131 research outputs found
Numerical computation of rare events via large deviation theory
An overview of rare events algorithms based on large deviation theory (LDT)
is presented. It covers a range of numerical schemes to compute the large
deviation minimizer in various setups, and discusses best practices, common
pitfalls, and implementation trade-offs. Generalizations, extensions, and
improvements of the minimum action methods are proposed. These algorithms are
tested on example problems which illustrate several common difficulties which
arise e.g. when the forcing is degenerate or multiplicative, or the systems are
infinite-dimensional. Generalizations to processes driven by non-Gaussian
noises or random initial data and parameters are also discussed, along with the
connection between the LDT-based approach reviewed here and other methods, such
as stochastic field theory and optimal control. Finally, the integration of
this approach in importance sampling methods using e.g. genealogical algorithms
is explored
Computability of differential equations
In this chapter, we provide a survey of results concerning the computability and computational complexity of differential equations. In particular, we study the conditions which ensure computability of the solution to an initial value problem for an ordinary differential equation (ODE) and analyze the computational complexity of a computable solution. We also present computability results concerning the asymptotic behaviors of ODEs as well as several classically important partial differential equations.info:eu-repo/semantics/acceptedVersio
Finite Mechanical Proxies for a Class of Reducible Continuum Systems
We present the exact finite reduction of a class of nonlinearly perturbed
wave equations, based on the Amann-Conley-Zehnder paradigm. By solving an
inverse eigenvalue problem, we establish an equivalence between the spectral
finite description derived from A-C-Z and a discrete mechanical model, a well
definite finite spring-mass system. By doing so, we decrypt the abstract
information encoded in the finite reduction and obtain a physically sound proxy
for the continuous problem.Comment: 15 pages, 3 figure
Boundary stabilization of focusing NLKG near unstable equilibria: radial case
We investigate the stability and stabilization of the cubic focusing
Klein-Gordon equation around static solutions on the closed ball in
. First we show that the system is linearly unstable near the
static solution for any dissipative boundary condition . Then by means of boundary controls (both open-loop
and closed-loop) we stabilize the system around this equilibrium exponentially
with rate less than , which is
sharp, provided that the radius of the ball satisfies
Dispersive homogenized models and coefficient formulas for waves in general periodic media
We analyze a homogenization limit for the linear wave equation of second
order. The spatial operator is assumed to be of divergence form with an
oscillatory coefficient matrix that is periodic with
characteristic length scale ; no spatial symmetry properties are
imposed. Classical homogenization theory allows to describe solutions
well by a non-dispersive wave equation on fixed time intervals
. Instead, when larger time intervals are considered, dispersive effects
are observed. In this contribution we present a well-posed weakly dispersive
equation with homogeneous coefficients such that its solutions
describe well on time intervals . More
precisely, we provide a norm and uniform error estimates of the form for . They are accompanied by computable formulas for all
coefficients in the effective models. We additionally provide an
-independent equation of third order that describes dispersion
along rays and we present numerical examples.Comment: 28 pages, 7 figure
Spectral Asymptotics of Elliptic Operators on Manifolds
The study of spectral properties of natural geometric elliptic partial
differential operators acting on smooth sections of vector bundles over
Riemannian manifolds is a central theme in global analysis, differential
geometry and mathematical physics. Instead of studying the spectrum of a
differential operator directly one usually studies its spectral functions,
that is, spectral traces of some functions of the operator, such as the
spectral zeta function \zeta(s)=\Tr L^{-s} and the heat trace
\Theta(t)=\Tr\exp(-tL). The kernel of the heat semigroup
, called the heat kernel, plays a major role in quantum field theory
and quantum gravity, index theorems, non-commutative geometry, integrable
systems and financial mathematics. We review some recent progress in the study
of spectral asymptotics. We study more general spectral functions, such as \Tr
f(tL), that we call quantum heat traces. Also, we define new invariants of
differential operators that depend not only on the their eigenvalues but also
on the eigenfunctions, and, therefore, contain much more information about the
geometry of the manifold. Furthermore, we study some new invariants, such as
\Tr\exp(-tL_+)\exp(-sL_-), that contain relative spectral information of two
differential operators. Finally we show how the convolution of the semigroups
of two different operators can be computed by using purely algebraic methods.Comment: 32 pages; Expanded version of a presentation at the One World
Mathematical Physics Seminar, November 22, 202
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