Computational Mathematics and Applications Seminar

Speaker Tobias Weinzierl will be talking about: 'Modern tasking approaches to simulate black holes (and other interesting phenomena): How can we make them fit to modern hardware?'

Over the past decade, my team has developed a simulation code for binary black hole mergers that runs on dynamically adaptive Cartesian meshes. 
Its dynamic adaptivity, coupled with multiple numerical schemes operating at different scales and non-deterministic loads from puncture sources, makes task-based parallelisation a natural choice:
Task stealing across fine-grained work units balances the load across many CPU cores, while treating tasks as atomic compute units should---in theory---allow us to deploy seamlessly to accelerators.

In practice, it is far from straightforward.

Fine-grained tasks clash with accelerators, which thrive on large, homogeneous data access patterns;
task bursts on the CPU overwhelm tasking systems and produce suboptimal execution schedules;
and when tasks span address spaces, expensive memory movements kill performance.
Surprisingly, many mainstream tasking frameworks even lack the features our domain demands, i.e. to express key task concepts.

Our application serves as a powerful lens for examining these challenges. 
While our code base extends to other wave phenomena, Lagrangian techniques, and multigrid solvers, they all reveal the same fundamental tension: 
modern hardware increasingly struggles to accommodate modern HPC concepts, and it even challenges the notion that one solution fits all hardware components.

The talk proposes practical workarounds and solutions to these shortcomings, while all solutions are designed, wherever possible, to be upstreamed into mainstream software building blocks or at least decoupled from our particular PDE solver, making them broadly applicable to the community.

This talk is hosted by Rutherford Appleton Laboratory and will take place @ Harwell Campus, Didcot, OX11 0QX
 

TBA

Speaker Prof Dr Maria Lukacova will talk about 'Numerical analysis of oscillatory solutions of compressible flows'

Oscillatory solutions of compressible flows arise in many practical situations.  An iconic example is the Kelvin-Helmholtz problem, where standard numerical methods yield oscillatory solutions. In such a situation,  standard tools of numerical analysis for partial differential equations are not applicable. 

We will show that structure-preserving numerical methods converge in general to generalised solutions, the so-called dissipative solutions. 
The latter describes the limits of oscillatory sequences. We will concentrate on the inviscid flows, the Euler equations of gas dynamics, and mention also the relevant results obtained for the viscous compressible flows, governed by the Navier-Stokes equations.

We discuss a concept of K-convergence that turns a weak convergence of numerical solutions into the strong convergence of
their empirical means to a dissipative solution. The latter satisfies a weak formulation of the Euler equations modulo the Reynolds turbulent stress.  We will also discuss suitable selection criteria to recover well-posedness of the Euler equations of gas dynamics. Theoretical results will be illustrated by a series of numerical simulations.  

TBA 

Professor Luis Nunes Vicente will talk about 'Reducing Sample Complexity in Stochastic Derivative-Free Optimization via Tail Bounds and Hypothesis Testing';

We introduce and analyze new probabilistic strategies for enforcing sufficient decrease conditions in stochastic derivative-free optimization, with the goal of reducing sample complexity and simplifying convergence analysis. First, we develop a new tail bound condition imposed on the estimated reduction in function value, which permits flexible selection of the power used in the sufficient decrease test, q in (1,2]. This approach allows us to reduce the number of samples per iteration from the standard O(delta^{−4}) to O(delta^{-2q}), assuming that the noise moment of order q/(q-1) is bounded. Second, we formulate the sufficient decrease condition as a sequential hypothesis testing problem, in which the algorithm adaptively collects samples until the evidence suffices to accept or reject a candidate step. This test provides statistical guarantees on decision errors and can further reduce the required sample size, particularly in the Gaussian noise setting, where it can approach O(delta^{−2-r}) when the decrease is of the order of delta^r. We incorporate both techniques into stochastic direct-search and trust-region methods for potentially non-smooth, noisy objective functions, and establish their global convergence rates and properties. 

This is joint work with Anjie Ding, Francesco Rinaldi, and Damiano Zeffiro.

TBA

Dr Lindon Roberts will talk about: 'Optimization Algorithms for Bilevel Learning with Applications to Imaging'

Many imaging problems, such as denoising or inpainting, can be expressed as variational regularization problems. These are optimization problems for which many suitable algorithms exist. We consider the problem of learning suitable regularizers for imaging problems from example (training) data, which can be formulated as a large-scale bilevel optimization problem. 

In this talk, I will introduce new deterministic and stochastic algorithms for bilevel optimization, which require no or minimal hyperparameter tuning while retaining convergence guarantees. 

This is joint work with Mohammad Sadegh Salehi and Matthias Ehrhardt (University of Bath), and Subhadip Mukherjee (IIT Kharagpur).

TBA 

TBA

Speaker Katharina Schratz will talk about 'Resonances as a computational tool'

A large toolbox of numerical schemes for dispersive equations has been established, based on different discretization techniques such as discretizing the variation-of-constants formula (e.g., exponential integrators) or splitting the full equation into a series of simpler subproblems (e.g., splitting methods). In many situations these classical schemes allow a precise and efficient approximation. This, however, drastically changes whenever non-smooth phenomena enter the scene such as for problems at low regularity and high oscillations. Classical schemes fail to capture the oscillatory nature of the solution, and this may lead to severe instabilities and loss of convergence. In this talk I present a new class of resonance based schemes. The key idea in the construction of the new schemes is to tackle and deeply embed the underlying nonlinear  structure of resonances into the numerical discretization. As in the continuous case, these terms are central to structure preservation and offer the new schemes strong geometric properties at low regularity.

TBA