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Related Experiment Video

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Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns

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Published on: August 30, 2013

A POSTERIORI ERROR ANALYSIS OF TWO STAGE COMPUTATION METHODS WITH APPLICATION TO EFFICIENT DISCRETIZATION AND THE

Jehanzeb Hameed Chaudhry¹, Don Estep², Simon Tavener³

¹Department of Mathematics & Statistics, The University of New Mexico, Albuquerque, NM 87131.

SIAM Journal on Numerical Analysis

|October 31, 2017

Summary

This study introduces a two-stage numerical method for solving initial value problems. The approach enhances accuracy and efficiency by combining coarse and fine discretization solutions, improving error estimation and parallel computing.

Keywords:

A posteriori error estimation Parareal algorithm adjoint problem cancellation of error efficient time step selection initial value problem parallel in time two stage computation

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Area of Science:

Numerical Analysis
Computational Mathematics
Scientific Computing

Background:

Initial value problems (IVPs) are fundamental in science and engineering.
Existing numerical methods face challenges in accuracy, efficiency, and error control.
Two-stage computational approaches offer potential for improved performance.

Purpose of the Study:

To develop a general framework for two-stage numerical methods for IVPs.
To introduce a robust a posteriori error analysis for these methods.
To enhance the efficiency and accuracy of solving complex computational problems.

Main Methods:

Formulation of a general two-stage computation strategy.
Development of a posteriori error analysis using computable residuals and adjoint problems.
Application to dual-weighted error estimation and the Parareal Algorithm.

Main Results:

A generalized error analysis accommodating variations in two-stage computations and adjoint problem formulations.
Computation of dual-weighted a posteriori error estimates.
Development of novel algorithms for efficient solutions considering error cancellation.

Conclusions:

The proposed two-stage approach provides a flexible and effective framework for numerical solutions of IVPs.
The a posteriori error analysis enables accurate error estimation and algorithm development.
The methods show promise for improving efficiency in parallel-in-time schemes and adjoint problem solutions.