In this article, we propose two numerical methods, the Gaussian Process (GP) method and the Fourier Features (FF) algorithm, to solve mean field games (MFGs). The GP algorithm approximates the solution of a MFG with maximum a posteriori probability estimators of GPs conditioned on the partial differential equation (PDE) system of the MFG at a finite number of sample points. The main bottleneck of the GP method is to compute the inverse of a square gram matrix, whose size is proportional to the number of sample points. To improve the performance, we introduce the FF method, whose insight comes from the recent trend of approximating positive definite kernels with random Fourier features. The FF algorithm seeks approximated solutions in the space generated by sampled Fourier features. In the FF method, the size of the matrix to be inverted depends only on the number of Fourier features selected, which is much less than the size of sample points. Hence, the FF method reduces the precomputation time, saves the memory, and achieves comparable accuracy to the GP method. We give the existence and the convergence proofs for both algorithms. The convergence argument of the GP method does not depend on any monotonicity condition, which suggests the potential applications of the GP method to solve MFGs with non-monotone couplings in future work. We show the efficacy of our algorithms through experiments on a stationary MFG with a non-local coupling and on a time-dependent planning problem. We believe that the FF method can also serve as an alternative algorithm to solve general PDEs.

Computing stationary states is an important topic for phase field crystal (PFC) models. Great efforts have been made for energy dissipation of the numerical schemes when using gradient flows. However, it is always time-consuming due to the requirement of small effective time steps. In this paper, we propose an adaptive accelerated proximal gradient method for finding the stationary states of PFC models. The energy dissipation is guaranteed and the convergence property is established for the discretized energy functional. Moreover, the connections between generalized proximal operator with classical (semi-)implicit and explicit schemes for gradient flow are given. Extensive numerical experiments, including two three dimensional periodic crystals in Landau-Brazovskii (LB) model and a two dimensional quasicrystal in Lifshitz-Petrich (LP) model, demonstrate that our approach has adaptive time steps which lead to significant acceleration over semi-implicit methods for computing complex structures. Furthermore, our result reveals a deep physical mechanism of the simple LB model via which the sigma phase is first discovered.

Sparse modeling/approximation of images plays an important role in image restoration. Instead of using a fixed system to sparsely model any input image, a more promising approach is using a system that is adaptive to the input image. A non-convex variational model is proposed in [1] for constructing a tight frame that is optimized for the input image, and an alternating scheme is used to solve the resulting non-convex optimization problem. Although it showed good empirical performance in image denoising, the proposed alternating iteration lacks convergence analysis. This paper aims at providing the convergence analysis of the method proposed in [1]. We first established the sub-sequence convergence property of the iteration scheme proposed in [1], i.e., there exists at least one convergent sub-sequence and any convergent sub-sequence converges to a stationary point of the minimization problem. Moreover, we showed that the original method can be modified to have sequence convergence, i.e., the modified algorithm generates a sequence that converges to a stationary point of the minimization problem.