In this paper, we establish the exponential convergence theory for the multipole and local expansions, shifting and translation operators for the Green’s function of 3-dimensional Laplace equation in layered media. An immediate application of the theory is to ensure the exponential convergence of the FMM which has been shown by the numerical results reported in [27]. As the Green’s function in layered media consists of free space and reaction field components and the theory for the free space components is well known, this paper will focus on the analysis for the reaction components. We first prove that the density functions in the integral representations of the reaction components are analytic and bounded in the right half complex wave number plane. Then, by using the Cagniard-de Hoop transform and contour deformations, estimates for the remainder terms of the truncated expansions are given, and, as a result, the exponential convergence for the expansions and translation operators is proven.

We present an arbitrary order discontinuous Galerkin finite element method for solving the biharmonic interface problem on the unfitted mesh. The approximation space is constructed by a patch reconstruction process with at most one degrees of freedom per element. The discrete problem is based on the symmetric interior penalty method and the jump conditions are weakly imposed by the Nitsche’s technique. The $C^2$-smooth interface is allowed to intersect elements in a very general fashion and the stability near the interface is naturally ensured by the patch reconstruction. We prove the optimal a priori error estimate under the energy norm and the $L^2$ norm. Numerical results are provided to verify the theoretical analysis.

In this paper, we study Riemannian optimization methods for the problem of nonnegative matrix completion that is to recover a nonnegative low rank matrix from its partial observed entries. With the underlying matrix incoherence conditions, we show that when the number $m$ of observed entries are sampled independently and uniformly without replacement, the inexact Riemannian gradient descent method can recover the underlying $n_{1}$-by-$n_{2}$ nonnegative matrix of rank $r$ provided that $m$ is of $\mathcal{O}(r^{2} s \log^2s )$, where $s = \max \{n_{1},n_{2} \}$. Numerical examples are given to illustrate that the nonnegativity property would be useful in the matrix recovery. In particular, we demonstrate the number of samples required to recover the underlying low rank matrix with using the nonnegativity property is smaller than that without using the property.

In this paper, a rough Heston model with variable volatility of volatility (vol-of-vol) is derived by modifying the generalized nonlinear Hawkes process and extending the scaling techniques. Then the nonlinear fractional Riccati equation for the characteristic function of the asset log-price is derived. The existence, uniqueness and regularity of the solution to the nonlinear fractional Riccati equation are proved and the equation is solved by the Adams methods. Finally the Fourier-cosine methods are combined with the Adams methods to price the options.