On adaptive filter structure and performance

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New Directions for Contact Integrators

Contact integrators are a family of geometric numerical schemes which guarantee the conservation of the contact structure. In this work we review the construction of both the variational and Hamiltonian versions of these methods. We illustrate some of the advantages of geometric integration in the dissipative setting by focusing on models inspired by recent studies in celestial mechanics and cosmology.Comment: To appear as Chapter 24 in GSI 2021, Springer LNCS 1282

Approximate controllability and observability measures in control systems design

The selection of systems of inputs and outputs (input and output structure) forms part of early system design, which is important since it preconditions the potential for control design. Existing methodologies for input, output structure selection rely on criteria expressing distance from uncontrollability (unobservability). The thesis introduces novel measures for evaluating and estimating the distance to uncontrollability and relatively unobservability. At first, the modal measuring approach is studied in detail, providing a framework for the ”best” structure selection. Although controllability (observability) is invariant under state feedback (output injection), the corresponding degrees expressing distance from uncontrollability (unobservability) are not. Hence, the thesis introduces new criteria for the distance problem from uncontrollability (unobservability) which is invariant under feedback transformations. The approach uses the restricted input-state (state-output) matrix pencil and then deploys exterior algebra that reduces the overall problem to the standard problem of distance of a set of polynomials from non-coprimeness. Results on the distance of the Sylvester Resultants from singularity provide the new measures. Since distance to singularity of the corresponding Sylvester matrix is the key in evaluating the distance to uncontrolability it is of the particular interest in the present work. In order to find the solution two novel methods are introduced in the thesis, namely the alternating projection algorithm and a structured singular value approach. A least-squares alternating projection algorithm, motivated by a factorisation result involving the Sylvester resultant matrix, is proposed for calculating the ”best” approximate GCD of a coprime polynomial set. The properties of the proposed algorithm are investigated and the method is compared with alternative optimisation techniques which can be employed to solve the problem. It is also shown that the problem of an approximate GCD calculation is equivalent tothe solution of a structured singular value (µ) problem arising in robust control for which numerous techniques are available. Motivated by the powerful concept of the structured singular values, the proposed method is extended to the special case of an implicit system that has a wide application in the behavioural analysis of complex systems. Moreover, µ-value approach has a potential application for the general distance problem to uncontrollability that is numerically hard to obtain. Overall, the proposed framework significantly simplifies and generalises the input-output structure selection procedure and evaluates alternative solutions for a variety of distance problems that appear in Control Theory

A total variation regularization based super-resolution reconstruction algorithm for digital video

Super-resolution (SR) reconstruction technique is capable of producing a high-resolution image from a sequence of low-resolution images. In this paper, we study an efficient SR algorithm for digital video. To effectively deal with the intractable problems in SR video reconstruction, such as inevitable motion estimation errors, noise, blurring, missing regions, and compression artifacts, the total variation (TV) regularization is employed in the reconstruction model. We use the fixed-point iteration method and preconditioning techniques to efficiently solve the associated nonlinear Euler-Lagrange equations of the corresponding variational problem in SR. The proposed algorithm has been tested in several cases of motion and degradation. It is also compared with the Laplacian regularization-based SR algorithm and other TV-based SR algorithms. Experimental results are presented to illustrate the effectiveness of the proposed algorithm.£.published_or_final_versio

Numerical solution of nonlinear elliptic partial differential equations by a generalized conjugate gradient method

We have studied previously a generalized conjugate gradient method for solving sparse positive-definite systems of linear equations arising from the discretization of elliptic partial-differential boundary-value problems. Here, extensions to the nonlinear case are considered. We split the original discretized operator into the sum of two operators, one of which corresponds to a more easily solvable system of equations, and accelerate the associated iteration based on this splitting by (nonlinear) conjugate gradients. The behavior of the method is illustrated for the minimal surface equation with splittings corresponding to nonlinear SSOR, to approximate factorization of the Jacobian matrix, and to elliptic operators suitable for use with fast direct methods. The results of numerical experiments are given as well for a mildy nonlinear example, for which, in the corresponding linear case, the finite termination property of the conjugate gradient algorithm is crucial. Wir haben früher eine verallgemeinerte Methode der konjugierten Gradienten studiert, um dünnbesetzte positiv definite Systeme von linearen Gleichungen zu lösen, die von der Diskretisierung von elliptischen partiellen Differential-Randwertproblemen herrühren. Wir betrachten hier die Verallgemeinerung auf den nichtlinearen Fall: Wir spalten den ursprünglichen diskretisierten Operator auf in eine Summe von zwei Operatoren. Einer von diesen Operatoren entspricht einem leicht lösbaren System von Gleichungen, und wir beschleunigen die aus dieser Spaltung hervorgehende Iteration mit (nichtlinearen) konjugierten Gradienten. Das Verhalten der Methode wird illustriert durch Anwendung auf die Minimalflächen-Gleichung, mit Spaltungen entsprechend dem nichtlinearen SSOR-Verfahren, der angenäherten Faktorisierung der Jacobi-Matrix, oder den elliptischen Operatoren, die sich für schnelle direkte Methoden eignen. Die Resultate von numerischen Experimenten für ein nur schwach nichtlineares Beispiel sind ebenfalls angegeben. Für den entsprechenden linearen Fall ist in diesem Fall die Konvergenz des konjugierten Gradienten-Algorithmus in einer endlichen Anzahl von Schritten wesentlich.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41643/1/607_2005_Article_BF02252030.pd

Strong Convergence of an Iterative Algorithm for Hierarchical Problems

We introduce the triple hierarchical problem over the solution set of the variational inequality problem and the fixed point set of a nonexpansive mapping. The strong convergence of the algorithm is proved under some mild conditions. Our results extend those of Yao et al., Iiduka, Ceng et al., and other authors