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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.

This thesis contains three main parts. The first part concerns the derivation of an asymptotic expansion for the solution of Stokes resolvent problem with a small perturbation of the domain. Firstly, we verify the continuity of the solution with respect to the small perturbation via the stability of the density function. Secondly, we derive the asymptotic expansion ofthe solution, after deriving the expansion of the density function. The procedure is based on potential theory for Stokes problem in connection with boundary integral equation method, and geometric properties of the perturbed boundary. The main objective of the second part on this report, is to present a schematic way to derive high-order asymptotic expansions for both eigenvalues and eigenfunctions for the Stokes operator caused by small perturbationsof the boundary. Also, we rigorously derive an asymptotic formula which is in some sense dual to the leading-order term in the asymptotic expansion of the perturbations in the Stokes eigenvalues due to interface changes of the inclusion. The implementation of the boundary element method requires the evaluation of integrals with a singular integrand. A reliable andaccurate calculation of these integrals can in some cases be crucial and difficult. In the third part of this report we propose a method of evaluation of singular integrals based on recursive reductions of the dimension of the integration domain. It leads to a representation of the integralas a linear combination of one-dimensional integrals whose integrand is regular and that can be evaluated numerically and even explicitly. The Maxwell equation is used as a model equation, but these results can be used for the Laplace and the Helmholtz equations in 3-D.For the discretization of the domain we use planar triangles, so we evaluate integrals over the product of two triangles. The technique we have developped requires to distinguish between several geometric configurations.; La premième partie a pour but l’établissement d’un développement asymptotique pour la solution du problème de Stokes avec une petite perturbation du domaine. Dans ce travail, nous avons appliqué la théorie du potentiel. On a écrit les solutions du problème non-perturbé et du problème perturbé sous forme des opérateurs intégraux. En calculant la différence, et en utilisant des propriétés liées aux noyaux des opérateurs on a établi un développement asymptotiquede la solution.L’objectif principal de la deuxième partie de ce rapport est de déterminer les termes d’ordre élevé de l’expansion asymptotique des valeurs propres et fonctions propres pour l’opérateur de Stokes dues aux changements d’interface de l’inclusion. Dans la troisième partie, nous proposons une méthode pour l’évaluation des integrales singulières provenant de la mise en oeuvre de la méthode des éléments finis de frontière en électromagnetisme. La méthodeque nous adoptons consiste en une réduction récursive de la dimension du domained’intégration et aboutit à une représentation de l’intégrale sous la forme d’une combinaison linéaire d’intégrales mono-dimensionnelles dont l’intégrand est régulier et qui peuvent s’évaluer numériquement mais aussi explicitement. Pour la discrétisation du domaine, destriangles plans sont utilisés ; par conséquent, nous évaluons des intégrales sur le produit de deux triangles. La technique que nous avons développée nécessite de distinguer entre diverses configurations géométriques.

Particle suspensions occur in many situations in nature and industry. In this master’s thesis, the motion of a single rigid spheroidal particle immersed in Stokes flow is studied numerically using a boundary integral method and a new specialized quadrature method known as quadrature by expansion (QBX). This method allows the spheroid to be massless or inertial, and placed in any kind of underlying Stokesian flow. A parameter study of the QBX method is presented, together with validation cases for spheroids in linear shear flow and quadratic flow. The QBX method is able to compute the force and torque on the spheroid as well as the resulting rigid body motion with small errors in a short time, typically less than one second per time step on a regular desktop computer. Novel results are presented for the motion of an inertial spheroid in quadratic flow, where in contrast to linear shear flow the shear rate is not constant. It is found that particle inertia induces a translational drift towards regions in the fluid with higher shear rate. Partikelsuspensioner förekommer i många sammanhang i naturen och industrin. I denna masteruppsats studeras rörelsen hos en enstaka stel sfäroidisk partikel i Stokesflöde numeriskt med hjälp av en randintegralmetod och en ny specialiserad kvadraturmetod som kallas quadrature by expansion (QBX). Metoden fungerar för masslösa eller tröga sfäroider, som kan placeras i ett godtyckligt underliggande Stokesflöde. En parameterstudie av QBX-metoden presenteras, tillsammans med valideringsfall för sfäroider i linjärt skjuvflöde och kvadratiskt flöde. QBX-metoden kan beräkna kraften och momentet på sfäroiden samt den resulterande stelkroppsrörelsen med små fel på kort tid, typiskt mindre än en sekund per tidssteg på en vanlig persondator. Nya resultat presenteras för rörelsen hos en trög sfäroid i kvadratiskt flöde, där skjuvningen till skillnad från linjärt skjuvflöde inte är konstant. Det visar sig att partikeltröghet medför en drift i sidled mot områden i fluiden med högre skjuvning.

Particle suspensions occur in many situations in nature and industry. In this master’s thesis, the motion of a single rigid spheroidal particle immersed in Stokes flow is studied numerically using a boundary integral method and a new specialized quadrature method known as quadrature by expansion (QBX). This method allows the spheroid to be massless or inertial, and placed in any kind of underlying Stokesian flow. A parameter study of the QBX method is presented, together with validation cases for spheroids in linear shear flow and quadratic flow. The QBX method is able to compute the force and torque on the spheroid as well as the resulting rigid body motion with small errors in a short time, typically less than one second per time step on a regular desktop computer. Novel results are presented for the motion of an inertial spheroid in quadratic flow, where in contrast to linear shear flow the shear rate is not constant. It is found that particle inertia induces a translational drift towards regions in the fluid with higher shear rate. Partikelsuspensioner förekommer i många sammanhang i naturen och industrin. I denna masteruppsats studeras rörelsen hos en enstaka stel sfäroidisk partikel i Stokesflöde numeriskt med hjälp av en randintegralmetod och en ny specialiserad kvadraturmetod som kallas quadrature by expansion (QBX). Metoden fungerar för masslösa eller tröga sfäroider, som kan placeras i ett godtyckligt underliggande Stokesflöde. En parameterstudie av QBX-metoden presenteras, tillsammans med valideringsfall för sfäroider i linjärt skjuvflöde och kvadratiskt flöde. QBX-metoden kan beräkna kraften och momentet på sfäroiden samt den resulterande stelkroppsrörelsen med små fel på kort tid, typiskt mindre än en sekund per tidssteg på en vanlig persondator. Nya resultat presenteras för rörelsen hos en trög sfäroid i kvadratiskt flöde, där skjuvningen till skillnad från linjärt skjuvflöde inte är konstant. Det visar sig att partikeltröghet medför en drift i sidled mot områden i fluiden med högre skjuvning.

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.

Purpose – The purpose of this paper is to present a modified version of the hybrid Finite Element Method-Dirichlet Boundary Condition Iteration method for the solution of open-boundary skin effect problems. Design/methodology/approach – The modification consists of overlapping the truncation and the integration boundaries of the standard method, so that the integral equation becomes singular as in the well-known Finite Element Method-Boundary Element Method (FEM-BEM) method. The new method is called FEM-SDBCI. Assuming an unknown Dirichlet condition on the truncation boundary, the global algebraic system is constituted by the sparse FEM equations and by the dense integral equations, in which singularities arise. Analytical formulas are provided to compute these singular integrals. The global system is solved by means of a Generalized Minimal Residual iterative procedure. Findings – The proposed method leads to slightly less accurate numerical results than FEM-BEM, but the latter requires much more computing time. Practical implications – Then FEM-SDBCI appears more appropriate than FEM-BEM for applications which require a shorter computing time, for example in the stochastic optimization of electromagnetic devices. Originality/value – Note that FEM-SDBCI assumes a Dirichlet condition on the truncation boundary, whereas FEM-BEM assumes a Neumann one.

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

This thesis summarizes certain boundary element methods applied to some initial and boundary value problems. Our model problem is the two-dimensional homogeneous heat conduction problem with vanishing initial data. We use the heat potential representation of the solution. The given boundary conditions, as well as the choice of the representation formula, yield various boundary integral equations. For the sake of simplicity, we use the direct boundary integral approach, where the unknown boundary density appearing in the boundary integral equation is a quantity of physical meaning. We consider two different sets of boundary conditions, the Dirichlet problem, where the boundary temperature is given and the Neumann problem, where the heat flux across the boundary is given. Even a nonlinear Neumann condition satisfying certain monotonicity and growth conditions is possible. The approach yields a nonlinear boundary integral equation of the second kind. In the stationary case, the model problem reduces to a potential problem with a nonlinear Neumann condition. We use the spaces of smoothest splines as trial functions. The nonlinearity is approximated by using the L²-orthogonal projection. The resulting collocation scheme retains the optimal L²-convergence. Numerical experiments are in agreement with this result. This approach generalizes to the time dependent case. The trial functions are tensor products of piecewise linear and piecewise constant splines. The proposed projection method uses interpolation with respect to the space variable and the orthogonal projection with respect to the time variable. Compared to the Galerkin method, this approach simplifies the realization of the discrete matrix equations. In addition, the rate of the convergence is of optimal order. On the other hand, the Dirichlet problem, where the boundary temperature is given, leads to a single layer heat operator equation of the first kind. In the first approach, we use tensor products of piecewise linear splines as trial functions with collocation at the nodal points. Stability and suboptimal L²-convergence of the method were proved in the case of a circular domain. Numerical experiments indicate the expected quadratic L²-convergence. Later, a Petrov-Galerkin approach was proposed, where the trial functions were tensor products of piecewise linear and piecewise constant splines. The resulting approximative scheme is stable and convergent. The analysis has been carried out in the cases of the single layer heat operator and the hypersingular heat operator. The rate of the convergence with respect to the L²-norm is also here of suboptimal order.

In this article we present and analyze the modified quadrature method for strongly elliptic boundary integral equations of negative order on smooth closed curves. The method employs a modification that extracts the singularity appearing in the kernel or in some of its derivatives. Moreover, the composite trapezoidal rule is used for approximation of the integral. The modified quadrature method is proved to have the maximal rate O(h−β+1) of convergence for operators of order β < 0 in general, and O(h−β+2) for a large class of operators appearing in applications. Some numerical experiments confirming our theoretical results are also presented.