[PERSON], [PERSON], and [PERSON] [26] focused on reduced order modeling of time-dependent PDEs with multiple parameters on POD approach to the reduced order model. [PERSON] and [PERSON] [27] used a compact POD to compute a reduced basis for an optimization-oriented reduced order model and applied this to compute a reduced basis for model reduction of static systems [28]. POD also has been used for various nonlinear dynamical systems. In [29], POD has been used with global optimum search framework to construct a temporally-local reduced model for nonlinear parabolic PDEs. An adaptive framework for POD was introduced in [30], when new snapshots are included for feedback control of dissipative nonlinear PDEs. In [31], a modification of data ensemble revision approach was developed and used with adaptive POD framework to construct reduced order models for output feedback control of distributed processes. The notion of sparse POD has been introduced and successfully used for constructing reduced order models of nonlinear parabolic PDEs with moving boundaries [32]. DEIM is a nonlinear technique for approximating nonlinear terms of a dynamical system to further decrease computational complexity of POD reduced systems. The DEIM approach was introduced in [33], and approximates a nonlinear function by combining projection with interpolation. This approach is a discrete variant of the empirical interpolation method (EIM) [34], which was originally described in a function space setting. The DEIM approximation was extended to nonlinear reduced models by localization (LDEIM) [35]. There are other nonlinear techniques for approximating nonlinear terms, such as the Gauss-Newton with approximated tensors (GNAT) method [36], the best points interpolation method [37] and missing point estimation [38]. There are many works combining a reduced model by POD with DEIM such as the photovoltaic module [39], the shallow water equations model [40], the Navier-Stokes equations [41] and drift-diffusion equations [42]. Gappy POD (GPOD) was first purposed by [PERSON] and [PERSON] [43]. It has been used as an approach to restore incomplete data. [PERSON], [PERSON], and [PERSON] [44] presented this in the context of fluid dynamic application. [PERSON] et al., [45] presented this in the context of solving the numerical simulation of nonlinear system. [PERSON] [46] extented GPOD to handle unsteady flow reconstruction problem. [PERSON] and [PERSON] [47] applied this method for various problems in aerospace engineering. [PERSON] and [PERSON] [48] used GPOD in the context of particle image velocimetry (PIV) data for subsonic cavity flow. In this work, new contributions include:  
* A novel model reduction algorithm that improves the accuracy of the existing standard model reduction technique that combines the orthogonal decomposition (POD) and discrete empirical interpolation method (DEIM) by using a modification of GPOD that is based on least-squares approximation to the nonlinear term. * Comparison of computational complexity of the proposed method and the standard methods in both theoretical and numerical aspects. * Numerical experiments that demonstrate the accuracy and efficiency of the proposed method for solving the parametrized Allen-Cahn equation for both homogeneous and non-homogeneous boundary conditions with different types of initial conditions. The remainder of this paper is organized as follow. In Section 2, the general form of Allend-Cahn equation and its corresponding discretization are given. Section 3 provides some background on POD and POD-DEIM approaches, as well as introduces the POD-GPOD approach. The accuracy and efficiency of the POD-GPOD framework are compared with the standard POD approach and POD-DEIM approach in Section 4 for simulating the solution of parametrized Allen-Cahn equation. Finally, conclusion and remarks are discussed in Section 5.