• Abstract

Consider the inverse acoustic scattering of time-harmonic point sources by a locally perturbed interface with bounded obstacles embedded in the lower half-space. A Newton-type iterative method is proposed to simultaneously reconstruct the locally rough interface and embedded obstacles by taking partial near-field measurements in the upper half-space. The method relies on a differentiability analysis of the scattering problem with respect to the locally rough interface and the embedded obstacle, which is established by introducing a kind of new shape derivatives and reducing the original model to an equivalent system of integral equations defined in a bounded domain. With a slight modification, the inversion method can be easily generalized to reconstruct local perturbations of a global rough interface. Finally, numerical results are presented to illustrate the effectiveness of the inversion method with the multi-frequency data.

• General Summary

• Abstract

Multiple solutions are common in various non-convex problems arising from industrial and scientific computing. Nonetheless, understanding the nontrivial solutions’ qualitative properties seems limited, partially due to the lack of efficient and reliable numerical methods. In this paper, we design a dedicated numerical method to explore these nontrivial solutions further. We first design an improved adaptive orthogonal basis deflation method by combining the adaptive orthogonal basis method with a bisection-deflation algorithm. We then apply the proposed new method to study the impact of domain changes on multiple solutions of certain nonlinear elliptic equations. When the domain varies from a circular disk to an elliptical disk, the corresponding functional value changes dramatically for some particular solutions, which indicates that these nontrivial solutions in the circular domain may become unstable in the elliptical domain. Moreover, several theoretical results on multiple solutions in the existing literature are verified. For the nonlinear sine-Gordon equation with parameter $λ,$ nontrivial solutions are found for $λ > λ_2,$ here $λ_2$ is the second eigenvalue of the corresponding linear eigenvalue problem. For the singularly perturbed Ginzburg-Landau equation, highly concentrated solutions are numerically verified, suggesting that their convergent limit is a delta function when the perturbation parameter goes to zero.

• General Summary

• Abstract

Stochastic gradient descent (SGD) methods have gained widespread popularity for solving large-scale optimization problems. However, the inherent variance in SGD often leads to slow convergence rates. We introduce a family of unbiased stochastic gradient estimators that encompasses existing estimators from the literature and identify a gradient estimator that not only maintains unbiasedness but also achieves minimal variance. Compared with the existing estimator used in SGD algorithms, the proposed estimator demonstrates a significant reduction in variance. By utilizing this stochastic gradient estimator to approximate the full gradient, we propose two mini-batch stochastic conjugate gradient algorithms with minimal variance. Under the assumptions of strong convexity and smoothness on the objective function, we prove that the two algorithms achieve linear convergence rates. Numerical experiments validate the effectiveness of the proposed gradient estimator in reducing variance and demonstrate that the two stochastic conjugate gradient algorithms exhibit accelerated convergence rates and enhanced stability.

• General Summary