An Application of a Modified Gappy Proper Orthogonal Decomposition on Complexity Reduction of Allen-Cahn Equation  
[PERSON]  
1 Department of mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi, Bangkok 10140, Thailand; [EMAIL_ADDRESS]  
[PERSON]  
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]. In particular, it has become a basic model equation for the diffuse interface approach developed to study phase transitions and interfacial dynamics in material science [6]. In a general model of Allen-Cahn equation used for describing phase transition in materials science, there is an important parameter called an \"interaction length\" [5]. The change of this parameter can significantly affect the numerical solutions and the behavior of the phase transition. In particular, when this parameter becomes very small (approaching zero), there will be a sharp interface limit (the mean curvature flow). This work aims to construct low-dimensional models for Allen-Cahn equation with different values of this interaction length parameter by using model reduction approach with common projection bases that can be generated only once and reused for all various parameter values. In the mathematical and computational field, model order reduction (MOR) is a technique for constructing a low dimensional model that preserve the necessary information from the full discretized system, while saving computational time and storage. The advantage of a reduced model often lies in the decrease of simulation time and the provision accurate results when compared to the high dimensional system. There are many MOR techniques such as balance truncation [7], Krylov subspace MOR methods [8], projection based MOR [9], and the Moment-matching method [10]. This work uses proper orthogonal decomposition (POD) as a starting point for constructing reduced-order model for Allen-Cahn equation. POD approach generally gives high accurate reduced models with much smaller dimensions by generating a set of optimal basis that represents a given data set with minimum error. However, it cannot truly reduce the computational complexity of nonlinear term. We therefore employ the discrete empirical interpolation method (DEIM), as well as a modified Gappy POD (GPOD) for efficiently constructing reduced order systems. POD is a popular method for model reduction, which was first proposed by [PERSON] [11]. It is also known as Karhunen-Loeve decomposition [12] or principal component analysis [13]. It provides a technique for analyzing multidimensional data. This method constructs an orthonormal basis for representing the given data in a certain least squares optimal sense. The basic properties of the POD method have been studied in [14] as it is applied to data compression and used as a model reduction technique for finite dimensional linear systems. [PERSON] [15] purposed sufficient conditions for preserving stability in POD model reduction. POD has been widely applied in many application such as data analysis, data compression and model reduction in various fields of engineering and science. Applications of POD include image processing [16], data compression, signal analysis [17], turbulence modeling [18], control of fluids [19], electrical power grids [20], and modeling and control of chemical reaction systems [21; 22]. [PERSON] et al. [23] presented a reduced-basis technique and a computational algorithm within the context for analyzing nonlinear structure. [PERSON] [24] demonstrated that the reduced basis method based on POD can efficiently approximate the solution of incompressible viscous flow. [PERSON] [25] demonstrated a reduced-order modeling approach using POD for simulation and control of incompressible flows.