Rayleigh–Ritz eigenvalue estimates for Hermitian matrices obey Cauchy interlacing, which has helpful implications for theory, applications, and algorithms. In contrast, few results about the Ritz values of non-Hermitian matrices are known, beyond their containment within the numerical range. To show that such Ritz values enjoy considerable structure, we establish regions within the numerical ra...

In this paper, we study the deep Ritz method for solving linear elasticity equation from a numerical analysis perspective. A modified formulation using $H^{1/2}(\Gamma_D)$ norm is introduced and analyzed in order to deal with (essential) Dirichlet boundary condition. We show that resulting provides best approximation among set of neural network (DNN) functions respect ``energy'' norm. Furthermo...

The objective of this paper is to present a comparative study of fitted-mesh finite difference method, Ritz-Galerkin finite element method and B-spline collocation method for a two-parameter singularly perturbed boundary value problems. Due to the small parameters ε and μ, the boundary layers arise. We have taken a piecewise-uniform fittedmesh to resolve the boundary layers and shown that fitte...

This paper describes an experimental study on the use of Ritz vectors for damage detection of a grid-type bridge model. A new procedure to extract Ritz vectors from experimental modal analysis is proposed and demonstrated using the test structure. Using appropriate load patterns, Ritz vectors can be made more sensitive to damage than modal vectors. The results indicate that the use of load-depe...

Abstract The parametric surrogate models for partial differential equations (PDEs) are a necessary component many applications in computational sciences, and the convolutional neural networks (CNNs) have proven to be an excellent tool generate these surrogates when fields present. CNNs commonly trained on labeled data based one-to-one sets of parameter-input PDE-output fields. Recently, residua...

a relationship based on the modified couple stress theory is developed to investigate the flexural sensitivity of an atomic force microscope (afm) with assembled cantilever probe (acp). this acp comprises a horizontal cantilever, two vertical extensions and two tips located at the free ends of the extensions which form a caliper. an approximate solution to the flexural vibration problem is obta...

Many inequality relations between real vector quantities can be succinctly expressed as “weak (sub)majorization” relations using the symbol ≺w. We explain these ideas and apply them in several areas, angles between subspaces, Ritz values, and graph Laplacian spectra, which we show are all surprisingly related. Let Θ(X ,Y) be the vector of principal angles in nondecreasing order between subspace...