Georgia Institute of Technology College of Sciences

Nonnegative inverse eigenvalue and singular value problems have been a research focus for decades. It is true that an inverse problem is trivial if the desired matrix is not restricted to any structure. This talk is to present two numerical procedures, based on a conquering procedure and an alternating projection process, to solve inverse eigenvalue and singular value problems for nonnegative matrices, respectively. In theory, we also discuss the existence of nonnegative matrices subject to prescribed eigenvalues and singular values. Though the focus of this talk is on inverse eigenvalue and singular value problems with nonnegative entries, the entire procedure can be straightforwardly applied to other types of structure with no difficulty.

The talk is the continuation of the previous one entitled "Structure-preserving numerical integration of ordinary and partial differential equations [8]" and is aimed to present both classical and more recent results regarding the numerical treatment of nonlinear differential equations, both for deterministic and stochastic problems. The perspective is that of introducing numerical methods which act as structure-preserving integrators, with special emphasys to numerically retaining dissipativity properties possessed by the problem.

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.