Reduced Basis Method Based on Fourier Transform for Time-Dependent Parameterized Nonlocal Problems (Communications in Mathematical Research, 2025)
In the paper, a reduced basis (RB) method for time-dependent nonlocal problems with a special parameterized fractional Laplace kernel function isproposed. Because of the lack of sparsity of discretized nonlocal systems compared to corresponding local partial differential equation (PDE) systems, modelreduction for nonlocal systems becomes more critical. The method of snapshotsand greedy (MOS-greedy) algorithm of RB method is developed for nonlocalproblems with random inputs, which provides an efficient and reliable approximation of the solution. A major challenge lies in the excessive influence of thetime domain on the model reduction process. To address this, the Fourier transform is applied to convert the original time-dependent parabolic equation intoa frequency-dependent elliptic equation, where variable frequencies are independent. This enables parallel computation for approximating the solution inthe frequency domain. Finally, the proposed MOS-greedy algorithm is appliedto the nonlocal diffusion problems. Numerical results demonstrate that it provides an accurate approximation of the full order problems and significantlyimproves computational efficiency.
Keywords: Nonlocal problems; reduced basis method; method of snapshots; greedy algorithm; Fourier transform.