Sparse operator compression of higher-order elliptic operators with rough coefficients

We introduce the sparse operator compression to compress a self-adjoint higher-order elliptic operator with rough coefficients and various boundary conditions. The operator compression is achieved by using localized basis functions, which are energy-minimizing functions on local patches. On a regular mesh with mesh size $h$, the localized basis functions have supports of diameter $O(h\log(1/h))$ and give optimal compression rate of the solution operator. We show that by using localized basis functions with supports of diameter $O(h\log(1/h))$, our method achieves the optimal compression rate of the solution operator. From the perspective of the generalized finite element method to solve elliptic equations, the localized basis functions have the optimal convergence rate $O(h^k)$ for a $(2k)$th-order elliptic problem in the energy norm. From the perspective of the sparse PCA, our results show that a large set of Mat\'{e}rn covariance functions can be approximated by a rank-$n$ operator with a localized basis and with the optimal accuracy

Numerics of boundary-domain integral and integro-differential equations for BVP with variable coefficient in 3D

This is the post-print version of the article. The official published version can be accessed from the links below - Copyright @ 2013 Springer-VerlagA numerical implementation of the direct boundary-domain integral and integro-differential equations, BDIDEs, for treatment of the Dirichlet problem for a scalar elliptic PDE with variable coefficient in a three-dimensional domain is discussed. The mesh-based discretisation of the BDIEs with tetrahedron domain elements in conjunction with collocation method leads to a system of linear algebraic equations (discretised BDIE). The involved fully populated matrices are approximated by means of the H-Matrix/adaptive cross approximation technique. Convergence of the method is investigated.This study is partially supported by the EPSRC grant EP/H020497/1:"Mathematical Analysis of Localised-Boundary-Domain Integral Equations for Variable-Coefficients Boundary Value Problems"

A Comparison of Numerical Methods used for\ud Finite Element Modelling of Soft Tissue\ud Deformation

Soft tissue deformation is often modelled using incompressible nonlinear elasticity, with solutions computed using the finite element method. There are a range of options available when using the finite element method, in particular, the polynomial degree of the basis functions used for interpolating position and pressure, and the type of element making up the mesh. We investigate the effect of these choices on the accuracy of the computed solution, using a selection of model problems motivated by typical deformations seen in soft tissue modelling. We set up model problems with discontinuous material properties (as is the case for the breast), steeply changing gradients in the body force (as found in contracting cardiac tissue), and discontinuous first derivatives in the solution at the boundary, caused by a discontinuous applied force (as in the breast during mammography). We find that the choice of pressure basis functions are vital in the presence of a material interface, higher-order schemes do not perform as well as may be expected when there are sharp gradients, and in general that it is important to take the expected regularity of the solution into account when choosing a numerical scheme

ENO-wavelet transforms for piecewise smooth functions

We have designed an adaptive essentially nonoscillatory (ENO)-wavelet transform for approximating discontinuous functions without oscillations near the discontinuities. Our approach is to apply the main idea from ENO schemes for numerical shock capturing to standard wavelet transforms. The crucial point is that the wavelet coefficients are computed without differencing function values across jumps. However, we accomplish this in a different way than in the standard ENO schemes. Whereas in the standard ENO schemes the stencils are adaptively chosen, in the ENO-wavelet transforms we adaptively change the function and use the same uniform stencils. The ENO-wavelet transform retains the essential properties and advantages of standard wavelet transforms such as concentrating the energy to the low frequencies, obtaining maximum accuracy, maintained up to the discontinuities, and having a multiresolution framework and fast algorithms, all without any edge artifacts. We have obtained a rigorous approximation error bound which shows that the error in the ENO-wavelet approximation depends only on the size of the derivative of the function away from the discontinuities. We will show some numerical examples to illustrate this error estimate

Graph Spectral Image Processing

Recent advent of graph signal processing (GSP) has spurred intensive studies of signals that live naturally on irregular data kernels described by graphs (e.g., social networks, wireless sensor networks). Though a digital image contains pixels that reside on a regularly sampled 2D grid, if one can design an appropriate underlying graph connecting pixels with weights that reflect the image structure, then one can interpret the image (or image patch) as a signal on a graph, and apply GSP tools for processing and analysis of the signal in graph spectral domain. In this article, we overview recent graph spectral techniques in GSP specifically for image / video processing. The topics covered include image compression, image restoration, image filtering and image segmentation

A rarefaction-tracking method for hyperbolic conservation laws

We present a numerical method for scalar conservation laws in one space dimension. The solution is approximated by local similarity solutions. While many commonly used approaches are based on shocks, the presented method uses rarefaction and compression waves. The solution is represented by particles that carry function values and move according to the method of characteristics. Between two neighboring particles, an interpolation is defined by an analytical similarity solution of the conservation law. An interaction of particles represents a collision of characteristics. The resulting shock is resolved by merging particles so that the total area under the function is conserved. The method is variation diminishing, nevertheless, it has no numerical dissipation away from shocks. Although shocks are not explicitly tracked, they can be located accurately. We present numerical examples, and outline specific applications and extensions of the approach.Comment: 21 pages, 7 figures. Similarity 2008 conference proceeding

Approximation of the critical buckling factor for composite panels

This article is concerned with the approximation of the critical buckling factor for thin composite plates. A new method to improve the approximation of this critical factor is applied based on its behavior with respect to lamination parameters and loading conditions. This method allows accurate approximation of the critical buckling factor for non-orthotropic laminates under complex combined loadings (including shear loading). The influence of the stacking sequence and loading conditions is extensively studied as well as properties of the critical buckling factor behavior (e.g concavity over tensor D or out-of-plane lamination parameters). Moreover, the critical buckling factor is numerically shown to be piecewise linear for orthotropic laminates under combined loading whenever shear remains low and it is also shown to be piecewise continuous in the general case. Based on the numerically observed behavior, a new scheme for the approximation is applied that separates each buckling mode and builds linear, polynomial or rational regressions for each mode. Results of this approach and applications to structural optimization are presented

High-frequency asymptotic compression of dense BEM matrices for general geometries without ray tracing

Wave propagation and scattering problems in acoustics are often solved with boundary element methods. They lead to a discretization matrix that is typically dense and large: its size and condition number grow with increasing frequency. Yet, high frequency scattering problems are intrinsically local in nature, which is well represented by highly localized rays bouncing around. Asymptotic methods can be used to reduce the size of the linear system, even making it frequency independent, by explicitly extracting the oscillatory properties from the solution using ray tracing or analogous techniques. However, ray tracing becomes expensive or even intractable in the presence of (multiple) scattering obstacles with complicated geometries. In this paper, we start from the same discretization that constructs the fully resolved large and dense matrix, and achieve asymptotic compression by explicitly localizing the Green's function instead. This results in a large but sparse matrix, with a faster associated matrix-vector product and, as numerical experiments indicate, a much improved condition number. Though an appropriate localisation of the Green's function also depends on asymptotic information unavailable for general geometries, we can construct it adaptively in a frequency sweep from small to large frequencies in a way which automatically takes into account a general incident wave. We show that the approach is robust with respect to non-convex, multiple and even near-trapping domains, though the compression rate is clearly lower in the latter case. Furthermore, in spite of its asymptotic nature, the method is robust with respect to low-order discretizations such as piecewise constants, linears or cubics, commonly used in applications. On the other hand, we do not decrease the total number of degrees of freedom compared to a conventional classical discretization. The combination of the ...Comment: 24 pages, 13 figure