Search

Non-equilibrium systems, or active systems, span virtually all disciplines of science. While the underlying processes driving these systems out of equilibrium (may that be thermal, chemical or another type of equilibrium) may be entirely unrelated, it is often the case that the same or a similar mathematical model is capable of describing either of the systems. The particular active systems we are concerned with here are (a) the microscopic interactions of active matter in colloidal physics, and (b) the epidemiological dynamics of infectious diseases in human populations. Aside from their manifestly out-of-equilibrium character, both of these systems comprise fundamentally stochastic processes. Thermal fluctuations at energy-scales comparable to the ones of colloidal dynamics lead to Brownian motion in the former. The latter is governed by both pathogen- and data-specific noise. Remarkably, applying a functional central limit theorem, we find that to linear order both of these active systems can be described by the same process – the Ornstein-Uhlenbeck process. This is despite the fact that colloidal physics is concerned with microscopic scales, while epidemiological models operate on the scale of human populations. Using this, we apply methods from Bayesian inference to quantify the respective underlying uncertainties from stochastic trajectories alone. In the following, we briefly introduce the two parts of this thesis on active systems: Active matter and Epidemiology. Abstract Active matter consists of mesoscopic self-propelled units, active particles, which on an individual level can metabolise energy into systematic movement. The mechanics and statistical mechanics of a suspension of active particles are determined by the traction (force per unit area) on their surfaces, arising from long-ranged hydrodynamic interactions at low Reynold's number. Here, we present a solution of the direct boundary integral equation for the traction on spherical active particles interacting with each other and their surroundings. For a single particle away from any boundaries this solution is exact, as both single- and double-layer integral operators can be simultaneously diagonalised in a basis of irreducible tensor spherical harmonics. The solution, thus, can be presented as an infinite number of linear relations between the harmonic coefficients of the traction and the velocity at the boundary of the particle. These generalise Stokes laws for the force and torque. In confinement, the dynamics of an active particle are uniquely defined by its mobility matrices, fully characterising hydrodynamic interactions due to the geometry of the system alone, and so-called propulsion tensors, arising from activity. Using Jacobi's method, we analytically derive iterative solutions for these quantities, once again starting from the direct boundary integral formulation of Stokes flow. We exemplify our results by explicitly providing the dynamics of an axisymmetric squirmer in the vicinity of a plane interface. For the computationally efficient simulation of many-body systems we develop a novel linear solver, based on a Krylov subspace method, that retains the true many-body character of hydrodynamic interactions. Finally, at non-zero temperature we consider thermal fluctuations of the suspending fluid, introducing stochasticity to the model above. We review and further develop a fast Bayesian method for parameter inference and model selection for a class of active systems that can be approximated by a stationary Ornstein-Uhlenbeck process. Such systems are said to be in a non-equilibrium steady state. A key aim of this study is to understand how one may distinguish between reversible and irreversible dynamics from information contained in stochastic trajectories alone. Abstract Epidemiological data is beset by uncertainties about the underlying epidemiological processes, and the surveillance process through which the data is acquired. We present a Bayesian inference methodology that quantifies these uncertainties, for epidemics that are modelled by non-stationary, continuous-time Markov population processes. The efficiency of the method derives from a functional central limit theorem approximation of the likelihood, akin to van Kampen's system size expansion in physics, and valid for large well-mixed populations. The Ornstein-Uhlenbeck process arising from this approximation can once again be studied using Bayesian inference, including maximum a posteriori parameter estimates, computation of the model evidence, and the determination of parameter sensitivities via the Fisher information matrix. Our methodology is implemented in PyRoss, an open-source platform for analysis of epidemiological compartment models.

There has been considerable attention given in recent years to the problem of extending finite and boundary element-based analysis of Helmholtz problems to higher frequencies. One approach is the Partition of Unity Method, which has been applied successfully to boundary integral solutions of Helmholtz problems, providing significant accuracy benefits while simultaneously reducing the required number of degrees of freedom for a given accuracy. These benefits accrue at the cost of the requirement to perform some numerically intensive calculations in order to evaluate boundary integrals of highly oscillatory functions. In this paper we adapt the numerical steepest descent method to evaluate these integrals for two-dimensional problems. The approach is successful in reducing the computational effort for most integrals encountered. The paper includes some numerical features that are important for successful practical implementation of the algorithm.

The domain decomposition method proposed by Schwarz [1] is gaining significance in view of the possibilities for parallel implementation. In this paper, we apply the domain decomposition method to the biharmonic equation in two overlapping discs. In each subdomain, a boundary integral formula is used to obtain the solution. Unlike the finite element or finite difference methods where the iterations have to be performed on the entire domain, the iterations of the boundary integral formulation need to be done only at the discretized points of the boundaries. Another aspect of the paper is that, in order to handle the operations with dense matrices arising from the boundary integral formulation, the discrete wavelet transform is used to compress the matrices which results in reduced computations without loss of accuracy. Examples involving a Laplace equation in an L-shaped region and a domain bounded by a cardioid and a circle are also illustrated.

In this paper we analyze a family of full discretizations of spline Galerkin methods for a class of systems of boundary integral equations of the first kind with logarithmic principal part. We prove the existence of an asymptotic expansion of the error of the Galerkin and the optimal order Galerkin collocation method. We finally derive asymptotic expansions for some common postprocessings of the solutions, both exactly and under the effect of additional discretization. Some examples where these techniques apply are provided.

Standard implementations of the fast multipole method, which compute fields due to point sources or dipoles, cannot be used to accurately evaluate the single- and double-layer potentials of potential theory close to the boundary, or on the boundary when the boundary curves back on itself. We describe the modifications necessary to accurately evaluate layer potentials in two dimensions, which include quadrature rules for the short-range contributions to the field, continuous multipole moments for long-range contributions, and a more complex bookkeeping procedure. We give formulae for second-, third-, and fourth-order methods. We show tests to verify the correctness of the method and numerical results which demonstrate the usefulness of the method for evaluating layer potentials near the boundary.

Three-dimensional Laplace problems with a boundary straight-edge singularity caused by two intersecting flat planes are considered. The solution in the neighbourhood of the straight edge can be expressed as an asymptotic expansion involving the eigenpairs of the analogous two-dimensional problem in polar coordinates, which have as coefficients the so-called edge flux intensity functions (EFIFs). The EFIFs are functions of the axial coordinate, the higher derivatives of which appear in an inner infinite series in the expansion. The objective of this work is to extend the singular function boundary integral method (SFBIM), developed for two-dimensional elliptic problems with point boundary singularities [G.C. Georgiou, L. Olson, G. Smyrlis, A singular function boundary integral method for the Laplace equation, Commun. Numer. Methods Eng. 12 (1996) 127-134] for solving the above problem and directly extracting the EFIFs. Approximating the latter by either piecewise constant or linear polynomials eliminates the inner infinite series in the local expansion and allows the straightforward extension of the SFBIM. As in the case of two-dimensional problems, the solution is approximated by the leading terms of the local asymptotic solution expansion. These terms are also used to weight the governing harmonic equation in the Galerkin sense. The resulting discretized equations are reduced to boundary integrals by means of the divergence theorem. The Dirichlet boundary conditions are then weakly enforced by means of Lagrange multipliers. The values of the latter are calculated together with the coefficients of the EFIFs. The SFBIM is applied to a test problem exhibiting fast convergence of order k + 1 (k being the order of the approximation of the EFIFs) in the L 2-norm and leading to accurate estimates for the EFIFs. © 2012 Elsevier Inc. All rights reserved. 219 3 1073 1081 Cited By :2

We present a Singular Function Boundary Integral Method (SFBIM) for solving elliptic problems with a boundary singularity. In this method the solution is approximated by the leading terms of the asymptotic solution expansion, which exists near the singular point and is known for many benchmark problems. The unknowns to be calculated are the singular coefficients, i.e. the coefficients in the asymptotic expansion, also called (generalized) stress intensity factors. The discretized Galerkin equations are reduced to boundary integrals by means of Green's theorem and the Dirichlet boundary conditions are weakly enforced by means of Lagrange multipliers, the values of which are introduced as additional unknowns in the resulting linear system. The method is described for two-dimensional Laplacian problems for which the analysis establishes exponential rates of convergence as the number of terms in the asymptotic expansion is increased. We also discuss the extension of the method to three-dimensional Laplacian problems with exhibits edge singularities. 1 2599 2608 Sponsors: The Netherlands Organization for Scientific Research (NWO) The Royal Netherlands Academy of Arts and Sciences (KNAW) Elsevier B.V. The University of Amsterdam Conference code: 83058 Cited By :1