The Optimisation and Numerical Analysis seminar is held regularly during termtime.
Tuesday 27 January 2026, 15:00-16:00
Location: Zoom
One of the key properties of convex optimisation problems is that every stationary point is a global optimum, and local nonlinear programming algorithms are thus global when the problem is convex. Generalisations of convexity have been proposed in order to leverage this powerful property in some nonconvex problems and formulate global optimality conditions. This talk proposes a new generalised convexity notion which we refer to as optima-invexity: the property that only one connected set of locally optimal solutions exists. This notion enables the analysis of the relations between stationary points of different types that is key to our approach. We present a study of optima-invexity of unconstrained and box-constrained quadratic programs and discuss how these results can be extended to general nonlinear programs. Finally, we outline algorithmic applications of optima-invexity to general nonconvex problems, and present an example of an invexity-guided algorithm.
Tuesday 10 February 2026, 14:00-15:00
Watson Building, B16
Scientific machine learning is a rapidly growing field, aiming to develop data-driven tools for problems traditionally addressed by numerical methods. One particularly active area is the prediction of solutions to differential equations. But do these machine learning techniques live up to their promise of providing fast, accurate, and general-purpose solvers?
In this talk, we critically examine the current capabilities and limitations of popular machine learning methods in this context, starting with the introduction of a common task framework for the objective evaluation of model performance. Drawing on a range of recent examples, we highlight both the genuine strengths and the practical shortcomings of data-driven approaches. We then explore how hybrid strategies - combining machine learning with classical numerical techniques - can offer a more robust alternative. In particular, we demonstrate how such approaches can accelerate Newton iterations in implicit methods for non-linear evolution equations and improve mesh relocation strategies in finite element solvers. These results suggest that perhaps a real strength in machine learning approaches in scientific computing is not to replace classical methodologies but rather to enhance them.
This is joint work with J. Bakarji, C. Budd, M. Cranmer, T. Deveney, J. A. Goldfeder, J. Germany, T. Jin, J. N. Kutz, P. Lio, E. H. Müller, A. Paganini, S. Riva, J. Rowbottom, A. S. Rude, K. Schratz, C.-B. Schönlieb, M. Tomasetto, P. M. Wyder, A. Yermakov, J. P. Williams, Y. Xiang, Y. Zhao and D. Zoro.
Tuesday 17 February 2026, 15:00-16:00
Watson Building, B16
The latent variable proximal point (LVPP) algorithm is an easy-to-implement solver for infinite-dimensional variational problems with pointwise constraints. It supports general discretisations, including high-order hp-FEM. Despite the flexibility, the method exhibits robust convergence and mesh-independent iteration counts in practice.
In this talk, we will derive LVPP as a saddle point reformulation of the Bregman proximal point algorithm. Although equivalent at the continuous level, the saddle point formulation is significantly more robust after discretization. We will then apply LVPP to solve obstacle, contact, fracture, and plasticity problems, as well as problems with more complex constraints such as quasi-variational inequalities, where the constraint depends implicitly on the unknown solution.
Papers: https://doi.org/10.1016/j.cma.2025.118181, https://arxiv.org/abs/2412.13733
Tuesday 24 February 2026, 14:00-15:00
Watson Building, B16
In uncertainty quantification (UQ), physical models with uncertain inputs are commonly represented as parametric partial differential equations (PDEs). That is, PDEs with inputs that are expressed as functions of parameters with an associated probability distribution. In forward UQ, one aims to understand how uncertainty in model inputs affects uncertainty in model solutions. Naive sampling methods require the repeated numerical solution of the original PDE for different samples of the inputs, but when the cost of solving the problem for just one sample is already expensive, obtaining accurate uncertainty assessments becomes infeasible. Fortunately, several families of numerical schemes have been developed to tackle this issue. So-called stochastic collocation is a popular approach because it is non-intrusive in nature, so compatible with exisiting space-time solvers, but yields a polynomial-based surrogate that can be cheaply evaluated for new parameter choices of interest. However, developing efficient and accurate algorithms that account for errors on the space and time domains as well as the parameter domain is highly challenging. Indeed, it is well known that standard polynomial-based parametric approximations can incur errors that grow in time. In this talk, we discuss an adaptive in time approach introduced in Kent at al [1] for parametric advection–diffusion problems. This builds a polynomial-based surrogate on the parameter domain that is adapted sequentially in time, using off-the-shelf adaptive time-stepping schemes with local error control.
[1] Benjamin M. Kent et al. Efficient Adaptive Stochastic Collocation Strategies for Advection–Diffusion Problems with Uncertain Inputs. (2023) https://doi.org/10.1007/s10915-023-02247-w
Tuesday 17 March 2026, 15:00-16:00
Watson Building, B16
Computing integrals with respect to the Gaussian measure is a common task across many disciplines. In this talk, I will discuss optimal numerical integration rules for this task. To formalise optimality, we fix a target function class and consider the L2-Sobolev space of dominating mixed smoothness.
In one dimension, the trapezoidal rule is asymptotically optimal in this space, up to a logarithmic factor. By contrast, Gauss–Hermite quadrature converges at only half this rate. A similar pattern holds for analytic functions.
We show that these results extend to higher dimensions: several quasi-Monte Carlo methods achieve the optimal rate, whereas the sparse-grid Gauss–Hermite rule based on attains only half the optimal rate.
This talk is based on the following joint work:
[1] Kazashi, Suzuki, and Goda (arXiv:2509.18712, 2025)
[2] Goda, Kazashi, and Tanaka. How Sharp Are Error Bounds? Lower Bounds on Quadrature Worst-Case Errors for Analytic Functions. SIAM Journal on Numerical Analysis (2024)
[3] Kazashi, Suzuki, and Goda. Suboptimality of Gauss--Hermite quadrature and optimality of the trapezoidal rule for functions with finite smoothness. SIAM Journal on Numerical Analysis (2023)
Tuesday 24 March 2026, 14:00-15:00
Watson Building, B16
We introduce a new class of graph kernels for machine learning with metric graphs based on tropical geometry and the graph topologies. Unlike traditional graph kernels that are defined by graph combinatorics (nodes, edges, subgraphs), our approach considers only the geometry and topology of the underlying metric space. A key property of our construction is its invariance under edge subdivision, making the kernels intrinsically well-suited for comparing graphs that represent different underlying spaces. Our kernels are efficient to compute and depend only on the graph genus rather than the size. In label-free settings, our kernels outperform existing methods, which we showcase on synthetic, benchmarking, and real-world road network data. Joint work with Yueqi Cao (KTH Sweden).
Wednesday 15 October 2025, 15:00-16:00
Arts, Room 103
Wednesday 22 October 2025, 14:00-15:00
Arts, Lecture Room 4
Fluid mechanics and magnetohydrodynamics often involve intricate differential and topological structures, such as vorticity and magnetic field knots, which are critical to the underlying physics. Numerical discretization errors can break these structures, leading to wrong solutions.
In this talk, we present two examples in topological (magneto)hydrodynamics: relaxation and dynamo. Relaxation addresses the evolution of magnetic fields from given initial conditions in plasma physics, focusing on the existence and properties of stationary states. Open questions, including the Parker hypothesis, highlight the role of magnetic field line topology, particularly knots, in constraining relaxation processes. Conversely, the dynamo problem examines the exponential growth of magnetic fields.
We emphasise the importance of structure-preserving numerical methods, specifically those that conserve helicity and topology. Using finite element de Rham complexes within the framework of finite element exterior calculus, we derive schemes that precisely preserve these structures, ensuring robust and physically meaningful simulations.
Wednesday 5 November 2025, 14:00-15:00
Watson Building, B16
In this talk, I will introduce tropical geometry - a variant of algebraic geometry which provides a geometric lens through which to view non-smooth optimisation problems, and that has become increasingly studied in applications such as computational biology, economics, and computer science. We will review various types of convexity which arise in tropical problems, and we propose a new gradient descent method for solving tropical optimisation problems. Theoretical results establish global solvability for tropically quasi-convex problems, while numerical experiments demonstrate the method's superior performance over classical descent for tropical optimisation problems which exhibit tropical quasi-convexity but not classical convexity. Notably, tropical gradient descent seamlessly integrates into advanced optimisation methods, such as Adam, offering improved overall performance.