purpose of this talk to analyze the behaviour of multi-value numerical
methods acting as structure-preserving integrators for the numerical
solution of ordinary and partial differential equations (PDEs), with
special emphasys to Hamiltonian problems and reaction-diffusion PDEs. As
regards Hamiltonian problems, we provide a rigorous long-term error
analyis obtained by means of backward error analysis arguments, leading
to sharp estimates for the parasitic solution components and for the
error in the Hamiltonian. As regards PDEs, we consider
structure-preservation properties in the numerical solution of
oscillatory problems based on reaction-diffusion equations, typically
modelling oscillatory biological systems, whose solutions oscillate both
in space and in time. Special purpose numerical methods able to
accurately retain the oscillatory behaviour are presented.
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
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.
so-called distance to instability, is a well-known measure of linear
dynamical system stability. Existing techniques compute this quantity
accurately but the cost is of the order of multiple SVDs of order n,
which makes the method suitable to middle size problems.
A new approach is presented, based on Newton's iteration applied to
pseudospectral abscissa, whose implementation is obtained by
discretization on differential equation for low-rank matrices,
particularly suited for large sparse matrices.
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.
from random observations via synthesis and analysis methods in
compressive signal processing (CSP). CSP is a new and exciting field
which arose as an efficient alternative to traditional signal
acquisition techniques. Using a (usually random) projection, signals
are measured directly in compressed form, and methods are then needed
to recover the signal from those measurements. Synthesis methods
attempt to identify the low-dimensional representation of the signal
directly, whereas analysis type methods reconstruct in signal space.
We also discuss special cases including provable near-optimal
reconstruction guarantees for total-variation
minimization and new techniques in super-resolution.