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Numerical Methods and Linear Algebra for PDE-Constrained Optimization Problems

Optimization problems subject to PDE constraints form a class of problems which may be applied to a wide range of scientific processes, including fluid flow control, medical imaging, biological and chemical processes, and electromagnetic inverse problems, to name a few. These problems involve minimizing some function arising from a particular physical objective, while at the same time obeying a system of PDEs which describe the process. It is crucial to be able to obtain accurate numerical solutions to such problems within a reasonable CPU time, in particular for time-dependent problems, for which the “all-at-once” solution can lead to extremely large linear systems once a suitable spatial discretization and time-stepping scheme have been applied. In this talk we consider iterative methods, in particular Krylov subspace methods, to solve such systems, accelerated by fast and robust preconditioners. As well as summarising some of the important applications of PDE-constrained optimization, we will describe how to derive efficient numerical methods and preconditioners for the accurate solution of a number of practical problems.

Making the most of observations in numerical weather prediction: a large nonlinear least-squares problem

Weather forecasts play a vital part in our lives, with major impacts on society and the economy. Forecasts are obtained by combining observations of the weather with computational predictions using a data assimilation process. Mathematically, this can be expressed as a large nonlinear least-squares problem, where the model and observational data are weighted by their respective uncertainties. Forecast accuracy depends upon correctly specified uncertainties, as measured by error statistics. Better estimates of uncertainty result in better forecasts. In this talk we will present recent research on characterising and treat observation uncertainty in an assimilation system. We will discuss efficient mathematical methods for estimating observation error statistics, effects on the conditioning of the least -squares problem and the results of implementing these methods in operational weather forecasting systems with remote-sensing data.

Computational methods for large-scale matrix equations and application to PDEs

Linear matrix equations such as the Lyapunov and Sylvester equations and their generalizations have classically played an important role in the analysis of dynamical systems, in control theory and in eigenvalue computation.  More recently, matrix equations have emerged as the natural linear algebra result of various discretization methods for the numerical treatment of (systems of) deterministic and stochastic partial differential equations ((S)PDEs), leading to new challenges and solution strategies. In this talk we will review some of the key methodologies for solving large scale linear matrix equations. We will also discuss recent applications where solving matrix equations yields particularly effective strategies.

The Geometric Occam razor implicit in deep learning

Deep learning has been very successful at discovering solutions to a wide variety of difficult problems in domains as diverse as image classification and language translation. From the outset, this is surprising, because deep neural networks are usually over-parameterized with a very large space of poor solutions. We will show, using backward error analysis, that there is a hidden mechanism built into gradient descent called implicit gradient regularization that guides deep learning toward solutions with low geometric model complexity.

An inverse problem in electromagnetism with partial data

We consider the problem of determining the permeability, permittivity, and conductivity in a time-harmonic Maxwell system from measurements of electric and magnetic fields taken on an arbitrarily small, open subset of the boundary. We prove that under suitable hypotheses the problem has a unique solution. This is joint work with Malcolm Brown and Juan Manuel Reyes Gonzales, https://doi.org/10.1016/j.jde.2016.01.002, funded by EPSRC grant EP/K024078/1.

An inverse problem in electromagnetism with partial data

We consider the problem of determining the permeability, permittivity, and conductivity in a time-harmonic Maxwell system from measurements of electric and magnetic fields taken on an arbitrarily small, open subset of the boundary. We prove that under suitable hypotheses the problem has a unique solution. This is joint work with Malcolm Brown and Juan Manuel Reyes Gonzales, https://doi.org/10.1016/j.jde.2016.01.002, funded by EPSRC grant EP/K024078/1.

Rational neural networks

The choice of the nonlinear activation function in deep learning architectures is crucial and heavily impacts the performance of a neural network. In this talk, we consider neural networks with rational activation functions. We then establish optimal bounds in terms of network complexity and prove that rational neural networks approximate smooth functions more efficiently than ReLU networks with exponentially smaller depth. The flexibility and smoothness of rational activation functions make them an attractive alternative to ReLU.

PDE-constrained optimization for multiscale particle dynamics

"There are many industrial and biological processes, such as beer brewing, nano-separation of colloids and bird flocking, which can be described by integro-PDEs. These PDEs describe the dynamics of a ‘particle’ density within a fluid bath, under the influence of diffusion, external forces, and particle interactions. They often include nonlinear, nonlocal boundary conditions. A key challenge is to optimize these types of processes, which requires tools from PDE-constrained optimization. In this talk I will introduce a numerical method to solve this class of optimal control problems, which combines pseudospectral methods and spectral elements with a Newton-Krylov algorithm. This provides a tool for the fast and accurate solution of the resulting optimality systems. In particular, this framework allows for the solution of (integro-)PDE models and optimal control problems on complex domains, which is a crucial feature in accurately describing various (industry) applications. Finally, some examples of current work and future industrial applications will be given. This is joint work with Ben Goddard and John Pearson. "

Nash Equilibria in certain two-choice multi-player games played on the ladder graph

In this research, we compute analytically the number of Nash Equilibria (NE) for a two-choice game played on a ladder graph and a circular ladder with 2n players, identified by the vertices of the graph. We do not fix the payoff parameters of the underlying two-player game, except for the requirement that a NE occurs if the players choose opposite strategies (anti-coordination game). The results show that for both, the ladder and circular ladder, the number of NE grows exponentially with (half) the number of players as NE(2n) ∼ C*ϕ^n, where ϕ = 1.618.. is the golden ratio and C is a constant. In addition, the value of the scaling factor C_{ladder} depends on the value of the payoff parameters. However, that is no longer true for the circular ladder (3-degree graph), that is C_{circ} is constant, which might suggest that the topology of the graph indeed plays an important role for setting the number of NE.

Stopping criterion for coarse grid solvers in the multigrid V-cycle method

"Multigrid methods are frequently used when solving systems of linear equations, applied either as standalone solvers or as preconditioners for iterative methods. Within each cycle, the approximation is computed using smoothing on fine levels and solving on the coarsest level. Solving on the coarsest level can be carried out by a direct solver based on the LU or Cholesky decomposition, i.e., the coarsest level problem is solved exactly up to a machine precision. When solving large-scale problems on parallel computers, using direct solvers can be ineffective, sometimes impossible to realize. In these settings it is preferable to use inexact solver, e.g., iterative Krylov subspace methods or direct methods based on low rank approximations; see, e.g., [M. Huber, Massively parallel and fault-tolerant multigrid solvers on peta-scale systems, Ph.D. Thesis, Technical University of Munich, Germany, 2019], [Buttari et al., Numerical Linear Algebra with Applications (2021)]. The solvers are typically stopped when the Euclidian norm of relative residual reaches specified tolerance. In practice, this tolerance is determined experimentally in order to balance the cost of the coarse grid solves and the total number of V-cycles required for convergence. In this talk, we present an approach to constructing stopping criteria for an inexact coarse grid solver in the multigrid V-cycle method without post-smoothing for symmetric positive definite problems. This approach enables controlling the slowdown in the rate of convergence occurring due to the use of an inexact solver. We discuss several stopping criteria derived using this approach and suggest a strategy for utilizing them in practice. The results are illustrated through numerical experiments. "

When do delays desynchronize a power grid? Master stability conditons for inertial oscillator networks

Time lags occur in a vast range of real-world dynamical systems due to finite reaction times. An important example are future power grids, where processing delays in the stability control may pose a challenge as renewable energies proliferate. This talk will introduce an analytical approach to determine the asymptotic stability of synchronous states in networks of coupled inertial oscillators with constant delay. Building on the master stability formalism, the technique provides necessary and sufficient delay master stability conditions. We will apply the results to a proposed smart grid model, illustrating how desynchronization depends on the delay and network topology.

A Posteriori Error Estimation for Stochastic Collocation Applied to Parametric Parabolic PDEs

"We investigate a posteriori error estimation for time-dependent parametric partial differential equations discretized using non-intrusive stochastic collocation finite element methods. In particular, we look to estimate the distinct contributions to the total approximation error stemming from both the parametric and time-stepping discretization schemes, in order to drive adaptive solution algorithms. The parametric error associated with the stochastic collocation method is estimated using a hierarchical method. The time-stepping algorithm is treated as a black-box with control imposed over the estimated local truncation error. Numerical results are presented for a time-dependent advection-diffusion problem with uncertain wind field. The evolution of error in time is analysed and the challenges of driving an adaptive-in-time stochastic collocation algorithm discussed. "