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.

Quantum inverse iteration algorithm for programmable quantum simulators

We propose a quantum inverse iteration algorithm, which can be used to estimate ground state properties of a programmable quantum device. The method relies on the inverse power iteration technique, where the sequential application of the Hamiltonian inverse to an initial state prepares the approximate ground state. To apply the inverse Hamiltonian operation, we write it as a sum of unitary evolution operators using the Fourier approximation approach. This allows to reformulate the protocol as separate measurements for the overlap of initial and propagated wavefunction. The algorithm thus crucially depends on the ability to run Hamiltonian dynamics with an available quantum device, and can be used for analog quantum simulators. We benchmark the performance using paradigmatic examples of quantum chemistry, corresponding to molecular hydrogen and beryllium hydride. Finally, we show its use for studying the ground state properties of relevant material science models, which can be simulated with existing devices, considering an example of the Bose-Hubbard atomic simulator.

Quick and Highly Efficient Modal Analysis Method Based on the Reanalysis Technique for Large Complex Structure and Topology Optimization

Modal analysis is widely used to investigate the dynamic characteristics of large and complex structures. For finite element models, iterative solvers are needed to precisely calculate eigenpairs or frequency and vibration mode. However, in cases such as large-scale analysis or reanalysis studies, or optimization design of a huge structure, computational cost may become too time consuming. This paper focuses on the quick structural modal analysis based on the reanalysis technique for large complex structures. Based on the stiffness and mass matrix of the analytical structures, a high precision and efficiency eigensolution is generated by the proposed modal analysis method (the Pseudo Random Independent and Coupling Inverse Iteration (PRICII) method), which combines the pseudo random number initialization, ICII (Independent and Coupling Inverse Iteration) strategy with the double Rayleigh–Ritz analysis. By comparing with the Subspace iteration method, Lanczos method, etc. the large-scale numerical examples show that the actual computational savings of the proposed method are usually higher than 75% with sufficient precision. Also, its applications in topology optimization are greatly effective.