DATA-DRIVEN MODELING AND SIMULATIONS OF SEISMIC WAVES

In recent decades, nonlocal models have been proved to be very effective in the study of complex processes and multiscale phenomena arising in many fields, such as quantum mechanics, geophysics, and cardiac electrophysiology. The fractional Laplacian(−Δ)/2 can be reviewed as nonlocal generalization of the classical Laplacian which has been widely used for the description of memory and hereditary properties of various material and process. However, the nonlocality property of fractional Laplacian introduces challenges in mathematical analysis and computation. Compared to the classical Laplacian, existing numerical methods for the fractional Laplacian still remain limited. The objectives of this research are to develop new numerical methods to solve nonlocal models with fractional Laplacian and apply them to study seismic wave modeling in both homogeneous and heterogeneous media. To this end, we have developed two classes of methods: meshfree pseudospectral method and operator factorization methods. Compared to the current state-of-the-art methods, both of them can achieve higher accuracy with less computational complexity. The operator factorization methods provide a general framework, allowing one to achieve better accuracy with high-degree Lagrange basis functions. The meshfree pseudospectral methods based on global radial basis functions can solve both classical and fractional Laplacians in a single scheme which are the first compatible methods for these two distinct operators. Numerical experiments have demonstrated the effectiveness of our methods on various nonlocal problems. Moreover, we present an extensive study of the variable-order Laplacian operator (−Δ)(x)/2 by using meshfree methods both analytically and numerically. Finally, we apply our numerical methods to solve seismic wave modeling and study the nonlocal effects of fractional wave equation --Abstract, p. i

On the Interpolation of Smooth Functions via Radial Basis Functions

We consider the interpolatory theory of bandlimited functions at both the integer lattice and at more general point sets in Rd by forming interpolants which lie in the linear span of translates of a single Radial Basis Function (RBF). Asymptotic behavior of the interpolants in terms of a given parameter associated with the RBF is considered; in these instances the original bandlimited function can be recovered in L2 and uniformly by a limiting process. Additionally, multivariate interpolation of nonuniform data is considered, and sufficient conditions are given on a family of RBFs which allow for recovery of multidimensional bandlimited functions. We also consider the rate of approximation that can be obtained in different cases. Sometimes, we may say something about the rate in terms of the RBF parameter mentioned above, while other times, we achieve rates based on a shrinking mesh size. The latter technique allows us to consider interpolation of Sobolev functions and their associated approximation rates as well