THE INVERSE EIGENPROBLEM OF CENTROSYMMETRIC MATRICES WITH A SUBMATRIX CONSTRAINT AND ITS APPROXIMATION

In this paper, we first consider the existence of and the general expression for the solution to the constrained inverse eigenproblem defined as follows: given a set of complex n-vectors fxigm i=1 and a set of complex numbers f¸igm i=1, and an s-by-s real matrix C0, find an n-by-n real centrosymmetric matrix C such that the s-by-s leading principal submatrix of C is C0, and fxigm i=1 and f¸igm i=1 are the eigenvectors and eigenvalues of C respectively. We then concerned with the best approximation problem for the constrained inverse problem whose solution set is nonempty. That is, given an arbitrary real n-by-n matrix ˜ C, find a matrix C which is the solution to the constrained inverse problem such that the distance between C and ˜ C is minimized in the Frobenius norm. We give an explicit solution and a numerical algorithm to the best approximation problem. Some illustrative experiments are also presented

An Inverse Eigenvalue Problem of Hermite-Hamilton Matrices in Structural Dynamic Model Updating

We first consider the following inverse eigenvalue problem: given X∈Cn×m and a diagonal matrix Λ∈Cm×m, find n×n Hermite-Hamilton matrices K and M such that KX=MXΛ. We then consider an optimal approximation problem: given n×n Hermitian matrices Ka and Ma, find a solution (K,M) of the above inverse problem such that ∥K-Ka∥2+∥M-Ma∥2=min⁡. By using the Moore-Penrose generalized inverse and the singular value decompositions, the solvability conditions and the representations of the general solution for the first problem are derived. The expression of the solution to the second problem is presented

Free and forced propagation of Bloch waves in viscoelastic beam lattices

Beam lattice materials can be characterized by a periodic microstructure realizing a geometrically regular pattern of elementary cells. Within this framework, governing the free and forced wave propagation by means of spectral design techniques and/or energy dissipation mechanisms is a major issue of theoretical interest with applications in aerospace, chemical, naval, biomedical engineering. The first part of the Thesis addresses the free propagation of Bloch waves in non-dissipative microstructured cellular materials. Focus is on the alternative formulations suited to describe the wave propagation in the bidimensional infinite material domain, according to the classic canons of linear solid or structural mechanics. Adopting the centrosymmetric tetrachiral cell as prototypical periodic topology, the frequency dispersion spectrum is obtained by applying the Floquet-Bloch theory. The dispersion spectrum resulting from a synthetic Lagrangian beam lattice formulation is compared with its counterpart derived from different continuous models (high-fidelity first-order heterogeneous and equivalent homogenized micropolar continua). Asymptotic perturbation-based approximations and numerical spectral solutions are compared and cross-validated. Adopting the low-frequency band gaps of the dispersion spectrum as functional targets, parametric analyses are carried out to highlight the descriptive limits of the synthetic models and to explore the enlarged parameter space described by high-fidelity models. The microstructural design or tuning of the mechanical properties of the cellular microstructure is employed to successfully verify the wave filtering functionality of the tetrachiral material. Alternatively, band gaps in the material spectrum can be opened at target center frequencies by using metamaterials with inertial resonators. Based on these motivations, in the second part of the Thesis, a general dynamic formulation is presented for determining the dispersion properties of viscoelastic metamaterials, equipped with local dissipative resonators. The linear mechanism of local resonance is realized by tuning periodic auxiliary masses, viscoelastically coupled with the beam lattice microstructure. As peculiar aspect, the viscoelastic coupling is derived by a mechanical formulation based on the Boltzmann superposition integral, whose kernel is approximated by a Prony series. Consequently, the free propagation of damped Bloch waves is governed by a linear homogeneous system of integro-differential equations of motion. Therefore, differential equations of motion with frequency-dependent coefficients are obtained by applying the bilateral Laplace transform. The corresponding complex-valued branches characterizing the dispersion spectrum are determined and parametrically analyzed. Particularly, the spectra corresponding to Taylor series approximations of the equation coefficients are investigated. The standard dynamic equations with linear viscous damping are recovered at the first order approximation. Increasing approximation orders determine non-negligible spectral effects, including the occurrence of pure damping spectral branches. Finally, the forced response to harmonic single frequency external forces in the frequency and the time domains is investigated. The response in the time domain is obtained by applying the inverse bilateral Laplace transform. The metamaterial responses to non-resonant, resonant and quasi-resonant external forces are compared and discussed from a qualitative and quantitative viewpoint

Iterative Methods to Solve the Generalized Coupled Sylvester-Conjugate Matrix Equations for Obtaining the Centrally Symmetric (Centrally Antisymmetric) Matrix Solutions

The iterative method is presented for obtaining the centrally symmetric (centrally antisymmetric) matrix pair (X,Y) solutions of the generalized coupled Sylvester-conjugate matrix equations A1X+B1Y=D1X¯E1+F1, A2Y+B2X=D2Y¯E2+F2. On the condition that the coupled matrix equations are consistent, we show that the solution pair (X*,Y*) can be obtained within finite iterative steps in the absence of round-off error for any initial value given centrally symmetric (centrally antisymmetric) matrix. Moreover, by choosing appropriate initial value, we can get the least Frobenius norm solution for the new generalized coupled Sylvester-conjugate linear matrix equations. Finally, some numerical examples are given to illustrate that the proposed iterative method is quite efficient

Theoretical and computational insights into the nonlinear optics of nanostructured bulk and 2D materials

In this thesis, a comprehensive analytical and computational study of linear and nonlinear optical response of nanostructured two-dimensional (2D) and bulk materials is presented. The new numerical methods developed in this thesis facilitate the efficient and accurate design of new artificial optical materials and novel nonlinear optical devices. Moreover, the presented results and conclusions can provide a deeper theoretical understanding of different resonant, nonlinear optical phenomena in photonic nanostructures. Two computational electromagnetic methods to calculate the interaction of light with linear and nonlinear diffraction gratings and more general periodic structures have been developed. An efficient formulation of the rigorous coupled-wave analysis (RCWA), a modal frequency domain method, for accurate near-field calculations and for complex oblique structures has been proposed. This method has been implemented into a powerful commercial software tool and applied to calculate diffraction in several nanophotonic devices highly relevant to practical applications. Beyond this commercial implementation, the RCWA has been extended to describe linear and nonlinear optical effects in nanostructured 2D materials, with second- and third-harmonic generation being the most important nonlinear processes. A key feature of this formulation is that it is independent of the height of the 2D material, and only requires knowledge of its linear and nonlinear optical properties. Using this method, plasmon resonances of nanostructured graphene have been investigated, and tuneable Fano resonances have been explored to increase the nonlinear efficiency of heterostructures containing transition metal dichalcogenide monolayers and nanopatterned slab waveguides. The second thrust of the thesis was devoted to extending the so-called generalised source method (GSM) to the area of nonlinear optics. In particular, its mathematical formulation has been extended to incorporate second- and third-order nonlinear optical effects, and the proposed nonlinear GSM has been used to design and optimise multi-resonant photonic devices made of nonlinear optical materials. In addition, this advanced computational method facilitated the understanding of strong nonlinear optical activity in plasmonic nanostructures, and explained the multipolar nonlinear optical response of certain nonlinear metasurfaces

From spline wavelet to sampling theory on circulant graphs and beyond– conceiving sparsity in graph signal processing

Graph Signal Processing (GSP), as the field concerned with the extension of classical signal processing concepts to the graph domain, is still at the beginning on the path toward providing a generalized theory of signal processing. As such, this thesis aspires to conceive the theory of sparse representations on graphs by traversing the cornerstones of wavelet and sampling theory on graphs. Beginning with the novel topic of graph spline wavelet theory, we introduce families of spline and e-spline wavelets, and associated filterbanks on circulant graphs, which lever- age an inherent vanishing moment property of circulant graph Laplacian matrices (and their parameterized generalizations), for the reproduction and annihilation of (exponen- tial) polynomial signals. Further, these families are shown to provide a stepping stone to generalized graph wavelet designs with adaptive (annihilation) properties. Circulant graphs, which serve as building blocks, facilitate intuitively equivalent signal processing concepts and operations, such that insights can be leveraged for and extended to more complex scenarios, including arbitrary undirected graphs, time-varying graphs, as well as associated signals with space- and time-variant properties, all the while retaining the focus on inducing sparse representations. Further, we shift from sparsity-inducing to sparsity-leveraging theory and present a novel sampling and graph coarsening framework for (wavelet-)sparse graph signals, inspired by Finite Rate of Innovation (FRI) theory and directly building upon (graph) spline wavelet theory. At its core, the introduced Graph-FRI-framework states that any K-sparse signal residing on the vertices of a circulant graph can be sampled and perfectly reconstructed from its dimensionality-reduced graph spectral representation of minimum size 2K, while the structure of an associated coarsened graph is simultaneously inferred. Extensions to arbitrary graphs can be enforced via suitable approximation schemes. Eventually, gained insights are unified in a graph-based image approximation framework which further leverages graph partitioning and re-labelling techniques for a maximally sparse graph wavelet representation.Open Acces

The inverse eigenproblem of centrosymmetric matrices with a submatrix constraint and its approximation

10.1137/S0895479803434185SIAM Journal on Matrix Analysis and Applications2641100-111