657 research outputs found
An implicit algorithm for validated enclosures of the solutions to variational equations for ODEs
We propose a new algorithm for computing validated bounds for the solutions
to the first order variational equations associated to ODEs. These validated
solutions are the kernel of numerics computer-assisted proofs in dynamical
systems literature. The method uses a high-order Taylor method as a predictor
step and an implicit method based on the Hermite-Obreshkov interpolation as a
corrector step. The proposed algorithm is an improvement of the -Lohner
algorithm proposed by Zgliczy\'nski and it provides sharper bounds.
As an application of the algorithm, we give a computer-assisted proof of the
existence of an attractor set in the R\"ossler system, and we show that the
attractor contains an invariant and uniformly hyperbolic subset on which the
dynamics is chaotic, that is, conjugated to subshift of finite type with
positive topological entropy.Comment: 33 pages, 11 figure
On solving Ordinary Differential Equations using Gaussian Processes
We describe a set of Gaussian Process based approaches that can be used to
solve non-linear Ordinary Differential Equations. We suggest an explicit
probabilistic solver and two implicit methods, one analogous to Picard
iteration and the other to gradient matching. All methods have greater accuracy
than previously suggested Gaussian Process approaches. We also suggest a
general approach that can yield error estimates from any standard ODE solver
A Survey on Continuous Time Computations
We provide an overview of theories of continuous time computation. These
theories allow us to understand both the hardness of questions related to
continuous time dynamical systems and the computational power of continuous
time analog models. We survey the existing models, summarizing results, and
point to relevant references in the literature
Delta-Complete Decision Procedures for Satisfiability over the Reals
We introduce the notion of "\delta-complete decision procedures" for solving
SMT problems over the real numbers, with the aim of handling a wide range of
nonlinear functions including transcendental functions and solutions of
Lipschitz-continuous ODEs. Given an SMT problem \varphi and a positive rational
number \delta, a \delta-complete decision procedure determines either that
\varphi is unsatisfiable, or that the "\delta-weakening" of \varphi is
satisfiable. Here, the \delta-weakening of \varphi is a variant of \varphi that
allows \delta-bounded numerical perturbations on \varphi. We prove the
existence of \delta-complete decision procedures for bounded SMT over reals
with functions mentioned above. For functions in Type 2 complexity class C,
under mild assumptions, the bounded \delta-SMT problem is in NP^C.
\delta-Complete decision procedures can exploit scalable numerical methods for
handling nonlinearity, and we propose to use this notion as an ideal
requirement for numerically-driven decision procedures. As a concrete example,
we formally analyze the DPLL framework, which integrates Interval
Constraint Propagation (ICP) in DPLL(T), and establish necessary and sufficient
conditions for its \delta-completeness. We discuss practical applications of
\delta-complete decision procedures for correctness-critical applications
including formal verification and theorem proving.Comment: A shorter version appears in IJCAR 201
Recursion Schemes, Discrete Differential Equations and Characterization of Polynomial Time Computations
This paper studies the expressive and computational power of discrete Ordinary Differential Equations (ODEs). It presents a new framework using discrete ODEs as a central tool for computation and algorithm design. We present the general theory of discrete ODEs for computation theory, we illustrate this with various examples of algorithms, and we provide several implicit characterizations of complexity and computability classes.
The proposed framework presents an original point of view on complexity and computation classes. It unifies several constructions that have been proposed for characterizing these classes including classical approaches in implicit complexity using restricted recursion schemes, as well as recent characterizations of computability and complexity by classes of continuous ordinary differential equations. It also helps understanding the relationships between analog computations and classical discrete models of computation theory.
At a more technical point of view, this paper points out the fundamental role of linear (discrete) ordinary differential equations and classical ODE tools such as changes of variables to capture computability and complexity measures, or as a tool for programming many algorithms
High order operator splitting methods based on an integral deferred correction framework
Integral deferred correction (IDC) methods have been shown to be an efficient
way to achieve arbitrary high order accuracy and possess good stability
properties. In this paper, we construct high order operator splitting schemes
using the IDC procedure to solve initial value problems (IVPs). We present
analysis to show that the IDC methods can correct for both the splitting and
numerical errors, lifting the order of accuracy by with each correction,
where is the order of accuracy of the method used to solve the correction
equation. We further apply this framework to solve partial differential
equations (PDEs). Numerical examples in two dimensions of linear and nonlinear
initial-boundary value problems are presented to demonstrate the performance of
the proposed IDC approach.Comment: 33 pages, 22 figure
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