An Application of a Modified Gappy Proper Orthogonal Decomposition on Complexity Reduction of Allen-Cahn Equation

Chutipong Dechanubeksa,Saifon Chaturantabut

doi:10.3390/a13060148

Abstract

This work considers model reduction techniques that can substantially decrease computational cost in simulating parmetrized Allen–Cahn equation. We first employ the proper orthogonal decomposition (POD) approach to reduce the number of unknowns in the full-order discretized system. Since POD cannot reduce the computational complexity of nonlinearity in Allen–Cahn equation, we also apply discrete empirical interpolation method (DEIM) to approximate the nonlinear term for a substantial reduction in overall simulation time. However, in general, the POD-DEIM approach is less accurate than the POD approach, since it further approximates the nonlinear term. To increase the accuracy of the POD-DEIM approach, this work introduces an extension of the DEIM approximation based on the concept of Gappy POD (GPOD), which is optimal in the least-squares sense. The POD-GPOD approach is tested and compared with the POD and POD-DEIM approaches on Allen–Cahn equation for both cases of fixed parameter value and varying parameter values. The modified GPOD approximation introduced in this work is demonstrated to improve accuracy of DEIM without sacrificing too much efficiency on the computational speedup, e.g., in one of our numerical tests, the POD-GPOD approach provides an approximate solution to the parmetrized Allen–Cahn equation 200 times faster than the full-order system with average error of order O ( 10 − 4 ) . The POD-GPOD approach is therefore shown to be a promising technique that compromises between the accuracy of POD approach and the efficiency of POD-DEIM approach.

Highlights

In the mathematical and computational field, model order reduction (MOR) is a technique for constructing efficient lower dimensional models for reducing the computational cost of large scale dynamical systems from partial differential equations (PDEs) in numerical simulation
We present the numerical tests that demonstrate the accuracy and efficiency of the resulting proper orthogonal decomposition (POD)-Gappy POD (GPOD) approach on approximating the solution of the parametrized Allen–Cahn equation when compared with the standard POD and POD-discrete empirical interpolation method (DEIM) approaches
We introduced a modification of DEIM that is based on the concept of GPOD, called POD-GPOD approach, which compromises between the accuracy of POD and computational efficiency of DEIM

Summary

Introduction

In the mathematical and computational field, model order reduction (MOR) is a technique for constructing efficient lower dimensional models for reducing the computational cost of large scale dynamical systems from partial differential equations (PDEs) in numerical simulation. Algorithms 2020, 13, 148 in metallic alloys [1] This equation is used in mathematical physics, to describe the process of phase separation. The Allen–Cahn equation has been widely used in many applications, such as image analysis [2,3], crystal growth [4], and motion by mean curvature flow [5]. It has become a basic model equation for the diffuse interface approach developed to study phase transitions and interfacial dynamics in material science [6]

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