2,086 research outputs found
On the convergence of spectral deferred correction methods
In this work we analyze the convergence properties of the Spectral Deferred
Correction (SDC) method originally proposed by Dutt et al. [BIT, 40 (2000), pp.
241--266]. The framework for this high-order ordinary differential equation
(ODE) solver is typically described wherein a low-order approximation (such as
forward or backward Euler) is lifted to higher order accuracy by applying the
same low-order method to an error equation and then adding in the resulting
defect to correct the solution. Our focus is not on solving the error equation
to increase the order of accuracy, but on rewriting the solver as an iterative
Picard integral equation solver. In doing so, our chief finding is that it is
not the low-order solver that picks up the order of accuracy with each
correction, but it is the underlying quadrature rule of the right hand side
function that is solely responsible for picking up additional orders of
accuracy. Our proofs point to a total of three sources of errors that SDC
methods carry: the error at the current time point, the error from the previous
iterate, and the numerical integration error that comes from the total number
of quadrature nodes used for integration. The second of these two sources of
errors is what separates SDC methods from Picard integral equation methods; our
findings indicate that as long as difference between the current and previous
iterate always gets multiplied by at least a constant multiple of the time step
size, then high-order accuracy can be found even if the underlying "solver" is
inconsistent the underlying ODE. From this vantage, we solidify the prospects
of extending spectral deferred correction methods to a larger class of solvers
to which we present some examples.Comment: 29 page
Implicit ODE solvers with good local error control for the transient analysis of Markov models
Obtaining the transient probability distribution vector of a continuous-time Markov chain (CTMC) using an implicit ordinary differential equation (ODE) solver tends to be advantageous in terms of run-time computational cost when the product of the maximum output rate of the CTMC and the largest time of interest is large. In this paper, we show that when applied to the transient analysis of CTMCs, many implicit ODE solvers are such that the linear systems involved in their steps can be solved by using iterative methods with strict control of the 1-norm of the error. This allows the development of implementations of those ODE solvers for the transient analysis of CTMCs that can be more efficient and more accurate than more standard implementations.Peer ReviewedPostprint (published version
A rational deferred correction approach to parabolic optimal control problems
The accurate and efficient solution of time-dependent PDE-constrained optimization problems is a challenging task, in large part due to the very high dimension of the matrix systems that need to be solved. We devise a new deferred correction method for coupled systems of time-dependent PDEs, allowing one to iteratively improve the accuracy of low-order time stepping schemes. We consider two variants of our method, a splitting and a coupling version, and analyze their convergence properties. We then test our approach on a number of PDE-constrained optimization problems. We obtain solution accuracies far superior to that achieved when solving a single discretized problem, in particular in cases where the accuracy is limited by the time discretization. Our approach allows for the direct reuse of existing solvers for the resulting matrix systems, as well as state-of-the-art preconditioning strategies
A multigrid perspective on the parallel full approximation scheme in space and time
For the numerical solution of time-dependent partial differential equations,
time-parallel methods have recently shown to provide a promising way to extend
prevailing strong-scaling limits of numerical codes. One of the most complex
methods in this field is the "Parallel Full Approximation Scheme in Space and
Time" (PFASST). PFASST already shows promising results for many use cases and
many more is work in progress. However, a solid and reliable mathematical
foundation is still missing. We show that under certain assumptions the PFASST
algorithm can be conveniently and rigorously described as a multigrid-in-time
method. Following this equivalence, first steps towards a comprehensive
analysis of PFASST using block-wise local Fourier analysis are taken. The
theoretical results are applied to examples of diffusive and advective type
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