University of Strathclyde

Physics-informed neural networks (PINNs) [2] are a solution method for solving boundary value problems based on differential equations (PDEs). The key idea of PINNs is to incorporate the residual of the PDE as well as boundary conditions into the loss function of the neural network. This provides a simple and mesh-free approach for solving problems relating to PDEs. However, a key limitation of PINNs is their lack of accuracy and efficiency when solving problems with larger domains and more complex, multi-scale solutions. 

In a more recent approach, Finite Basis Physics-Informed Neural Networks (FBPINNs) [1], the authors use ideas from domain decomposition to accelerate the learning process of PINNs and improve their accuracy in this setting. In this talk, we show how Schwarz-like additive, multiplicative, and hybrid iteration methods for training FBPINNs can be developed. Furthermore, we will present numerical experiments on the influence on convergence and accuracy of these different variants. 

This is joint work with Alexander Heinlein (Delft) and Benjamin Moseley (Oxford).

References 
1.    [1]  B. Moseley, A. Markham, and T. Nissen-Meyer. Finite basis physics- informed neural networks (FBPINNs): a scalable domain decomposition approach for solving differential equations. arXiv:2107.07871, 2021. 
2.    [2]  M. Raissi, P. Perdikaris, and G. E. Karniadakis. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378:686–707, 2019.

Most of SDE models in epidemics, ecology, biology, finance etc. are highly nonlinear and do not have explicit solutions. Monte Carlo simulations have played a more and more important role. This talk will point out several well-known numerical schemes may fail to preserve the positivity or moment of the solutions to SDE models. Reliable numerical schemes are therefore required to be designed so that the corresponding Monte Carlo simulations can be trusted. The talk will then concentrate on new numerical schemes for the well-known stochastic Lotka--Volterra model for interacting multi-species. This model has some typical features: highly nonlinear, positive solution and multi-dimensional. The known numerical methods including the tamed/truncated Euler-Maruyama (EM) applied to it do not preserve its positivity. The aim of this talk is to modify the truncated EM to establish a new positive preserving truncated EM (PPTEM).

The development of effective solvers for high frequency wave propagation problems, such as those described by the Helmholtz equation, presents significant challenges. One promising class of solvers for such problems are parallel domain decomposition methods, however, an appropriate coarse space is typically required in order to obtain robust behaviour (scalable with respect to the number of domains, weakly dependant on the wave number but also on the heterogeneity of the physical parameters). In this talk we introduce a coarse space based on generalised eigenproblems in the overlap (GenEO) for the Helmholtz equation. Numerical results within FreeFEM demonstrate convergence that is effectively independent of the wave number and contrast in the heterogeneous coefficient as well as good performance for minimal overlap.

In this talk, we will show how sharp bounds on the moments of the solutions to some stochastic heat equations can lead to various qualitative properties of the solutions. A major part of the method consists of approximating the solution by “independent quantities”. These quantities together with the moments bounds give us sharp almost sure properties of the solution.

In this talk I will review recent results on the analysis of shock-capturing-type methods applied to convection-dominated problems. The method of choice is a variant of the Algebraic Flux-Correction (AFC) scheme. This scheme has received some attention over the last two decades due to its very satisfactory numerical performance. Despite this attention, until very recently there was no stability and convergence analysis for it. Thus, the purpose of the works reviewed in this talk was to bridge that gap. The first step towards the full analysis of the method is a rewriting of it as a nonlinear edge-based diffusion method. This writing makes it possible to present a unified analysis of the different variants of it. So, minimal assumptions on the components of the method are stated in such a way that the resulting scheme satisfies the Discrete Maximum Principle (DMP) and is convergence. One property that will be discussed in detail is the linearity preservation. This property has been linked to the good performance of methods of this kind. We will discuss in detail its role and the impact of it in the overall convergence of the method. Time permitting, some results on a posteriori error estimation will also be presented. 
This talk will gather contributions with A. Allendes (UTFSM, Chile), E. Burman (UCL, UK), V. John (WIAS, Berlin), F. Karakatsani (Chester, UK), P. Knobloch (Prague, Czech Republic), and 
R. Rankin (U. of Nottingham, China).

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 the first part of the talk, I will describe a fluid-dynamical model for a "anti-surfactant" solution (such as salt dissolved in water) whose surface tension is an increasing function of bulk solvent concentration. In particular, I will show that this model is consistent with the standard model for surfactants, and predicts a novel instability for anti-surfactants not present for surfactants. Some further details are given in the recent paper by Conn et al. Phys. Rev. E 93 043121 (2016).

In the second part of the talk, I will formulate and analyse the governing equations for the flow of a thixotropic or antithixotropic fluid in a slowly varying channel. These equations are equivalent to the equations of classical lubrication theory for a Newtonian fluid, but incorporate the evolving microstructure of the fluid, described in terms of a scalar structure parameter. If time permits, I will seek draw some conclusions relevant to thixotropic flow in porous media. Some further details are given in the forthcoming paper by Pritchard et al. to appear in J Non-Newt. Fluid Mech (2016).

We propose a strategy which allows computing eigenvalue enclosures for the Maxwell operator by means of the finite element method. The origins of this strategy can be traced back to over 20 years ago. One of its main features lies in the fact that it can be implemented on any type of regular mesh (structured or otherwise) and any type of elements (nodal or otherwise). In the first part of the talk we formulate a general framework which is free from spectral pollution and allows estimation of eigenfunctions.

We then prove the convergence of the method, which implies precise convergence rates for nodal finite elements. Various numerical experiments on benchmark geometries, with and without symmetries, are reported.

We consider the problem of taking a matrix A and finding diagonal matrices D and E such that the rows and columns of B = DAE satisfy some specific constraints. Examples of constraints are that

* the row and column sums of B should all equal one;
* the norms of the rows and columns of B should all be equal;
* the row and column sums of B should take values specified by vectors p and q.

Simple iterative algorithms for solving these problems have been known for nearly a century. We provide a simple framework for describing these algorithms that allow us to develop robust convergence results and describe a straightforward approach to accelerate the rate of convergence.

We describe some of the diverse applications of balancing with examples from preconditioning, clustering, network analysis and psephology.

This is joint work with Kerem Akartunali (Strathclyde), Daniel Ruiz (ENSEEIHT, Toulouse) and Bora Ucar (ENS, Lyon).

In this talk I shall describe two rather different, but not entirely unrelated,

problems involving thin-film flow of a viscous fluid which I have found of interest

and which may have some application to a number of practical situations,

including condensation in heat exchangers and microfluidics.

The first problem,

which is joint work with Adam Leslie and Brian Duffy at the University of Strathclyde,

concerns the steady three-dimensional flow of a thin, slowly varying ring of fluid

on either the outside or the inside of a uniformly rotating large horizontal cylinder.

Specifically, we study ``full-ring'' solutions, corresponding to a ring of continuous,

finite and non-zero thickness that extends all the way around the cylinder.

These full-ring solutions may be thought of as a three-dimensional generalisation of

the ``full-film'' solutions described by Moffatt (1977) for the corresponding two-dimensional problem.

We describe the behaviour of both the critical and non-critical full-ring solutions.

In particular,

we show that, while for most values of the rotation speed and the load the azimuthal velocity is

in the same direction as the rotation of the cylinder, there is a region of parameter space close

to the critical solution for sufficiently small rotation speed in which backflow occurs in a

small region on the upward-moving side of the cylinder.

The second problem,

which is joint work with Phil Trinh and Howard Stone at Princeton University,

concerns a rigid plate moving steadily on the free surface of a thin film of fluid.

Specifically, we study two problems

involving a rigid flat (but not, in general, horizontal) plate:

the pinned problem, in which the upstream end of plate is pinned at a fixed position,

the fluid pressure at the upstream end of the plate takes a prescribed value and there is a free surface downstream of the plate, and

the free problem, in which the plate is freely floating and there are free surfaces both upstream and downstream of the plate.

For both problems, the motion of the fluid and the position of the plate

(and, in particular, its angle of tilt to the horizontal) depend in a non-trivial manner on the

competing effects of the relative motion of the plate and the substrate,

the surface tension of the free surface, and of the viscosity of the fluid,

together with the value of the prescribed pressure in the pinned case.

Specifically, for the pinned problem we show that,

depending on the value of an appropriately defined capillary number and on the value of the

prescribed fluid pressure, there can be either none, one, two or three equilibrium solutions

with non-zero tilt angle.

Furthermore, for the free problem we show that the solutions

with a horizontal plate (i.e.\ zero tilt angle) conjectured by Moriarty and Terrill (1996)

do not, in general, exist, and in fact there is a unique equilibrium solution with,

in general, a non-zero tilt angle for all values of the capillary number.

Finally, if time permits some preliminary results for an elastic plate will be presented.

Part of this work was undertaken while I was a

Visiting Fellow in the Department of Mechanical and Aerospace Engineering

in the School of Engineering and Applied Science at Princeton University, Princeton, USA.

Another part of this work was undertaken while I was a

Visiting Fellow in the Oxford Centre for Collaborative Applied Mathematics (OCCAM),

University of Oxford, United Kingdom.

This publication was based on work supported in part by Award No KUK-C1-013-04,

made by King Abdullah University of Science and Technology (KAUST).