Orthonormal basis

About: Orthonormal basis is a research topic. Over the lifetime, 6014 publications have been published within this topic receiving 174416 citations.

On the representation of operators in bases of compactly supported wavelets

Abstract: This paper describes exact and explicit representations of the differential operators, ${{d^n } / {dx^n }}$, $n = 1,2, \cdots $, in orthonormal bases of compactly supported wavelets as well as the representations of the Hilbert transform and fractional derivatives. The method of computing these representations is directly applicable to multidimensional convolution operators.Also, sparse representations of shift operators in orthonormal bases of compactly supported wavelets are discussed and a fast algorithm requiring $O(N\log N)$ operations for computing the wavelet coefficients of all N circulant shifts of a vector of the length $N = 2^n $ is constructed. As an example of an application of this algorithm, it is shown that the storage requirements of the fast algorithm for applying the standard form of a pseudodifferential operator to a vector (see [G. Beylkin, R. R. Coifman, and V. Rokhlin, Comm. Pure. Appl. Math., 44 (1991), pp. 141–183]) may be reduced from $O(N)$ to $O(\log ^2 N)$ significant entries.

Sub-Cell Shock Capturing for Discontinuous Galerkin Methods

Abstract: A shock capturing strategy for higher order Discontinuous Galerkin approximations of scalar conservation laws is presented. We show how the original explicit artificial viscosity methods proposed over fifty years ago for finite volume methods, can be used very eectively in the context of high order approximations. Rather than relying on the dissipation inherent in Discontinuous Galerkin approximations, we add an artificial viscosity term which is aimed at eliminating the high frequencies in the solution, thus eliminating Gibbs-type oscillations. We note that the amount of viscosity required for stability is determined by the resolution of the approximating space and therefore decreases with the order of the approximating polynomial. Unlike classical finite volume artificial viscosity methods, where the shock is spread over several computational cells, we show that the proposed approach is capable of capturing the shock as a sharp, but smooth profile, which is typically contained within one element. The method is complemented with a shock detection algorithm which is based on the rate of decay of the expansion coecients of the solution when this is expressed in a hierarchical orthonormal basis. For the Euler equations, we consider and discuss the performance of several forms of the artificial viscosity term.

Spectral compression of mesh geometry

Abstract: We show how spectral methods may be applied to 3D mesh data to obtain compact representations. This is achieved by projecting the mesh geometry onto an orthonormal basis derived from the mesh topology. To reduce complexity, the mesh is partitioned into a number of balanced submeshes with minimal interaction, each of which are compressed independently. Our methods may be used for compression and progressive transmission of 3D content, and are shown to be vastly superior to existing methods using spatial techniques, if slight loss can be tolerated.

Compressed Sensing and Redundant Dictionaries

Abstract: This paper extends the concept of compressed sensing to signals that are not sparse in an orthonormal basis but rather in a redundant dictionary. It is shown that a matrix, which is a composition of a random matrix of certain type and a deterministic dictionary, has small restricted isometry constants. Thus, signals that are sparse with respect to the dictionary can be recovered via basis pursuit (BP) from a small number of random measurements. Further, thresholding is investigated as recovery algorithm for compressed sensing, and conditions are provided that guarantee reconstruction with high probability. The different schemes are compared by numerical experiments.

Model Order Reduction: Theory, Research Aspects and Applications

Abstract: Basic Concepts.- to Model Order Reduction.- Linear Systems, Eigenvalues, and Projection.- Theory.- Structure-Preserving Model Order Reduction of RCL Circuit Equations.- A Unified Krylov Projection Framework for Structure-Preserving Model Reduction.- Model Reduction via Proper Orthogonal Decomposition.- PMTBR: A Family of Approximate Principal-components-like Reduction Algorithms.- A Survey on Model Reduction of Coupled Systems.- Space Mapping and Defect Correction.- Modal Approximation and Computation of Dominant Poles.- Some Preconditioning Techniques for Saddle Point Problems.- Time Variant Balancing and Nonlinear Balanced Realizations.- Singular Value Analysis and Balanced Realizations for Nonlinear Systems.- Research Aspects and Applications.- Matrix Functions.- Model Reduction of Interconnected Systems.- Quadratic Inverse Eigenvalue Problem and Its Applications to Model Updating - An Overview.- Data-Driven Model Order Reduction Using Orthonormal Vector Fitting.- Model-Order Reduction of High-Speed Interconnects Using Integrated Congruence Transform.- Model Order Reduction for MEMS: Methodology and Computational Environment for Electro-Thermal Models.- Model Order Reduction of Large RC Circuits.- Reduced Order Models of On-Chip Passive Components and Interconnects, Workbench and Test Structures.