Inverse problems in quantifying mechanical properties in nanomaterials

Abstract

In this paper, we study three inverse problems on quantifying mechanical properties of nanomaterials, including an inverse source problem, an inverse coefficient problem and an inverse spectral problem. These problems are of significant importance to nanosciences, especially on developing functional devices, measurements, optimal designs and their applications. According to engineering applications, we mainly have two methods on quantifying mechanical properties of nanomaterials: static and dynamical methods. In this paper, we first investigate an inverse random source problem, inverse coefficient problem and inverse spectral problem based upon these two methods. Specifically, first convergence result is obtained with overwhelming probability for the inverse random source and coefficient problem. For the inverse spectral problem, we propose an iterative method to solve the approximated optimization problem. Numerical examples are presented to illustrate the validity and effectiveness of the proposed method. Several new applications of these inverse problems and future work are mentioned at the end of this paper.