Wednesdays at 4:15 PM in 380C.
Kohn-Sham density functional theory (KSDFT) is the most widely used electronic structure theory for molecules and condensed matter systems. Although KSDFT is often stated as a nonlinear eigenvalue problem, an alternative formulation of the problem, which is more convenient for understanding the convergence of numerical algorithms for solving this type of problem, is based on a nonlinear map known as the Kohn-Sham map. The solution to the KSDFT problem is a fixed point of this nonlinear map. The simplest way to solve the KSDFT problem is to apply a fixed point iteration to the nonlinear equation defined by the Kohn-Sham map. This is commonly known as the self-consistent field (SCF) iteration in the condensed matter physics and chemistry communities. However, this simple approach often fails to converge. The difficulty of reaching convergence can be seen from the analysis of the Jacobian matrix of the Kohn-Sham map, which we will present in this talk. The Jacobian matrix is directly related to the dielectric matrix or the linear response operator in the condense matter community. We will show the different behaviors of insulating and metallic systems in terms of the spectral property of the Jacobian matrix. A particularly difficult case for SCF iteration is systems with mixed insulating and metallic nature, such as metal padded with vacuum, or metallic slabs. We discuss how to use these properties to approximate the Jacobian matrix and to develop effective preconditioners to accelerate the convergence of the SCF iteration. In particular, we introduce a new technique called elliptic preconditioner, which unifies the treatment of large scale metallic and insulating systems at low temperature. Numerical results show that the elliptic preconditioner can effectively accelerate the SCF convergence of metallic systems, insulating systems, and systems of mixed metallic and insulating nature. (Joint work with Chao Yang)
From habitat degradation and climate change to spatial spread of invasive species, dispersal plays a central role in determining how organisms cope with a changing environment. How should organisms disperse “optimally” in spatially heterogeneous environments? In this talk I will discuss some recent development of game theoretic approach on the evolution of dispersal via Lotka-Volterra competition models.
Abstract: We will add a divergence-free drift with increasing magnitude to the fractional Laplacian on a bounded smooth domain, and discuss the behavior of the principal eigenvalue for the Dirichlet problem.
Abstract. I will discuss estimates of Schroedinger perturbations of integral kernels, including Gaussian estimates and a recent sharp 4G Theorem. This is a joint work with W. Hansen, T. Jakubowski, K. Szczypkowski and S. Sydor.
The Burgers-Hilbert equation arises as a model equation for the motion of a vortex patch or vorticity discontinuity in a two-dimensional, inviscid, incompressible fluid flow, and it describes the effect of nonlinear steepening on an interface or wave that oscillates at a constant background frequency. For small amplitudes, these oscillations delay wave breaking. We will explain how non-standard normal form methods can be used to prove an enhanced life-span of small smooth solutions of the Burgers-Hilbert equation in comparison with the inviscid Burgers equation. These normal form methods can be applied to other quasilinear wave equations, for which the Burgers-Hilbert equation provides a useful test case. This is joint work with M. Ifrim, D. Tataru, and D. Wong.
We introduce a new variational method for the numerical homogenization of divergence form elliptic, parabolic and hyperbolic equations with arbitrary rough ($L^\infty$) coefficients. Our method does not rely on concepts of ergodicity or scale-separation but on compactness properties of the solution space and a new variational approach to homogenization. The approximation space is generated by an interpolation basis (over scattered points forming a mesh of resolution $H$) minimizing the $L^2$ norm of source terms obtained by applying the original operator to interpolated functions; its (pre-)computation involves minimizing $\mathcal{O}(H^{-d})$ quadratic (cell) problems on (super-)localized sub-domains of size $\mathcal{O}(H \ln (1/ H))$. The resulting localized linear systems remain sparse and banded. The resulting interpolation basis functions are biharmonic (for $d\leq 3$ and polyharmonic for $d\geq 4$) and can be seen as a generalization of polyharmonic splines to differential operators with arbitrary rough coefficients. The accuracy of the method ($\mathcal{O}(H)$ in energy norm and independent from aspect ratios of the mesh formed by the scattered points) is established via the introduction of a new class of higher-order Poincar\'{e} inequalities. The method bypasses (pre-)computations on the full domain and naturally generalizes to time dependent problems, it also provides a natural solution to the inverse problem of recovering the solution of a divergence form elliptic equation from a finite number of point measurements. This is a joint work with Lei Zhang (Jiaotong University) and Leonid Berlyand (PSU).
Nonlinear elliptic and parabolic PDEs have applications to image processing, first arrival times in wave propagation, homogenization, mathematical finance, stochastic control and games theory. In order to capture geometric features such as folds and corners, and avoid artificial singularities which arise from bad representations of the operators, it is important ti use convergent numerical schemes. In many cases these equations are considered too difficult to solve, which is why linearized models or other approximations are commonly used. Progress has recently been made in building solvers for a class of Geometric PDEs. I'll discuss a few important geometric PDEs which can be solved using a numerical method called Wide Stencil finite difference schemes: Monge-Ampere, Convex Envelope, Infinity Laplace, Mean Curvature, and others. Focusing on the Monge-Ampere equation, which is the seminal geometric PDE, I'll show how naive schemes can work well for smooth solutions, but break down in the singular case. Several groups of researchers have proposed numerical schemes which fail to converge, or converge only in the case of smooth solutions. I'll present a convergent solver which which is fast: comparable to solving the Laplace equation a few times. The most effective notion of weak solutions for fully nonlinear elliptic equations is that of viscosity solutions, developed by Crandall, Ishii, and Lions. Viscosity solutions enjoy strong stability properties, and allow for uniform convergence of approximations, using the Barles-Souganidis theorem. This theory is used to prove convergence of the finite difference method. The talk will be accessible to graduate students.
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