Decision trees usefully represent the sparse, high dimensional and noisy nature of chemical data from experiments. Having learned a function from this data, we may want to thereafter optimise the function, e.g. for picking the best catalyst for a chemical process. This work studies a mixed-integer non-linear optimisation problem involving: (i) gradient boosted trees modelling catalyst behaviour, (ii) penalty functions mitigating risk, and (iii) penalties enforcing chemical composition constraints. We develop several heuristic methods to find feasible solutions, and an exact, branch and bound algorithm that leverages structural properties of the gradient boost trees and penalty functions. We computationally test our methods on an industrial instance from BASF.

This work was completed in collaboration with Mr Miten Mistry and Dr Dimitris Letsios at Imperial College London and Dr Robert Lee and Dr Gerhard Krennrich from BASF.

# Past Computational Mathematics and Applications Seminar

We present discrete methods for computing low-rank approximations of time-dependent tensors that are the solution of a differential equation. The approximation format can be Tucker, tensor trains, MPS or hierarchical tensors. We will consider two types of discrete integrators: projection methods based on quasi-optimal metric projection, and splitting methods based on inexact solutions of substeps. For both approaches we show numerically and theoretically that their behaviour is superior compared to standard methods applied to the so-called gauged equations. In particular, the error bounds are robust in the presence of small singular values of the tensor’s matricisations. Based on joint work with Emil Kieri, Christian Lubich, and Hanna Walach.

Solving the Stokes equation by an optimal domain decomposition method derived algebraically involves the use of non standard interface conditions whose discretisation is not trivial. For this reason the use of approximation methods such as hybrid discontinuous Galerkin appears as an appropriate strategy: on the one hand they provide the best compromise in terms of the number of degrees of freedom in between standard continuous and discontinuous Galerkin methods, and on the other hand the degrees of freedom used in the non standard interface conditions are naturally defined at the boundary between elements. In this work we introduce the coupling between a well chosen discretisation method (hybrid discontinuous Galerkin) and a novel and efficient domain decomposition method to solve the Stokes system. We present the detailed analysis of the hybrid discontinuous Galerkin method for the Stokes problem with non standard boundary conditions. This analysis is supported by numerical evidence. In addition, the advantage of the new preconditioners over more classical choices is also supported by numerical experiments.

This work was done in collaboration with G. Barrenechea, M. Bosy (Univ. Strathclyde) and F. Nataf, P-H Tournier (Univ of Paris VI)

In this talk we describe an approach to approximate the truncated singular value decomposition of a large matrix by first decomposing the matrix into a sum of Kronecker products. Our approach can be used to more efficiently approximate a large number of singular values and vectors than other well known schemes, such as iterative algorithms based on the Golub-Kahan bidiagonalization or randomized matrix algorithms. We provide theoretical results and numerical experiments to demonstrate accuracy of our approximation, and show how the approximation can be used to solve large scale ill-posed inverse problems, either as an approximate filtering method, or as a preconditioner to accelerate iterative algorithms.

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.

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.