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; andmethods 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 particularbranches, 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
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978-0-08-095598-8 (9780080955988)
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PrefaceContents1 OverviewI Methods of Operator Approximation in System Modelling2 Nonlinear Operator Approximation with Preassigned Accuracy2.1 Introduction2.2 Generic formulation of the problem2.3 Operator approximation in space C([0; 1]):2.4 Operator approximation in Banach spaces by polynomial operators2.5 Approximation on compact sets in topological vector spaces2.6 Approximation on noncompact sets in Hilbert spaces2.7 Special results for maps into Banach spaces2.8 Concluding remarks3 Interpolation of Nonlinear Operators 653.1 Introduction3.2 Lagrange interpolation in Banach spaces3.3 Weak interpolation of nonlinear operators3.4 Some related results3.5 Concluding remarks4 Realistic Operators and their Approximation4.1 Introduction4.2 Formalization of concepts related to description of real-world objects4.3 Approximation of R¡continuous operators4.4 Concluding remarks5 Methods of Best Approximation for Nonlinear Operators5.1 Introduction5.2 Best Approximation of nonlinear operators in Banach spaces: Deterministic case5.3 Estimation of mean and covariance matrix for random vectors5.4 Best Hadamard-quadratic approximation5.5 Best polynomial approximation5.6 Best causal approximation5.7 Best hybrid approximations5.8 Concluding remarksII Optimal Estimation of Random Vectors6 Computational Methods for Optimal Filtering of Stochastic Signals6.1 Introduction6.2 Optimal linear Filtering in Finite dimensional vector spaces6.3 Optimal linear Filtering in Hilbert spaces6.4 Optimal causal linear Filtering with piecewise constant memory6.5 Optimal causal polynomial Filtering with arbitrarily variable memory 6.6 Optimal nonlinear Filtering with no memory constraint6.7 Concluding remarks7 Computational Methods for Optimal Compression andReconstruction of Random Data7.1 Introduction7.2 Standard Principal Component Analysis and Karhunen-Loeeve transform (PCA{KLT)7.3 Rank-constrained matrix approximations 7.4 Generic PCA{KLT7.5 Optimal hybrid transform based on Hadamard-quadratic approximation7.6 Optimal transform formed by a combination of nonlinear operators 7.7 Optimal generalized hybrid transform7.8 Concluding remarksBibliographyIndex
