Developing Clean Technology through Approximate Solutions of Mathematical Models

In this paper, the role of mathematical modeling in the development of clean technology has been considered. One method each for obtaining approximate solutions of mathematical models by ordinary differential equations and partial differential equations respectively arising from the modeling of systems and physical phenomena has been considered. The construction of continuous hybrid methods for the numerical approximation of the solutions of initial value problems of ordinary differential equations as well as homotopy analysis method, an approximate analytical method, for the solution of nonlinear partial differential equations are discussed

WavePacket: A Matlab package for numerical quantum dynamics. II: Open quantum systems, optimal control, and model reduction

WavePacket is an open-source program package for numeric simulations in quantum dynamics. It can solve time-independent or time-dependent linear Schr\"odinger and Liouville-von Neumann-equations in one or more dimensions. Also coupled equations can be treated, which allows, e.g., to simulate molecular quantum dynamics beyond the Born-Oppenheimer approximation. Optionally accounting for the interaction with external electric fields within the semi-classical dipole approximation, WavePacket can be used to simulate experiments involving tailored light pulses in photo-induced physics or chemistry. Being highly versatile and offering visualization of quantum dynamics 'on the fly', WavePacket is well suited for teaching or research projects in atomic, molecular and optical physics as well as in physical or theoretical chemistry. Building on the previous Part I which dealt with closed quantum systems and discrete variable representations, the present Part II focuses on the dynamics of open quantum systems, with Lindblad operators modeling dissipation and dephasing. This part also describes the WavePacket function for optimal control of quantum dynamics, building on rapid monotonically convergent iteration methods. Furthermore, two different approaches to dimension reduction implemented in WavePacket are documented here. In the first one, a balancing transformation based on the concepts of controllability and observability Gramians is used to identify states that are neither well controllable nor well observable. Those states are either truncated or averaged out. In the other approach, the H2-error for a given reduced dimensionality is minimized by H2 optimal model reduction techniques, utilizing a bilinear iterative rational Krylov algorithm

Efficient Numerical Solution of Large Scale Algebraic Matrix Equations in PDE Control and Model Order Reduction

Matrix Lyapunov and Riccati equations are an important tool in mathematical systems theory. They are the key ingredients in balancing based model order reduction techniques and linear quadratic regulator problems. For small and moderately sized problems these equations are solved by techniques with at least cubic complexity which prohibits their usage in large scale applications. Around the year 2000 solvers for large scale problems have been introduced. The basic idea there is to compute a low rank decomposition of the quadratic and dense solution matrix and in turn reduce the memory and computational complexity of the algorithms. In this thesis efficiency enhancing techniques for the low rank alternating directions implicit iteration based solution of large scale matrix equations are introduced and discussed. Also the applicability in the context of real world systems is demonstrated. The thesis is structured in seven central chapters. After the introduction chapter 2 introduces the basic concepts and notations needed as fundamental tools for the remainder of the thesis. The next chapter then introduces a collection of test examples spanning from easily scalable academic test systems to badly conditioned technical applications which are used to demonstrate the features of the solvers. Chapter four and five describe the basic solvers and the modifications taken to make them applicable to an even larger class of problems. The following two chapters treat the application of the solvers in the context of model order reduction and linear quadratic optimal control of PDEs. The final chapter then presents the extensive numerical testing undertaken with the solvers proposed in the prior chapters. Some conclusions and an appendix complete the thesis

A numerical comparison of solvers for large-scale, continuous-time algebraic Riccati equations and LQR problems

In this paper, we discuss numerical methods for solving large-scale continuous-time algebraic Riccati equations. These methods have been the focus of intensive research in recent years, and significant progress has been made in both the theoretical understanding and efficient implementation of various competing algorithms. There are several goals of this manuscript: first, to gather in one place an overview of different approaches for solving large-scale Riccati equations, and to point to the recent advances in each of them. Second, to analyze and compare the main computational ingredients of these algorithms, to detect their strong points and their potential bottlenecks. And finally, to compare the effective implementations of all methods on a set of relevant benchmark examples, giving an indication of their relative performance

Krylov subspace techniques for model reduction and the solution of linear matrix equations

This thesis focuses on the model reduction of linear systems and the solution of large scale linear matrix equations using computationally efficient Krylov subspace techniques. Most approaches for model reduction involve the computation and factorization of large matrices. However Krylov subspace techniques have the advantage that they involve only matrix-vector multiplications in the large dimension, which makes them a better choice for model reduction of large scale systems. The standard Arnoldi/Lanczos algorithms are well-used Krylov techniques that compute orthogonal bases to Krylov subspaces and, by using a projection process on to the Krylov subspace, produce a reduced order model that interpolates the actual system and its derivatives at infinity. An extension is the rational Arnoldi/Lanczos algorithm which computes orthogonal bases to the union of Krylov subspaces and results in a reduced order model that interpolates the actual system and its derivatives at a predefined set of interpolation points. This thesis concentrates on the rational Krylov method for model reduction. In the rational Krylov method an important issue is the selection of interpolation points for which various techniques are available in the literature with different selection criteria. One of these techniques selects the interpolation points such that the approximation satisfies the necessary conditions for H2 optimal approximation. However it is possible to have more than one approximation for which the necessary optimality conditions are satisfied. In this thesis, some conditions on the interpolation points are derived, that enable us to compute all approximations that satisfy the necessary optimality conditions and hence identify the global minimizer to the H2 optimal model reduction problem. It is shown that for an H2 optimal approximation that interpolates at m interpolation points, the interpolation points are the simultaneous solution of m multivariate polynomial equations in m unknowns. This condition reduces to the computation of zeros of a linear system, for a first order approximation. In case of second order approximation the condition is to compute the simultaneous solution of two bivariate polynomial equations. These two cases are analyzed in detail and it is shown that a global minimizer to the H2 optimal model reduction problem can be identified. Furthermore, a computationally efficient iterative algorithm is also proposed for the H2 optimal model reduction problem that converges to a local minimizer. In addition to the effect of interpolation points on the accuracy of the rational interpolating approximation, an ordinary choice of interpolation points may result in a reduced order model that loses the useful properties such as stability, passivity, minimum-phase and bounded real character as well as structure of the actual system. Recently in the literature it is shown that the rational interpolating approximations can be parameterized in terms of a free low dimensional parameter in order to preserve the stability of the actual system in the reduced order approximation. This idea is extended in this thesis to preserve other properties and combinations of them. Also the concept of parameterization is applied to the minimal residual method, two-sided rational Arnoldi method and H2 optimal approximation in order to improve the accuracy of the interpolating approximation. The rational Krylov method has also been used in the literature to compute low rank approximate solutions of the Sylvester and Lyapunov equations, which are useful for model reduction. The approach involves the computation of two set of basis vectors in which each vector is orthogonalized with all previous vectors. This orthogonalization becomes computationally expensive and requires high storage capacity as the number of basis vectors increases. In this thesis, a restart scheme is proposed which restarts without requiring that the new vectors are orthogonal to the previous vectors. Instead, a set of two new orthogonal basis vectors are computed. This reduces the computational burden of orthogonalization and the requirement of storage capacity. It is shown that in case of Lyapunov equations, the approximate solution obtained through the restart scheme approaches monotonically to the actual solution