We analyze a fully discrete numerical scheme for solving a parabolic PDE on a moving surface. The method is based on a diffuse interface approach that involves a level set description of the moving surface. Under suitable conditions on the spatial grid size, the time step and the interface width we obtain stability and error bounds with respect to natural norms. Test calculations are presented that confirm our analysis.

# Past Computational Mathematics and Applications Seminar

Spline curves represent a simple and efficient tool for data interpolation in Euclidean space. During the past decades, however, more and more applications have emerged that require interpolation in (often high-dimensional) nonlinear spaces such as Riemannian manifolds. An example is the generation of motion sequences in computer graphics, where the animated figure represents a curve in a Riemannian space of shapes. Two particularly useful spline interpolation methods derive from a variational principle: linear splines minimize the average squared velocity and cubic splines minimize the average squared acceleration among all interpolating curves. Those variational principles and their discrete analogues can be used to define continuous and discretized spline curves on (possibly infinite-dimensional) Riemannian manifolds. However, it turns out that well-posedness of cubic splines is much more intricate on nonlinear and high-dimensional spaces and requires quite strong conditions on the underlying manifold. We will analyse and discuss linear and cubic splines as well as their discrete counterparts on Riemannian manifolds and show a few applications.

Maintenance activities help prevent costly power generator breakdowns but because generators under maintenance are typically unavailable, the impact of maintenance schedules is significant and their cost must be accounted for when planning maintenance. In this paper we address the generator maintenance scheduling problem in hydropower systems. While this problem has been widely studied, specific operating conditions of hydroelectric systems have received less attention. We present a mixed-integer linear programming model that considers the time windows of the maintenance activities, as well as the nonlinearities and disjunctions of the hydroelectric production functions. Because the resulting model is hard to solve, we also propose an extended formulation, a set reduction approach that uses logical conditions for excluding unnecessary set elements from the model, and valid inequalities. Computational experiments using a variety of instances adapted from a real hydropower system in Canada support the conclusion that the extended formulation with set reduction achieves the best results in terms of computational time and optimality gap. This is joint work with Jesus Rodriguez, Pascal Cote and Guy Desaulniers.

Because of atmospheric turbulence, images of objects in outer space acquired via ground-based telescopes are usually blurry. One way to estimate the blurring kernel or point spread function (PSF) is to make use of the aberration of wavefront received at the telescope, i.e., the phase. However only the low-resolution wavefront gradients can be collected by wavefront sensors. In this talk, I will discuss how to use regularization methods to reconstruct high-resolution phase gradients and then use them to recover the phase and the PSF in high accuracy. I will end by relating the problem to high-resolution image reconstruction and methods for solving it.

Joint work with Rui Zhao and research supported by HKRGC.

Several optimization problems combine nonlinear constraints with the integrality of a subset of variables. For an important class of problems called Mixed Integer Second-Order Cone Optimization (MISOCO), with applications in facility location, robust optimization, and finance, among others, these nonlinear constraints are second-order (or Lorentz) cones.

For such problems, as for many discrete optimization problems, it is crucial to understand the properties of the union of two disjoint sets of feasible solutions. To this end, we apply the disjunctive programming paradigm to MISOCO and present conditions under which the convex hull of two disjoint sets can be obtained by intersecting the feasible set with a specially constructed second-order cone. Computational results show that such cone has a positive impact on the solution of MISOCO problems.

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).

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.