Hengguang Li

Serge Nicaise, Hengguang Li, Anna Mazzucato

We consider the regularity of a mixed boundary value problem for the Laplace operator on a polyhedral domain, where Ventcel boundary conditions are imposed on one face of the polyhedron and Dirichlet boundary conditions are imposed on the complement of that face in the boundary. We establish improved regularity estimates for the trace of the variational solution on the Ventcel face, and use them to derive a decomposition of the solution into a regular and a singular part that belongs to suitable weighted Sobolev spaces. This decomposition, in turn, via interpolation estimates both in the interior as well as on the Ventcel face, allows us to perform an a priori error analysis for the Finite Element approximation of the solution on anisotropic graded meshes. Numerical tests support the theoretical analysis.

Constantin Bacuta, Hengguang Li, Victor Nistor

We study families of strongly elliptic, second order differential operators with singular coefficients on domains with conical points. We obtain uniform estimates on their inverses and on the regularity of the solutions to the associated Poisson problem with mixed boundary conditions. The coefficients and the solutions belong to (suitable) weighted Sobolev spaces. The space of coefficients is a Banach space that contains, in particular, the space of smooth functions. Hence, our results extend classical well-posedness results for strongly elliptic equations in domains with conical points to problems with singular coefficients. We furthermore provide precise uniform estimates on the norms of the solution operators.

Eugenie Hunsicker, Hengguang Li, Victor Nistor et al.

Let $V$ be a potential on $\RR^3$ that is smooth everywhere except at a discrete set $\maS$ of points, where it has singularities of the form $Z/ρ^2$, with $ρ(x) = |x - p|$ for $x$ close to $p$ and $Z$ continuous on $\RR^3$ with $Z(p) > -1/4$ for $p \in \maS$. Also assume that $ρ$ and $Z$ are smooth outside $\maS$ and $Z$ is smooth in polar coordinates around each singular point. We either assume that $V$ is periodic or that the set $\maS$ is finite and $V$ extends to a smooth function on the radial compactification of $\RR^3$ that is bounded outside a compact set containing $\maS$. In the periodic case, we let $Λ$ be the periodicity lattice and define $\TT := \RR^3/ Λ$. We obtain regularity results in weighted Sobolev space for the eigenfunctions of the Schrödinger-type operator $H = -Δ+ V$ acting on $L^2(\TT)$, as well as for the induced $\vt k$--Hamiltonians $\Hk$ obtained by restricting the action of $H$ to Bloch waves. Under some additional assumptions, we extend these regularity and solvability results to the non-periodic case. We sketch some applications to approximation of eigenfunctions and eigenvalues that will be studied in more detail in a second paper.

Hengguang Li, Victor Nistor, Yu Qiao

Let $Ω\subset \RR^d$, $d \geqslant 1$, be a bounded domain with piecewise smooth boundary $\partial Ω$ and let $U$ be an open subset of a Banach space $Y$. Motivated by questions in "Uncertainty Quantification," we consider a parametric family $P = (P_y)_{y \in U}$ of uniformly strongly elliptic, second order partial differential operators $P_y$ on $Ω$. We allow jump discontinuities in the coefficients. We establish a regularity result for the solution $u: Ω\times U \to \RR$ of the parametric, elliptic boundary value/transmission problem $P_y u_y = f_y$, $y \in U$, with mixed Dirichlet-Neumann boundary conditions in the case when the boundary and the interface are smooth and in the general case for $d=2$. Our regularity and well-posedness results are formulated in a scale of broken weighted Sobolev spaces $\hat\maK^{m+1}_{a+1}(Ω)$ of Babuška-Kondrat'ev type in $Ω$, possibly augmented by some locally constant functions. This implies that the parametric, elliptic PDEs $(P_y)_{y \in U}$ admit a shift theorem that is uniform in the parameter $y\in U$. In turn, this then leads to $h^m$-quasi-optimal rates of convergence (i.e. algebraic orders of convergence) for the Galerkin approximations of the solution $u$, where the approximation spaces are defined using the "polynomial chaos expansion" of $u$ with respect to a suitable family of tensorized Lagrange polynomials, following the method developed by Cohen, Devore, and Schwab (2010).

Eugenie Hunsicker, Hengguang Li, Victor Nistor et al.

Let $V$ be a {\em periodic} potential on $\RR^3$ that is smooth everywhere except at a discrete set $\maS$ of points, where it has singularities of the form $Z/ρ^2$, with $ρ(x) = |x - p|$ for $x$ close to $p$ and $Z$ is continuous, $Z(p) > -1/4$ for $p \in \maS$. We also assume that $ρ$ and $Z$ are smooth outside $\maS$ and $Z$ is smooth in polar coordinates around each singular point. Let us denote by $Λ$ the periodicity lattice and set $\TT := \RR^3/ Λ$. In the first paper of this series \cite{HLNU1}, we obtained regularity results in weighted Sobolev space for the eigenfunctions of the Schrödinger-type operator $H = -Δ+ V$ acting on $L^2(\TT)$, as well as for the induced $\vt k$--Hamiltonians $\Hk$ obtained by resticting the action of $H$ to Bloch waves. In this paper we present two related applications: one to the Finite Element approximation of the solution of $(L+\Hk) v = f$ and one to the numerical approximation of the eigenvalues, $λ$, and eigenfunctions, $u$, of $\Hk$. We give optimal, higher order convergence results for approximation spaces defined piecewise polynomials. Our numerical tests are in good agreement with the theoretical results.

Young Ju Lee, Hengguang Li

We study the regularity and finite element approximation of the axisymmetric Stokes problem on a polygonal domain $Ω$. In particular, taking into account the singular coefficients in the equation and non-smoothness of the domain, we establish the well-posedness and full regularity of the solution in new weighted Sobolev spaces $\maK^m_{μ, 1}(Ω)$. Using our a priori results, we give a specific construction of graded meshes on which the Taylor-Hood mixed method approximates singular solutions at the optimal convergence rate. Numerical tests are presented to confirm the theoretical results in the paper.

Hengguang Li

Consider the Poisson equation with the Dirichlet boundary condition on a three-dimensional polyhedral domain. For singular solutions from the non-smoothness of the domain boundary, we propose new anisotropic tetrahedral mesh refinement algorithms to improve the convergence of finite element approximation. The proposed algorithm is simple, explicit, and requires less geometric conditions on the mesh and on the domain. Then, we develop interpolation error estimates in suitable weighted spaces for the anisotropic mesh. These estimates can be used to design optimal finite element methods approximating singular solutions. We report numerical test results to validate the method.

Gerard Awanou, Hengguang Li, Eric Malitz

The purpose of this paper is to analyze an efficient method for the solution of the nonlinear system resulting from the discretization of the elliptic Monge-Ampère equation by a $C^0$ interior penalty method with Lagrange finite elements. We consider the two-grid method for nonlinear equations which consists in solving the discrete nonlinear system on a coarse mesh and using that solution as initial guess for one iteration of Newton's method on a finer mesh. Thus both steps are inexpensive. We give quasi-optimal $W^{1,\infty}$ error estimates for the discretization and estimate the difference between the interior penalty solution and the two-grid numerical solution. Numerical experiments confirm the computational efficiency of the approach compared to Newton's method on the fine mesh.

Aixiang Chen, Bingchuan Chen, Xiaolong Chai et al.

SGD (Stochastic Gradient Descent) is a popular algorithm for large scale optimization problems due to its low iterative cost. However, SGD can not achieve linear convergence rate as FGD (Full Gradient Descent) because of the inherent gradient variance. To attack the problem, mini-batch SGD was proposed to get a trade-off in terms of convergence rate and iteration cost. In this paper, a general CVI (Convergence-Variance Inequality) equation is presented to state formally the interaction of convergence rate and gradient variance. Then a novel algorithm named SSAG (Stochastic Stratified Average Gradient) is introduced to reduce gradient variance based on two techniques, stratified sampling and averaging over iterations that is a key idea in SAG (Stochastic Average Gradient). Furthermore, SSAG can achieve linear convergence rate of $\mathcal {O}((1-\fracμ{8CL})^k)$ at smaller storage and iterative costs, where $C\geq 2$ is the category number of training data. This convergence rate depends mainly on the variance between classes, but not on the variance within the classes. In the case of $C\ll N$ ($N$ is the training data size), SSAG's convergence rate is much better than SAG's convergence rate of $\mathcal {O}((1-\fracμ{8NL})^k)$. Our experimental results show SSAG outperforms SAG and many other algorithms.