Adaptive error control during gradient search for an elliptic optimization problem

Abstract

In this article we describe a cost effective adaptive procedure for optimization of a quantity of interest of a solution of an elliptic problem with respect to parameters in the data, using a gradient search approach. The numerical error in both the quantity of interest and the computed gradient may affect the progression of the search algorithm, while the errors generally change at each step during the search algorithm. We address this by using an accurate a posteriori estimate for the error in a quantity of interest that indicates the effect of error on the computed gradient and so provides a measure for how to refine the discretization as the search proceeds. Specifically, we devise an adaptive algorithm to refine and unrefine the finite element mesh at each step in the search algorithm. We give basic examples and apply this technique to a model of a healing wound.

Keywords:

• a posteriori error estimate
• conjugate gradient
• gradient search
• finite element method
• partial differential equations
• parameter optimization
• adaptive error control

AMS Subject Classifications:

• 49N99
• 65N50
• 65N30
• 90C30
• 35J65

Acknowledgements

Estep's work is supported in part by the Defense Threat Reduction Agency (HDTRA1-09-1-0036), Department of Energy (DE-FG02-04ER25620, DE-FG02-05ER25699, DE-FC02-07ER54909, DE-SC0001724, DE-SC0005304, INL00120133), Lawrence Livermore National Laboratory (B573139, B584647, B590495), the National Aeronautics and Space Administration (NNG04GH63G), the National Institutes of Health (5R01GM096192-02), the National Science Foundation (DMS-0107832, DMS-0715135, DGE-0221595003, MSPA-CSE-0434354, ECCS-0700559, DMS-1016268, DMS-FRG-1065046) and Idaho National Laboratory (00069249, 00115474). Lee's work is supported in part by the Department of Energy (DE-FG02-05ER25699) and the National Science Foundation (DMS-0107832).