Computational Methods for Modeling of Nonlinear Systems

In this book, we study theoretical and practical aspects of computing methods for mathematical modelling of nonlinear systems. A number of computing techniques are considered, such as methods of operator approximation with any given accuracy; operator interpolation techniques including a non-Lagrange interpolation; methods of system representation subject to constraints associated with concepts of causality, memory and stationarity; methods of system representation with an accuracy that is the best within a given class of models; methods of covariance matrix estimation;
methods for low-rank matrix approximations; hybrid methods based on a combination of iterative procedures and best operator approximation; and
methods for information compression and filtering under condition that a filter model should satisfy restrictions associated with causality and different types of memory.

As a result, the book represents a blend of new methods in general computational analysis,
and specific, but also generic, techniques for study of systems theory ant its particular
branches, such as optimal filtering and information compression.

- Best operator approximation,
- Non-Lagrange interpolation,
- Generic Karhunen-Loeve transform
- Generalised low-rank matrix approximation
- Optimal data compression
- Optimal nonlinear filtering

Preface

Contents

1 Overview

I Methods of Operator Approximation in System Modelling

2 Nonlinear Operator Approximation with Preassigned Accuracy

2.1 Introduction

2.2 Generic formulation of the problem

2.3 Operator approximation in space C([0; 1]):

2.4 Operator approximation in Banach spaces by polynomial operators

2.5 Approximation on compact sets in topological vector spaces

2.6 Approximation on noncompact sets in Hilbert spaces

2.7 Special results for maps into Banach spaces

2.8 Concluding remarks

3 Interpolation of Nonlinear Operators 65

3.1 Introduction

3.2 Lagrange interpolation in Banach spaces

3.3 Weak interpolation of nonlinear operators

3.4 Some related results

3.5 Concluding remarks

4 Realistic Operators and their Approximation

4.1 Introduction

4.2 Formalization of concepts related to description of real-world objects

4.3 Approximation of R¡continuous operators

4.4 Concluding remarks

5 Methods of Best Approximation for Nonlinear Operators

5.1 Introduction

5.2 Best Approximation of nonlinear operators in Banach spaces: Deterministic case

5.3 Estimation of mean and covariance matrix for random vectors

5.4 Best Hadamard-quadratic approximation

5.5 Best polynomial approximation

5.6 Best causal approximation

5.7 Best hybrid approximations

5.8 Concluding remarks

II Optimal Estimation of Random Vectors

6 Computational Methods for Optimal Filtering of Stochastic Signals

6.1 Introduction

6.2 Optimal linear Filtering in Finite dimensional vector spaces

6.3 Optimal linear Filtering in Hilbert spaces

6.4 Optimal causal linear Filtering with piecewise constant memory

6.5 Optimal causal polynomial Filtering with arbitrarily variable memory

6.6 Optimal nonlinear Filtering with no memory constraint

6.7 Concluding remarks

7 Computational Methods for Optimal Compression and
Reconstruction of Random Data

7.1 Introduction

7.2 Standard Principal Component Analysis and Karhunen-Loeeve transform (PCA{KLT)

7.3 Rank-constrained matrix approximations

7.4 Generic PCA{KLT

7.5 Optimal hybrid transform based on Hadamard-quadratic approximation

7.6 Optimal transform formed by a combination of nonlinear operators

7.7 Optimal generalized hybrid transform

7.8 Concluding remarks

Bibliography

Index