5,668 research outputs found
Finite element approximation of high-dimensional transport-dominated diffusion problems
High-dimensional partial differential equations with nonnegative characteristic form arise in numerous mathematical models in science. In problems of this kind, the computational challenge of beating the exponential growth of complexity as a function of dimension is exacerbated by the fact that the problem may be transport-dominated. We develop the analysis of stabilised sparse finite element methods for such high-dimensional, non-self-adjoint and possibly degenerate partial differential equations.\ud
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(Presented as an invited lecture under the title "Computational multiscale modelling: Fokker-Planck equations and their numerical analysis" at the Foundations of Computational Mathematics conference in Santander, Spain, 30 June - 9 July, 2005.
Convergence of the stochastic Euler scheme for locally Lipschitz coefficients
Stochastic differential equations are often simulated with the Monte Carlo
Euler method. Convergence of this method is well understood in the case of
globally Lipschitz continuous coefficients of the stochastic differential
equation. The important case of superlinearly growing coefficients, however,
has remained an open question. The main difficulty is that numerically weak
convergence fails to hold in many cases of superlinearly growing coefficients.
In this paper we overcome this difficulty and establish convergence of the
Monte Carlo Euler method for a large class of one-dimensional stochastic
differential equations whose drift functions have at most polynomial growth.Comment: Published at http://www.springerlink.com/content/g076w80730811vv3 in
the Foundations of Computational Mathematics 201
Hodge Theory on Metric Spaces
Hodge theory is a beautiful synthesis of geometry, topology, and analysis,
which has been developed in the setting of Riemannian manifolds. On the other
hand, spaces of images, which are important in the mathematical foundations of
vision and pattern recognition, do not fit this framework. This motivates us to
develop a version of Hodge theory on metric spaces with a probability measure.
We believe that this constitutes a step towards understanding the geometry of
vision.
The appendix by Anthony Baker provides a separable, compact metric space with
infinite dimensional \alpha-scale homology.Comment: appendix by Anthony W. Baker, 48 pages, AMS-LaTeX. v2: final version,
to appear in Foundations of Computational Mathematics. Minor changes and
addition
Metrics for generalized persistence modules
We consider the question of defining interleaving metrics on generalized
persistence modules over arbitrary preordered sets. Our constructions are
functorial, which implies a form of stability for these metrics. We describe a
large class of examples, inverse-image persistence modules, which occur
whenever a topological space is mapped to a metric space. Several standard
theories of persistence and their stability can be described in this framework.
This includes the classical case of sublevelset persistent homology. We
introduce a distinction between `soft' and `hard' stability theorems. While our
treatment is direct and elementary, the approach can be explained abstractly in
terms of monoidal functors.Comment: Final version; no changes from previous version. Published online Oct
2014 in Foundations of Computational Mathematics. Print version to appea
Upwinding in finite element systems of differential forms
We provide a notion of finite element system, that enables the construction spaces of differential forms, which can be used for the numerical solution of variationally posed partial difeerential equations. Within this framework, we introduce a form of upwinding, with the aim of stabilizing methods for the purposes of computational uid dynamics, in the vanishing viscosity regime.
Published as the Smale Prize Lecture in: Foundations of computational mathematics, Budapest 2011, London Mathematical Society Lecture Note Series, 403, Cambridge University Press, 2013
The foundations of spectral computations via the Solvability Complexity Index hierarchy: Part I
The problem of computing spectra of operators is arguably one of the most
investigated areas of computational mathematics. Recent progress and the
current paper reveal that, unlike the finite-dimensional case,
infinite-dimensional problems yield a highly intricate infinite classification
theory determining which spectral problems can be solved and with which type of
algorithms. Classifying spectral problems and providing optimal algorithms is
uncharted territory in the foundations of computational mathematics. This paper
is the first of a two-part series establishing the foundations of computational
spectral theory through the Solvability Complexity Index (SCI) hierarchy and
has three purposes. First, we establish answers to many longstanding open
questions on the existence of algorithms. We show that for large classes of
partial differential operators on unbounded domains, spectra can be computed
with error control from point sampling operator coefficients. Further results
include computing spectra of operators on graphs with error control, the
spectral gap problem, spectral classifications, and discrete spectra,
multiplicities and eigenspaces. Second, these classifications determine which
types of problems can be used in computer-assisted proofs. The theory for this
is virtually non-existent, and we provide some of the first results in this
infinite classification theory. Third, our proofs are constructive, yielding a
library of new algorithms and techniques that handle problems that before were
out of reach. We show several examples on contemporary problems in the physical
sciences. Our approach is closely related to Smale's program on the foundations
of computational mathematics initiated in the 1980s, as many spectral problems
can only be computed via several limits, a phenomenon shared with the
foundations of polynomial root finding with rational maps, as proved by
McMullen
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