Consider an inverse problem that aims to identify key statistical properties of the profile for the unknown random perfectly conducting grating structure by boundary measurements of the diffracted fields in transverse magnetic polarization. The method proposed in this paper is based on a novel combination of the Monte Carlo technique, a continuation method and the Karhunen-Loève expansion for the uncertainty quantification of the random structure. Numerical results are presented to demonstrate the effectiveness of the proposed method.

The concept of $tt^∗$ geometric structure was introduced by physicists (see [4, 10] and references therein), and then studied firstly in mathematics by C. Hertling [28]. It is believed that the $tt^∗$ geometric structure contains the whole genus 0 information of a two dimensional topological field theory. In this paper, we propose the LG/CY correspondence conjecture for  $tt^∗$ geometry and obtain the following result. Let $f ∈ \mathbb{C}[z_0,...,z_{n+1}]$ be a nondegenerate homogeneous polynomial of degree $n$+2, then it defines a Calabi-Yau model represented by a Calabi-Yau hypersurface $X_f$ in $\mathbb{CP}^{n+1}$ or a Landau-Ginzburg model represented by a hypersurface singularity ($\mathbb{C}^{n+2}, f$), both can be written as a $tt^∗$ structure. We proved that there exists a $tt^∗$ substructure on Landau-Ginzburg side, which should correspond to the $tt^∗$ structure from variation of Hodge structures in Calabi-Yau side. We build the isomorphism of almost all structures in $tt^∗$ geometries between these two models except the isomorphism between real structures.

In this paper, we consider the modified one-dimensional Schrödinger equation:
$$(D_t-F(D))u=λ|u|^2u,$$

where F(ξ) is a second order constant coefficients classical elliptic symbol, and with smooth initial datum of size $ε≪1$. We prove that the solution is global-in-time, combining the vector fields method and a semiclassical analysis method introduced by Delort. Moreover, we get a one term asymptotic expansion for $u$ when $t→+∞$.

In this work, we study the gradient projection method for solving a class of stochastic control problems by using a mesh free approximation approach to implement spatial dimension approximation. Our main contribution is to extend the existing gradient projection method to moderate high-dimensional space. The moving least square method and the general radial basis function interpolation method are introduced as showcase methods to demonstrate our computational framework, and rigorous numerical analysis is provided to prove the convergence of our meshfree approximation approach. We also present several numerical experiments to validate the theoretical results of our approach and demonstrate the performance meshfree approximation in solving stochastic optimal control problems.