Abstract: In this talk, we consider computational aspects of different quantum dmechanical problems in form of a linear or nonlinear eigenvalue problem. After introducing the density-matrix formalism, whose elements can be identified with elements of the Grassmann manifold and that allows to elegantly deal with clusters of eigenvalues and arbitrary multiplicities, we will proceed with three applications of the formalism. First, we will discuss a posteriori error estimators for the density-matrix error for linear Schrödinger eigenvalue problems. In the second part, we consider ab-initio Born-Oppenheimer Molecular Dynamics (BOMD) at the level of Kohn-Sham Density Functional Theory (DFT) and exploit the geometric structure of the Grassmann manifold computationally by making use of the Grassmann exponential and logarithm to provide accurate initial guesses of the resolution of the nonlinear DFT-equations at each time-step. Finally, we show how the reduced basis method allows breaking the curse of dimensionality for the ground-state computation of an array of excited Rydberg atoms where the dimension of the Hilbert space grows exponentially with the number of particles.

Abstract:In this talk, we will discuss the global existence of classical solutions to the incompressible isotropic Hookean elastodynamics in three-dimensional thin domain with periodic boundary condition. This work is joint with Yuan Cai (Fudan University)

Abstract: In this talk, I will first review some problems and known results on the long time dynamics of NLS. Then I will introduce our recent progress in the mass critical setting, including the existence and uniqueness of multi-bubble blow-up solutions, Bourgain-Wang type solutions and multi solitons. This talk is based on the joint works with Prof. Cao Daomin, Prof. Rockner Michael, and Prof. Zhang Deng.

Abstract:

Abstract: This talk concerns the construction of high-order energy-decaying numerical methods for gradient flows of evolving surfaces with curvature-dependent energy functionals. The semidiscrete evolving surface FEM is derived based on the calculus of variation of the semidiscrete surface energy functional. This makes the semidiscrete problem naturally inherit the energy decay structure. With this property, the semidiscrete problem is furthermore formulated as a gradient flow system of ODEs. The averaged vector-field spectral collocation method is used for time discretization of the ODEs to preserve energy decay at the fully discrete level while achieving high-order accuracy in time. We will also present some numerical examples to illustrate the accuracy and energy diminishing property.

Abstract: Computational hemodynamics is being developed as an alternative approach for assisting clinical diagnosis and treatment planning for liver diseases. The technology is non-invasive, but the computational time could be high when the full geometry of the blood vessels is taken into account. Existing approaches use either one-dimensional model of the artery or simplified three-dimensional tubular geometry in order to reduce the computational time, but the accuracy is sometime compromised, for example, when simulating blood flows in arteries with plaque. In this work, we study a highly parallel method for the transient incompressible Navier-Stokes equations for the simulation of the blood flows in the full three-dimensional patient-specific hepatic artery, portal vein and hepatic vein. As applications, we also simulate the flow in a patient with hepatectomy and calculate the portal pressure gradient. One of the advantages of simulating blood flows in all hepatic vessels is that it provides a direct estimate of the portal pressure gradient, which is a gold standard value to assess the portal hypertension. Moreover, the robustness and scalability of the algorithm are also investigated. A 83% parallel efficiency is achieved for solving a problem with 7 million elements on a supercomputer with more than 1000 processor cores.

Abstract: In many applications, the information about the number of eigenvalues inside a given region, rather than the eigenvalues, is required. In this talk, I will introduce a contour-integral based method for this purpose. Our method is motivated by two findings. There exist methods for estimating the number of eigenvalues inside a region in the complex plane, but our method is able to compute the number exactly. Our method has a good potential to be implemented on a high-performance parallel architecture. Numerical experiments will be reported to show the viability of our method.

Abstract: TBA

Abstract: TBA

Abstract: TBA

Abstract: TBA

Abstract: TBA

Abstract: TBA

Abstract: TBA

Abstract: TBA

Abstract: TBA

Abstract: TBA

Abstract: TBA

Abstract: TBA

Abstract: TBA

Abstract: TBA