Georgia Institute of Technology College of Sciences

Recent experimental and numerical observations have shown the significance of the Basset--Boussinesq memory term on the dynamics of small spherical rigid particles (or inertial particles) suspended in an ambient fluid flow. These observations suggest an algebraic decay to an asymptotic state, as opposed to the exponential convergence in the absence of the memory term. I discuss the governing equations of motion for the inertial particles, i.e. the Maxey-Riley equation, including a fractional order derivative in time. Then I show that the observed algebraic decay is a universal property of the Maxey--Riley equation. Specifically, the particle velocity decays algebraically in time to a limit that is O(\epsilon)-close to the fluid velocity, where 0<\epsilon<<1 is proportional to the square of the ratio of the particle radius to the fluid characteristic length-scale. These results follows from a sharp analytic upper bound that we derive for the particle velocity.

Observations of high energy density environments, from supernovae implosions/explosions to inertial confinement fusion, are determined by many different physical effects acting concurrently. For example, one set of equations will describe material motion, while another set will describe the spatial flow of energy. The relevant spatial and temporal scales can vary substantially. Since direct measurement is difficult if not impossible, and the relevant physics happen concurrently, computer simulation becomes an important tool to understand how emergent behavior depends on the constituent laws governing the evolution of the system. Further, computer simulation can provide a means to use observation to constrain underlying physical models. This talk shall examine the challenges associated with developing computational multiphysics simulation. In particular this talk will outline some of the physics, the relevant mathematical models, the associated algorithmic challenges, some of which are driven by emerging compute architectures. The problem as a whole can be formidable and an effective solution couples many disciplines together.

An alternating direction approximate Newton method (ADAN) is developedfor solving inverse problems of the form$\min \{\phi(Bu) +1/2\norm{Au-f}_2^2\}$,where $\phi$ is a convex function, possibly nonsmooth,and $A$ and $B$ are matrices.Problems of this form arise in image reconstruction where$A$ is the matrix describing the imaging device, $f$ is themeasured data, $\phi$ is a regularization term, and $B$ is aderivative operator. The proposed algorithm is designed tohandle applications where $A$ is a large, dense ill conditionmatrix. The algorithm is based on the alternating directionmethod of multipliers (ADMM) and an approximation to Newton's method in which Newton's Hessian is replaced by a Barzilai-Borwein approximation. It is shown that ADAN converges to a solutionof the inverse problem; neither a line search nor an estimateof problem parameters, such as a Lipschitz constant, are required.Numerical results are provided using test problems fromparallel magnetic resonance imaging (PMRI).ADAN performed better than the other schemes that were tested.

Weak Galerkin finite element method is a new and efficient numerical method for solving PDEs which was first proposed by Junping Wang and Xiu Ye in 2011. The main idea of WG method is to introduce weak differential operators and apply them to the corresponding variational formulations to solve PDEs. In this talk, I will focus on the WG methods for biharmonic equations, maxwell equations and div-curl equations.