Cumulative reports and publications through December 31, 1990

This document contains a complete list of ICASE reports. Since ICASE reports are intended to be preprints of articles that will appear in journals or conference proceedings, the published reference is included when it is available

Time- or Space-Dependent Coefficient Recovery in Parabolic Partial Differential Equation for Sensor Array in the Biological Computing

This study presents numerical schemes for solving a parabolic partial differential equation with a time- or space-dependent coefficient subject to an extra measurement. Through the extra measurement, the inverse problem is transformed into an equivalent nonlinear equation which is much simpler to handle. By the variational iteration method, we obtain the exact solution and the unknown coefficients. The results of numerical experiments and stable experiments imply that the variational iteration method is very suitable to solve these inverse problems

Cumulative reports and publications through December 31, 1988

This document contains a complete list of ICASE Reports. Since ICASE Reports are intended to be preprints of articles that will appear in journals or conference proceedings, the published reference is included when it is available

Cumulative reports and publications through December 31, 1989

A complete list of reports from the Institute for Computer Applications in Science and Engineering (ICASE) is presented. The major categories of the current ICASE research program are: numerical methods, with particular emphasis on the development and analysis of basic numerical algorithms; control and parameter identification problems, with emphasis on effectual numerical methods; computational problems in engineering and the physical sciences, particularly fluid dynamics, acoustics, structural analysis, and chemistry; computer systems and software, especially vector and parallel computers, microcomputers, and data management. Since ICASE reports are intended to be preprints of articles that will appear in journals or conference proceedings, the published reference is included when it is available

Bivariate pseudospectral collocation algorithms for nonlinear partial differential equations.

Doctor of Philosophy in Applied Matheatics. University of KwaZulu-Natal, Pietermaritzburg 2016.Abstract available in PDF file

A numerical framework for solving PDE-constrained optimization problems from multiscale particle dynamics

In this thesis, we develop accurate and efficient numerical methods for solving partial differential equation (PDE) constrained optimization problems arising from multiscale particle dynamics, with the aim of producing a desired time-dependent state at the minimal cost. A PDE-constrained optimization problem seeks to move one or more state variables towards a desired state under the influence of one or more control variables, and a set of constraints that are described by PDEs governing the behaviour of the variables. In particular, we consider problems constrained by one-dimensional and two-dimensional advection-diffusion problems with a non-local integral term, such as the associated mean-field limit Fokker-Planck equation of the noisy Hegselmann-Krause opinion dynamics model. We include additional bound constraints on the control variable for the opinion dynamics problem. Lastly, we consider constraints described by a two-dimensional robot swarming model made up of a system of advection-diffusion equations with additional linear and integral terms. We derive continuous Lagrangian first-order optimality conditions for these problems and solve the resulting systems numerically for the optimized state and control variables. Each of these problems, combined with Dirichlet, no-flux, or periodic boundary conditions, present unique challenges that require versatility of the numerical methods devised. Our numerical framework is based on a novel combination of four main components: (i) a discretization scheme, in both space and time, with the choice of pseudospectral or fi nite difference methods; (ii) a forward problem solver that is implemented via a differential-algebraic equation solver; (iii) an optimization problem solver that is a choice between a fi xed-point solver, with or without Armijo-Wolfe line search conditions, a Newton-Krylov algorithm, or a multiple shooting scheme, and; (iv) a primal-dual active set strategy to tackle additional bound constraints on the control variable. Pseudospectral methods efficiently produce highly accurate solutions by exploiting smoothness in the solutions, and are designed to perform very well with dense, small matrix systems. For a number of problems, we take advantage of the exponential convergence of pseudospectral methods by discretising in this way not only in space, but also in time. The alternative fi nite difference method performs comparatively well when non-smooth bound constraints are added to the optimization problem. A differential{algebraic equation solver works out the discretized PDE on the interior of the domain, and applies the boundary conditions as algebraic equations. This ensures generalizability of the numerical method, as one does not need to explicitly adapt the numerical method for different boundary conditions, only to specify different algebraic constraints that correspond to the boundary conditions. A general fixed-point or sweeping method solves the system of equations iteratively, and does not require the analytic computation of the Jacobian. We improve the computational speed of the fi xed-point solver by including an adaptive Armijo-Wolfe type line search algorithm for fixed-point problems. This combination is applicable to problems with additional bound constraints as well as to other systems for which the regularity of the solution is not sufficient to be exploited by the spectral-in-space-and-time nature of the Newton-Krylov approach. The recently devised Newton-Krylov scheme is a higher-order, more efficient optimization solver which efficiently describes the PDEs and the associated Jacobian on the discrete level, as well as solving the resulting Newton system efficiently via a bespoke preconditioner. However, it requires the computation of the Jacobian, and could potentially be more challenging to adapt to more general problems. Multiple shooting solves an initial-value problem on sections of the time interval and imposes matching conditions to form a solution on the whole interval. The primal-dual active set strategy is used for solving our non-linear and non-local optimization problems obtained from opinion dynamics problems, with pointwise non-equality constraints. This thesis provides a numerical framework that is versatile and generalizable for solving complex PDE-constrained optimization problems from multiscale particle dynamic

Computational modelling and optimal control of interacting particle systems: connecting dynamic density functional theory and PDE-constrained optimization

Processes that can be described by systems of interacting particles are ubiquitous in nature, society, and industry, ranging from animal flocking, the spread of diseases, and formation of opinions to nano-filtration, brewing, and printing. In real-world applications it is often relevant to not only model a process of interest, but to also optimize it in order to achieve a desired outcome with minimal resources, such as time, money, or energy. Mathematically, the dynamics of interacting particle systems can be described using Dynamic Density Functional Theory (DDFT). The resulting models are nonlinear, nonlocal partial differential equations (PDEs) that include convolution integral terms. Such terms also enter the naturally arising no-flux boundary conditions. Due to the nonlocal, nonlinear nature of such problems they are challenging both to analyse and solve numerically. In order to optimize processes that are modelled by PDEs, one can apply tools from PDE-constrained optimization. The aim here is to drive a quantity of interest towards a target state by varying a control variable. This is constrained by a PDE describing the process of interest, in which the control enters as a model parameter. Such problems can be tackled by deriving and solving the (first-order) optimality system, which couples the PDE model with a second PDE and an algebraic equation. Solving such a system numerically is challenging, since large matrices arise in its discretization, for which efficient solution strategies have to be found. Most work in PDE-constrained optimization addresses problems in which the control is applied linearly, and which are constrained by local, often linear PDEs, since introducing nonlinearity significantly increases the complexity in both the analysis and numerical solution of the optimization problem. However, in order to optimize real-world processes described by nonlinear, nonlocal DDFT models, one has to develop an optimal control framework for such models. The aim is to drive the particles to some desired distribution by applying control either linearly, through a particle source, or bilinearly, though an advective field. The optimization process is constrained by the DDFT model that describes how the particles move under the influence of advection, diffusion, external forces, and particle–particle interactions. In order to tackle this, the (first-order) optimality system is derived, which, since it involves nonlinear (integro-)PDEs that are coupled nonlocally in space and time, is significantly harder than in the standard case. Novel numerical methods are developed, effectively combining pseudospectral methods and iterative solvers, to efficiently and accurately solve such a system. In a next step this framework is extended so that it can capture and optimize industrially relevant processes, such as brewing and nano-filtration. In order to do so, extensions to both the DDFT model and the numerical method are made. Firstly, since industrial processes often involve tubes, funnels, channels, or tanks of various shapes, the PDE model itself, as well as the optimization problem, need to be solved on complicated domains. This is achieved by developing a novel spectral element approach that is compatible with both the PDE solver and the optimal control framework. Secondly, many industrial processes, such as nano-filtration, involve more than one type of particle. Therefore, the DDFT model is extended to describe multiple particle species. Finally, depending on the application of interest, additional physical effects need to be included in the model. In this thesis, to model sedimentation processes in brewing, the model is modified to capture volume exclusion effects

Geometric Numerical Integration

The subject of this workshop was numerical methods that preserve geometric properties of the flow of an ordinary or partial differential equation. This was complemented by the question as to how structure preservation affects the long-time behaviour of numerical methods