Modelling techniques for biological systems

The objective of this investigation has been to develop and evaluate techniques which are appropriate to the modelling and simulation of biological reaction system behaviour. The model used as the basis for analysis of modelling and simulation techniques is a reduced version of the biological model proposed by the IAWPRC Task Group for mathematical modell ing in wastewater treatment design. This limited model has the advantage of being easily manageable in terms of analysis and presentation of the simulation techniQues whilst at the same time incorporating a range of features encountered with biological growth applications in general. Because a model may incorporate a number of different components and large number of biological conversion processes, a convenient method of presentation was found to be a matrix format. The matrix representation ensures clarity as to what compounds, processes and react ion terms are to be incorporated and allows easy comparison of different models. In addition, it facilitates transforming the model into a computer program. Simulation of the system response first involves specifying the reactor configuration and flow patterns. With this information fixed, mass balances for each compound in each reactor can be completed. These mass balances constitute a set of simultaneous non-linear differential and algebraic eQuations which, when solved, characterise the system behaviour

Adapting the interior point method for the solution of LPs on serial, coarse grain parallel and massively parallel computers

In this paper we describe a unified scheme for implementing an interior point algorithm (IPM) over a range of computer architectures. In the inner iteration of the IPM a search direction is computed using Newton's method. Computationally this involves solving a sparse symmetric positive definite (SSPD) system of equations. The choice of direct and indirect methods for the solution of this system, and the design of data structures to take advantage of serial, coarse grain parallel and massively parallel computer architectures, are considered in detail. We put forward arguments as to why integration of the system within a sparse simplex solver is important and outline how the system is designed to achieve this integration

Jacobian-free diagonal Newton's method for solving nonlinear systems with singular Jacobian

The basic requirement of Newton’s method in solving systems of nonlinear equations is, the Jacobian must be non-singular. This condition restricts to some extent the application of Newton method. In this paper we present a modification of Newton’s method for systems of nonlinear equations where the Jacobian is singular. This is made possible by approximating the Jacobian inverse into a diagonal matrix by means of variational techniques. The anticipation of our approach is to bypass the point in which the Jacobian is singular. The local convergence of the proposed method has been proven under suitable assumptions. Numerical experiments are carried out which show that, the proposed method is very encouraging

Fast and stable contour integration for high order divided differences via elliptic functions

In this paper, we will present a new method for evaluating high order divided differences for certain classes of analytic, possibly, operator valued functions. This is a classical problem in numerical mathematics but also arises in new applications such as, e.g., the use of generalized convolution quadrature to solve retarded potential integral equations. The functions which we will consider are allowed to grow exponentially to the left complex half plane, polynomially to the right half plane and have an oscillatory behaviour with increasing imaginary part. The interpolation points are scattered in a large real interval. Our approach is based on the representation of divided differences as contour integral and we will employ a subtle parameterization of the contour in combination with a quadrature approximation by the trapezoidal rule

Sparse representation-based SAR imaging

There is increasing interest in using synthetic aperture radar (SAR) images in automated target recognition and decision-making tasks. The success of such tasks depends on how well the reconstructed SAR images exhibit certain features of the underlying scene. Based on the observation that typical underlying scenes usually exhibit sparsity in terms of such features, we develop an image formation method which formulates the SAR imaging problem as a sparse signal representation problem. Sparse signal representation, which has mostly been exploited in real-valued problems, has many capabilities such as superresolution and feature enhancement for various reconstruction and recognition tasks. However, for problems of complex-valued nature, such as SAR, a key challenge is how to choose the dictionary and the representation scheme for effective sparse representation. Since we are usually interested in features of the magnitude of the SAR reflectivity field, our new approach is designed to sparsely represent the magnitude of the complex-valued scattered field. This turns the image reconstruction problem into a joint optimization problem over the representation of magnitude and phase of the underlying field reflectivities. We develop the mathematical framework for this method and propose an iterative solution for the corresponding joint optimization problem. Our experimental results demonstrate the superiority of this method over previous approaches in terms of both producing high quality SAR images as well as exhibiting robustness to uncertain or limited data

Sparse representation-based synthetic aperture radar imaging

There is increasing interest in using synthetic aperture radar (SAR) images in automated target recognition and decision-making tasks. The success of such tasks depends on how well the reconstructed SAR images exhibit certain features of the underlying scene. Based on the observation that typical underlying scenes usually exhibit sparsity in terms of such features, we develop an image formation method which formulates the SAR imaging problem as a sparse signal representation problem. Sparse signal representation, which has mostly been exploited in real-valued problems, has many capabilities such as superresolution and feature enhancement for various reconstruction and recognition tasks. However, for problems of complex-valued nature, such as SAR, a key challenge is how to choose the dictionary and the representation scheme for effective sparse representation. Since we are usually interested in features of the magnitude of the SAR reflectivity field, our new approach is designed to sparsely represent the magnitude of the complex-valued scattered field. This turns the image reconstruction problem into a joint optimization problem over the representation of magnitude and phase of the underlying field reflectivities. We develop the mathematical framework for this method and propose an iterative solution for the corresponding joint optimization problem. Our experimental results demonstrate the superiority of this method over previous approaches in terms of both producing high quality SAR images as well as exhibiting robustness to uncertain or limited data

Data Mining with Newton\u27s Method.

Capable and well-organized data mining algorithms are essential and fundamental to helpful, useful, and successful knowledge discovery in databases. We discuss several data mining algorithms including genetic algorithms (GAs). In addition, we propose a modified multivariate Newton\u27s method (NM) approach to data mining of technical data. Several strategies are employed to stabilize Newton\u27s method to pathological function behavior. NM is compared to GAs and to the simplex evolutionary operation algorithm (EVOP). We find that GAs, NM, and EVOP all perform efficiently for well-behaved global optimization functions with NM providing an exponential improvement in convergence rate. For local optimization problems, we find that GAs and EVOP do not provide the desired convergence rate, accuracy, or precision compared to NM for technical data. We find that GAs are favored for their simplicity while NM would be favored for its performance

Kontinuierliche Optimierung und Industrieanwendungen

[no abstract available