1,202 research outputs found
Time complexity and convergence analysis of domain theoretic Picard method
We present an implementation of the domain-theoretic Picard method for solving initial value problems (IVPs) introduced by Edalat and Pattinson [1]. Compared to Edalat and Pattinson's implementation, our algorithm uses a more efficient arithmetic based on an arbitrary precision floating-point library. Despite the additional overestimations due to floating-point rounding, we obtain a similar bound on the convergence rate of the produced approximations. Moreover, our convergence analysis is detailed enough to allow a static optimisation in the growth of the precision used in successive Picard iterations. Such optimisation greatly improves the efficiency of the solving process. Although a similar optimisation could be performed dynamically without our analysis, a static one gives us a significant advantage: we are able to predict the time it will take the solver to obtain an approximation of a certain (arbitrarily high) quality
Recursive Solution of Initial Value Problems with Temporal Discretization
We construct a continuous domain for temporal discretization of differential
equations. By using this domain, and the domain of Lipschitz maps, we formulate
a generalization of the Euler operator, which exhibits second-order
convergence. We prove computability of the operator within the framework of
effectively given domains. The operator only requires the vector field of the
differential equation to be Lipschitz continuous, in contrast to the related
operators in the literature which require the vector field to be at least
continuously differentiable. Within the same framework, we also analyze
temporal discretization and computability of another variant of the Euler
operator formulated according to Runge-Kutta theory. We prove that, compared
with this variant, the second-order operator that we formulate directly, not
only imposes weaker assumptions on the vector field, but also exhibits superior
convergence rate. We implement the first-order, second-order, and Runge-Kutta
Euler operators using arbitrary-precision interval arithmetic, and report on
some experiments. The experiments confirm our theoretical results. In
particular, we observe the superior convergence rate of our second-order
operator compared with the Runge-Kutta Euler and the common (first-order) Euler
operators.Comment: 50 pages, 6 figure
Making big steps in trajectories
We consider the solution of initial value problems within the context of
hybrid systems and emphasise the use of high precision approximations (in
software for exact real arithmetic). We propose a novel algorithm for the
computation of trajectories up to the area where discontinuous jumps appear,
applicable for holomorphic flow functions. Examples with a prototypical
implementation illustrate that the algorithm might provide results with higher
precision than well-known ODE solvers at a similar computation time
Proof mining in metric fixed point theory and ergodic theory
In this survey we present some recent applications of proof mining to the
fixed point theory of (asymptotically) nonexpansive mappings and to the
metastability (in the sense of Terence Tao) of ergodic averages in uniformly
convex Banach spaces.Comment: appeared as OWP 2009-05, Oberwolfach Preprints; 71 page
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