Research in applied mathematics, numerical analysis, and computer science

Research conducted at the Institute for Computer Applications in Science and Engineering (ICASE) in applied mathematics, numerical analysis, and computer science is summarized and abstracts of published reports are presented. The major categories of the ICASE research program are: (1) numerical methods, with particular emphasis on the development and analysis of basic numerical algorithms; (2) control and parameter identification; (3) computational problems in engineering and the physical sciences, particularly fluid dynamics, acoustics, and structural analysis; and (4) computer systems and software, especially vector and parallel computers

On the solution of the Riccati differential equation arising from the LQ optimal control problem

In this paper we consider the matrix Riccati differential equation (RDE) that arises from linear-quadratic (LQ) optimal control problems. In particular, we establish explicit closed formulae for the solution of the RDE with a terminal condition using particular solutions of the associated algebraic Riccati equation. We discuss how these formulae change as assumptions are progressively weakened. An application to LQ optimal control is briefly analysed

Iterative and doubling algorithms for Riccati-type matrix equations: a comparative introduction

We review a family of algorithms for Lyapunov- and Riccati-type equations which are all related to each other by the idea of \emph{doubling}: they construct the iterate $Q_k = X_{2^k}$ of another naturally-arising fixed-point iteration $(X_h)$ via a sort of repeated squaring. The equations we consider are Stein equations $X - A^*XA=Q$, Lyapunov equations $A^*X+XA+Q=0$, discrete-time algebraic Riccati equations $X=Q+A^*X(I+GX)^{-1}A$, continuous-time algebraic Riccati equations $Q+A^*X+XA-XGX=0$, palindromic quadratic matrix equations $A+QY+A^*Y^2=0$, and nonlinear matrix equations $X+A^*X^{-1}A=Q$. We draw comparisons among these algorithms, highlight the connections between them and to other algorithms such as subspace iteration, and discuss open issues in their theory.Comment: Review article for GAMM Mitteilunge

A structure-preserving doubling algorithm for Lur'e equations

We introduce a numerical method for the numerical solution of the Lur'e equations, a system of matrix equations that arises, for instance, in linear-quadratic infinite time horizon optimal control. We focus on small-scale, dense problems. Via a Cayley transformation, the problem is transformed to the discrete-time case, and the structural infinite eigenvalues of the associated matrix pencil are deflated. The deflated problem is associated with a symplectic pencil with several Jordan blocks of eigenvalue 1 and even size, which arise from the nontrivial Kronecker chains at infinity of the original problem. For the solution of this modified problem, we use the structure-preserving doubling algorithm. Implementation issues such as the choice of the parameter γ in the Cayley transform are discussed. The most interesting feature of this method, with respect to the competing approaches, is the absence of arbitrary rank decisions, which may be ill-posed and numerically troublesome. The numerical examples presented confirm the effectiveness of this method

Activities of the Institute for Computer Applications in Science and Engineering (ICASE)

Research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, numerical analysis, and computer science during the period October 1, 1984 through March 31, 1985 is summarized

Efficient sensitivity analysis of chaotic systems and applications to control and data assimilation

Sensitivity analysis is indispensable for aeronautical engineering applications that require optimisation, such as flow control and aircraft design. The adjoint method is the standard approach for sensitivity analysis, but it cannot be used for chaotic systems. This is due to the high sensitivity of the system trajectory to input perturbations; a characteristic of many turbulent systems. Although the instantaneous outputs are sensitive to input perturbations, the sensitivities of time-averaged outputs are well-defined for uniformly hyperbolic systems, but existing methods to compute them cannot be used. Recently, a set of alternative approaches based on the shadowing property of dynamical systems was proposed to compute sensitivities. These approaches are computationally expensive, however. In this thesis, the Multiple Shooting Shadowing (MSS) [1] approach is used, and the main aim is to develop computational tools to allow for the implementation of MSS to large systems. The major contributor to the cost of MSS is the solution of a linear matrix system. The matrix has a large condition number, and this leads to very slow convergence rates for existing iterative solvers. A preconditioner was derived to suppress the condition number, thereby accelerating the convergence rate. It was demonstrated that for the chaotic 1D Kuramoto Sivashinsky equation (KSE), the rate of convergence was almost independent of the #DOF and the trajectory length. Most importantly, the developed solution method relies only on matrix-vector products. The adjoint version of the preconditioned MSS algorithm was then coupled with a gradient descent method to compute a feedback control matrix for the KSE. The adopted formulation allowed all matrix elements to be computed simultaneously. Within a single iteration, a stabilising matrix was computed. Comparisons with standard linear quadratic theory (LQR) showed remarkable similarities (but also some differences) in the computed feedback control kernels. A preconditioned data assimilation algorithm was then derived for state estimation purposes. The preconditioner was again shown to accelerate the rate of convergence significantly. Accurate state estimations were computed for the Lorenz system.Open Acces

Research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, numerical analysis and computer science

This report summarizes research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, numerical analysis, and computer science during the period April l, 1988 through September 30, 1988

Optimal control of linear stochastic evolution equations in Hilbert spaces and uniform observability

summary:In this paper we study the existence of the optimal (minimizing) control for a tracking problem, as well as a quadratic cost problem subject to linear stochastic evolution equations with unbounded coefficients in the drift. The backward differential Riccati equation (BDRE) associated with these problems (see \cite {chen}, for finite dimensional stochastic equations or \cite {UC}, for infinite dimensional equations with bounded coefficients) is in general different from the conventional BDRE (see \cite {1990}, \cite {ukl}). Under stabilizability and uniform observability conditions and assuming that the control weight-costs are uniformly positive, we establish that BDRE has a unique, uniformly positive, bounded on ${\mathbf R}_{+}$ and stabilizing solution. Using this result we find the optimal control and the optimal cost. It is known \cite {ukl} that uniform observability does not imply detectability and consequently our results are different from those obtained under detectability conditions (see \cite {1990})

Summary of research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, numerical analysis and computer science

Research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, numerical analysis, and computer science during the period October 1, 1988 through March 31, 1989 is summarized