearities in the problem and utilising adaptive mesh refinement.

An in-house two-grid solution strategy for the forward and adjoint problems, that significantly reduces memory requirements and CPU time, is proposed and investigated computationally.

Furthermore, parallelisation as well as an adaptive-step gradient update for the control are employed to further improve efficiency.

Along with a detailed description of our proposed solution method together with its implementation we present a number of computational results that demonstrate and evaluate our algorithms with respect to accuracy and efficiency.

A highlight of the present work is simulation results on the optimal control of phase field formulations of geometric evolution laws in 3-D which would be computationally infeasible without the solution strategies proposed in the present work.

Adjoint derivatives reveal the sensitivity of a computer program's output to changes in its inputs. These derivatives are useful as a building block for optimisation, uncertainty quantification, noise estimation, inverse design, etc., in many industrial and scientific applications that use PDE solvers or other codes.

Algorithmic differentiation (AD) is an established method to transform a given computation into its corresponding adjoint computation. One of the key challenges in this process is the efficiency of the resulting adjoint computation. This becomes especially pressing with the increasing use of shared-memory parallelism on multi- and many-core architectures, for which AD support is currently insufficient.

In this talk, I will present an overview of challenges and solutions for the differentiation of shared-memory-parallel code, using two examples: an unstructured-mesh CFD solver, and a structured-mesh stencil kernel, both parallelised with OpenMP. I will show how AD can be used to generate adjoint solvers that scale as well as their underlying original solvers on CPUs and a KNC XeonPhi. The talk will conclude with some recent efforts in using AD and formal verification tools to check the correctness of manually optimised adjoint solvers.

Gauss invented Gaussian quadrature following an approach entirely different from the one we now find in textbooks. I will describe leisurely the contents of Gauss's original memoir on quadrature, an impressive piece of mathematics, based on continued fractions, Padé approximation, generating functions, the hypergeometric series and more.

The talk will describe accelerated algorithms for computing full or partial matrix factorizations such as the eigenvalue decomposition, the QR factorization, etc. The key technical novelty is the use of randomized projections to reduce the effective dimensionality of intermediate steps in the computation. The resulting algorithms execute faster on modern hardware than traditional algorithms, and are particularly well suited for processing very large data sets.

The algorithms described are supported by a rigorous mathematical analysis that exploits recent work in random matrix theory. The talk will briefly review some representative theoretical results.

For many applications in science and engineering, the ability to efficiently and accurately approximate solutions to elliptic PDEs dictates what physical phenomena can be simulated numerically. In this seminar, we present a high-order accurate discretization technique for variable coefficient PDEs with smooth coefficients. The technique comes with a nested dissection inspired direct solver that scales linearly or nearly linearly with respect to the number of unknowns. Unlike the application of nested dissection methods to classic discretization techniques, the constant prefactors do not grow with the order of the discretization. The discretization is robust even for problems with highly oscillatory solutions. For example, a problem 100 wavelengths in size can be solved to 9 digits of accuracy with 3.7 million unknowns on a desktop computer. The precomputation of the direct solver takes 6 minutes on a desktop computer. Then applying the computed solver takes 3 seconds. The recent application of the algorithm to inverse media scattering also will be presented.

Structural optimization can be interpreted as the attempt to find the best mechanical structure to support specific load cases respecting some possible constraints. Within this context, topology optimization aims to obtain the connectivity, shape and location of voids inside a prescribed structural design domain. The methods for the design of stiff lightweight structures are well established and can already be used in a specific range of industries where such structures are important, e.g., in aerospace and automobile industries.

In this seminar, we will go through the basic engineering concepts used to quantify and analyze the computational models of mechanical structures. After presenting the motivation, the methods and mathematical tools used in structural topology optimization will be discussed. In our method, an implicit level set function is used to describe the structural boundaries. The optimization problem is approximated by linearization of the objective and constraint equations via Taylor’s expansion. Shape sensitivities are used to evaluate the change in the structural performance due to a shape movement and to feed the mathematical optimiser in an iterative procedure. Recent developments comprising multiscale and Multiphysics problems will be presented and a specific application proposal including acoustic-structure interaction will be discussed.

We describe a convergence acceleration technique for generic optimization problems. Our scheme computes estimates of the optimum from a nonlinear average of the iterates produced by any optimization method. The weights in this average are computed via a simple linear system, whose solution can be updated online. This acceleration scheme runs in parallel to the base algorithm, providing improved estimates of the solution on the fly, while the original optimization method is running. Numerical experiments are detailed on classical classification problems.

Abstract: We study nonuniform sampling in shift-invariant spaces whose generator is a totally positive function. For a subclass of such generators the sampling theorems can be formulated in analogy to the theorems of Beurling and Landau for bandlimited functions. These results are optimal and validate the heuristic reasonings in the engineering literature. In contrast to the cardinal series, the reconstruction procedures for sampling in a shift-invariant space with a totally positive generator are local and thus accessible to numerical linear algebra.

A subtle connection between sampling in shift-invariant spaces and the theory of Gabor frames leads to new and optimal results for Gabor frames. We show that the set of phase-space shifts of $g$ (totally positive with a Gaussian part) with respect to a rectangular lattice forms a frame, if and only if the density of the lattice is strictly larger than 1. This solves an open problem going backto Daubechies in 1990 for the class of totally positive functions of Gaussian type.

Almost all real-world applications involve a degree of uncertainty. This may be the result of noisy measurements, restrictions on observability, or simply unforeseen events. Since many models in both engineering and the natural sciences make use of partial differential equations (PDEs), it is natural to consider PDEs with random inputs. In this context, passing from modelling and simulation to optimization or control results in stochastic PDE-constrained optimization problems. This leads to a number of theoretical, algorithmic, and numerical challenges.

From a mathematical standpoint, the solution of the underlying PDE is a random field, which in turn makes the quantity of interest or the objective function an implicitly defined random variable. In order to minimize this distributed objective, one can use, e.g., stochastic order constraints, a distributionally robust approach, or risk measures. In this talk, we will make use of risk measures.

After motivating the approach via a model for the mitigation of an airborne pollutant, we build up an analytical framework and introduce some useful risk measures. This allows us to prove the existence of solutions and derive optimality conditions. We then present several approximation schemes for handling non-smooth risk measures in order to leverage existing numerical methods from PDE-constrained optimization. Finally, we discuss solutions techniques and illustrate our results with numerical examples.