MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

Numerical Methods for the Chemical Master Equation

The dynamics of biochemical networks can be described by a Markov jump process on a high-dimensional state space, with the corresponding probability distribution being the solution of the Chemical Master Equation (CME). In this thesis, adaptive wavelet methods for the time-dependent and stationary CME, as well as for the approximation of committor probabilities are devised. The methods are illustrated on multi-dimensional models with metastable solutions and large state spaces

Computational Tools for Large-Scale Linear Systems

While the theoretical analysis of linear dynamical systems with finite state-spaces is a mature topic, in situations where the underlying model has a large number of dimensions, modelers must turn to computational tools to better visualize and analyze the dynamic behavior of interest. In these situations, we are confronted with the Curse of Dimensionality: computational and storage complexity grows exponentially in the number of dimensions. This doctoral project focuses on two main classes of large-scale linear systems which arise in system biology. The Chemical Master Equation (CME) is a Fokker-Planck equation which describes the evolution of the probability mass function of a countable state space Markov process. Each state of the CME is labelled with an ordered S-tuple corresponding to one configuration of a well-mixed chemical system, where S is the number of distinct chemical species of interest. Even in cases where one only considers a projection of the CME to a finite subset of the states, one still must contend with the Curse of Dimensionality: the computational complexity grows exponentially in the number of chemical species. This dissertation describes a computational methodology for efficient solution of the CME which, in the best cases, will scale linearly in the number of chemical species. The second main class of high-dimensional problems requiring computational tools are coupled linear reaction-diffusion equations. For this class of models, we focus primarily on the computation of certain high-dimensional matrices which describe in a quantitative sense the input-to-state and state-to-output relationships. We describe algorithms for extracting useful information stored in these matrices and use this information to efficiently compute both reduced order models and open-loop control laws for steering the full system. A key feature of this approach is that the method is completely simulation or experiment free, in fact, in our numerical experiments, the computation of a reduced model or open-loop control law is an order of magnitude faster on a laptop than simulation of the full system on a 32 core node of a high-performance cluster. In both projects, the enabling computational technology is the recently proposed Tensor Train (TT) structured low-parametric representation of high-dimensional data. The TT-format effectively exploits low-rank structure of the "unfolding matrices" for compression and computational efficiency. Formally, the computational complexity of basic TT arithmetics scale linearly in the number of dimensions, potentially circumventing the curse of dimensionality. To demonstrate the effectiveness of this approach, we performed numerous numerical experiments whose results are reported here

Machine-Learning Methods for Computational Science and Engineering

The re-kindled fascination in machine learning (ML), observed over the last few decades, has also percolated into natural sciences and engineering. ML algorithms are now used in scientific computing, as well as in data-mining and processing. In this paper, we provide a review of the state-of-the-art in ML for computational science and engineering. We discuss ways of using ML to speed up or improve the quality of simulation techniques such as computational fluid dynamics, molecular dynamics, and structural analysis. We explore the ability of ML to produce computationally efficient surrogate models of physical applications that circumvent the need for the more expensive simulation techniques entirely. We also discuss how ML can be used to process large amounts of data, using as examples many different scientific fields, such as engineering, medicine, astronomy and computing. Finally, we review how ML has been used to create more realistic and responsive virtual reality applications

Machine-Learning Methods for Computational Science and Engineering

The re-kindled fascination in machine learning (ML), observed over the last few decades, has also percolated into natural sciences and engineering. ML algorithms are now used in scientific computing, as well as in data-mining and processing. In this paper, we provide a review of the state-of-the-art in ML for computational science and engineering. We discuss ways of using ML to speed up or improve the quality of simulation techniques such as computational fluid dynamics, molecular dynamics, and structural analysis. We explore the ability of ML to produce computationally efficient surrogate models of physical applications that circumvent the need for the more expensive simulation techniques entirely. We also discuss how ML can be used to process large amounts of data, using as examples many different scientific fields, such as engineering, medicine, astronomy and computing. Finally, we review how ML has been used to create more realistic and responsive virtual reality applications