The Hierarchical Subspace Iteration Method for Laplace–Beltrami Eigenproblems

Sparse eigenproblems are important for various applications in computer graphics. The spectrum and eigenfunctions of the Laplace–Beltrami operator, for example, are fundamental for methods in shape analysis and mesh processing. The Subspace Iteration Method is a robust solver for these problems. In practice, however, Lanczos schemes are often faster. In this article, we introduce the Hierarchical Subspace Iteration Method (HSIM) , a novel solver for sparse eigenproblems that operates on a hierarchy of nested vector spaces. The hierarchy is constructed such that on the coarsest space all eigenpairs can be computed with a dense eigensolver. HSIM uses these eigenpairs as initialization and iterates from coarse to fine over the hierarchy. On each level, subspace iterations, initialized with the solution from the previous level, are used to approximate the eigenpairs. This approach substantially reduces the number of iterations needed on the finest grid compared to the non-hierarchical Subspace Iteration Method. Our experiments show that HSIM can solve Laplace–Beltrami eigenproblems on meshes faster than state-of-the-art methods based on Lanczos iterations, preconditioned conjugate gradients, and subspace iterations.

Dynamic Characteristics of a Double-Pylon Cable-Stayed Bridge with Steel Truss Girder and Single-Cable Plane

The dynamic characteristics are closely linked to the seismic stability and wind-resistant of the bridge. But different bridge types have different dynamic characteristics. In order to study the dynamic characteristics of a double-pylon cable-stayed bridge with a single-cable plane and steel truss girder whose main span is the longest in the world, the dynamic load test was done, and the finite element and the subspace iteration methods were used to analyze the vibration mode of the bridge. The influence of different structural parameters on the dynamic characteristics of the bridge was analyzed. The changed structural parameters are cable layout, stiffness of steel truss girder, stiffness of stayed cables, stiffness of pylons, the concentration of dead load, number and location of auxiliary piers, and structural system. The results show that the bending and torsion resistance of the double-pylon cable-stayed bridge with a single-cable plane and steel truss girder is weak. The torsional stiffness of the cable-stayed bridge with a double-cable plane is stronger than that of the cable-stayed bridge with a single-cable plane. The seismic stability and wind-resistant of the bridge can be improved by using light dead load, improving the stiffness of pylon and girder, and adding auxiliary piers scientifically. However, the change of cable stiffness has a complex influence on the dynamic characteristics of the bridge. The conclusion can offer references for the construction, maintenance, and design of the same type of bridges.

Inexact Inverse Subspace Iteration with Preconditioning Applied to Quadratic Matrix Polynomials

An inexact variant of inverse subspace iteration is used to find a small invariant pair of a large quadratic matrix polynomial. It is shown that linear convergence is preserved provided the inner iteration is performed with increasing accuracy. A preconditioned block GMRES solver is employed as inner iteration. The preconditioner uses the strategy of “tuning” which prevents the inner iteration from increasing and therefore results in a substantial saving in costs. The accuracy of the computed invariant pair can be improved by the addition of a post-processing step involving very few iterations of Newton’s method. The effectiveness of the proposed approach is demonstrated by numerical experiments.

An FFT-based approach for Bloch wave analysis: application to polycrystals

A method based on the Fast Fourier Transform is proposed to obtain the dispersion relation of acoustic waves in heterogeneous periodic media with arbitrary microstructures. The microstructure is explicitly considered using a voxelized Representative Volume Element (RVE). The dispersion diagram is obtained solving an eigenvalue problem for Bloch waves in Fourier space. To this aim, two linear operators representing stiffness and mass are defined through the use of differential operators in Fourier space. The smallest eigenvalues are obtained using the implicitly restarted Lanczos and the subspace iteration methods, and the required inverse of the stiffness operator is done using the conjugate gradient with a preconditioner. The method is used to study the propagation of acoustic waves in elastic polycrystals, showing the strong effect of crystal anistropy and polycrystaline texture on the propagation. It is shown that the method combines the simplicity of classical Fourier series analysis with the versatility of Finite Elements to account for complex geometries proving an efficient and general approach which allows the use of large RVEs in 3D.

Smoothed-Adaptive Perturbed Inverse Iteration for Elliptic Eigenvalue Problems

We present a perturbed subspace iteration algorithm to approximate the lowermost eigenvalue cluster of an elliptic eigenvalue problem. As a prototype, we consider the Laplace eigenvalue problem posed in a polygonal domain. The algorithm is motivated by the analysis of inexact (perturbed) inverse iteration algorithms in numerical linear algebra. We couple the perturbed inverse iteration approach with mesh refinement strategy based on residual estimators. We demonstrate our approach on model problems in two and three dimensions.

A finite-difference iterative solver of the Helmholtz equation for frequency-domain seismic wave modeling and full waveform inversion

We present a finite difference iterative solver of the Helmholtz equation for seismic modeling and inversion in the frequency-domain. The iterative solver involves the shifted Laplacian operator and two-level pre-conditioners. It is based on the application of the pre-conditioners to the Krylov subspace stabilized biconjugate gradient method. A critical factor for the iterative solver is the introduction of a new pre-conditioner into the Krylov subspace iteration method to solve the linear system resulting from the discretization of the Helmholtz equation. This new pre-conditioner is based upon a reformulation of an integral equation-based convergent Born series for the Lippmann-Schwinger equation to an equivalent differential equation. We demonstrate that the proposed iterative solver combined with the novel pre-conditioner when incorporated with the finite difference method accelerates the convergence of the Krylov subspace iteration method for frequency-domain seismic wave modeling. A comparison of a direct solver, a one-level Krylov subspace iterative solver and the proposed two-level iterative solver verified the accuracy and accelerated convergence of the new scheme. Extensive tests in full waveform inversion demonstrate the solver applicability to full waveform inversion applications.