arXiv subjects

Zhengjian Bai

Publications and source records attributed to Zhengjian Bai.

3 recordsLinked to original sources

arXiv2026

Precision-induced Adaptive Randomized Low-Rank Approximation for SVD and Matrix Inversion

Singular value decomposition (SVD) and matrix inversion are ubiquitous in scientific computing. Both tasks are computationally demanding for large scale matrices. Existing algorithms can approximatively solve these problems with a given rank, which however is unknown in practice and requires considerable cost for tuning. In this paper, we tackle the SVD and matrix inversion problems from a new angle, where the optimal rank for the approximate solution is explicitly guided by the distribution of the singular values. Under the framework, we propose a precision-induced random re-normalization procedure for the considered problems without the need of guessing a good rank. The new algorithms built upon the procedure simultaneously calculate the optimal rank for the task at a desired precision level and lead to the corresponding approximate solution with a substantially reduced computational cost. The promising performance of the new algorithms is supported by both theory and numerical examples.

math.NA↗

arXiv2021

Stable determination of an elastic medium scatterer by a single far-field measurement and beyond

We are concerned with the time-harmonic elastic scattering due to an inhomogeneous elastic material inclusion located inside a uniformly homogeneous isotropic medium. We establish a sharp stability estimate of logarithmic type in determining the support of the elastic scatterer, independent of its material content, by a single far-field measurement when the support is a convex polyhedral domain in $\mathbb{R}^n$, $n=2,3$. Our argument in establishing the stability result is localized around a corner of the medium scatterer. This enables us to further establish a byproduct result by proving that if a generic medium scatterer, not necessary to be a polyhedral shape, possesses a corner, then there exists a positive lower bound of the scattered far-field patterns. The latter result indicates that if an elastic material object possesses a corner on its support, then it scatters every incident wave stably and invisibility phenomenon does not occur.

math.AP↗

arXiv2021

Effective medium theory for embedded obstacles in elasticity with applications to inverse problems

We consider the time-harmonic elastic wave scattering from a general (possibly anisotropic) inhomogeneous medium with an embedded impenetrable obstacle. We show that the impenetrable obstacle can be effectively approximated by an isotropic elastic medium with a particular choice of material parameters. We derive sharp estimates to rigorously verify such an effective approximation. Our study is strongly motivated by the related studies of two challenging inverse elastic problems including the inverse boundary problem with partial data and the inverse scattering problem of recovering mediums with buried obstacles. The proposed effective medium theory readily yields some interesting applications of practical significance to these inverse problems.

math.AP↗