arXiv subjects

Zhiping Mao

Publications and source records attributed to Zhiping Mao.

23 records · Page 2Linked to original sources

arXiv2018

Nonlocal flocking dynamics: Learning the fractional order of PDEs from particle simulations

Flocking refers to collective behavior of a large number of interacting entities, where the interactions between discrete individuals produce collective motion on the large scale. We employ an agent-based model to describe the microscopic dynamics of each individual in a flock, and use a fractional PDE to model the evolution of macroscopic quantities of interest. The macroscopic models with phenomenological interaction functions are derived by applying the continuum hypothesis to the microscopic model. Instead of specifying the fPDEs with an ad hoc fractional order for nonlocal flocking dynamics, we learn the effective nonlocal influence function in fPDEs directly from particle trajectories generated by the agent-based simulations. We demonstrate how the learning framework is used to connect the discrete agent-based model to the continuum fPDEs in 1D and 2D nonlocal flocking dynamics. In particular, a Cucker-Smale particle model is employed to describe the microscale dynamics of each individual, while Euler equations with nonlocal interaction terms are used to compute the evolution of macroscale quantities. The trajectories generated by the particle simulations mimic the field data of tracking logs that can be obtained experimentally. They can be used to learn the fractional order of the influence function using a Gaussian process regression model implemented with the Bayesian optimization. We show that the numerical solution of the learned Euler equations solved by the finite volume scheme can yield correct density distributions consistent with the collective behavior of the agent-based system. The proposed method offers new insights on how to scale the discrete agent-based models to the continuum-based PDE models, and could serve as a paradigm on extracting effective governing equations for nonlocal flocking dynamics directly from particle trajectories.

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arXiv2018

Multi-domain Spectral Collocation Method for Variable-Order Nonlinear Fractional Differential Equations

Spectral and spectral element methods using Galerkin type formulations are efficient for solving linear fractional PDEs (FPDEs) of constant order but are not efficient in solving nonlinear FPDEs and cannot handle FPDEs with variable-order. In this paper, we present a multi-domain spectral collocation method that addresses these limitations. We consider FPDEs in the Riemann-Liouville sense, and employ Jacobi Lagrangian interpolants to represent the solution in each element. We provide variable-order differentiation formulas, which can be computed efficiently for the multi-domain discretization taking into account the nonlocal interactions. We enforce the interface continuity conditions by matching the solution values at the element boundaries via the Lagrangian interpolants, and in addition we minimize the jump in (integer) fluxes using a penalty method. We analyze numerically the effect of the penalty parameter on the condition number of the global differentiation matrix and on the stability and convergence of the penalty collocation scheme. We demonstrate the effectiveness of the new method for the fractional Helmholtz equation of constant and variable-order using $h-p$ refinement for different values of the penalty parameter. We also solve the fractional Burgers equation with constant and variable-order and compare with solutions obtained with a single domain spectral collocation method.

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arXiv2018

A fast solver for spectral element approximation applied to fractional differential equations using hierarchical matrix approximation

We develop a fast solver for the spectral element method (SEM) applied to the two-sided fractional diffusion equation on uniform, geometric and graded meshes. By approximating the singular kernel with a degenerate kernel, we construct a hierarchical matrix (H-matrix) to represent the stiffness matrix of the SEM and provide error estimates verified numerically. We can solve efficiently the H-matrix approximation problem using a hierarchical LU decomposition method, which reduces the computational cost to $O(R^2 N_d \log^2N) +O(R^3 N_d \log N)$, where $R$ it is the rank of submatrices of the H-matrix approximation, $N_d$ is the total number of degrees of freedom and $N$ is the number of elements. However, we lose the high accuracy of the SEM. Thus, we solve the corresponding preconditioned system by using the H-matrix approximation problem as a preconditioner, recovering the high order accuracy of the SEM. The condition number of the preconditioned system is independent of the polynomial degree $P$ and grows with the number of elements, but at modest values of the rank $R$ is below order 10 in our experiments, which represents a reduction of more than 11 orders of magnitude from the unpreconditioned system; this reduction is higher in the two-sided fractional derivative compared to one-sided fractional derivative. The corresponding cost is $O(R^2 N_d \log^2 N)+O(R^3 N_d \log N)+O(N_d^2)$. Moreover, by using a structured mesh (uniform or geometric mesh), we can further reduce the computational cost to $O(R^2 N_d\log^2 N) +O(R^3 N_d \log N)+ O(P^2 N\log N)$ for the preconditioned system. We present several numerical tests to illustrate the proposed algorithm using $h$ and $p$ refinements.

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arXiv2018

A spectral penalty method for two-sided fractional differential equations with general boundary conditions

We consider spectral approximations to the conservative form of the two-sided Riemann-Liouville (R-L) and Caputo fractional differential equations (FDEs) with nonhomogeneous Dirichlet (fractional and classical, respectively) and Neumann (fractional) boundary conditions. In particular, we develop a spectral penalty method (SPM) by using the Jacobi poly-fractonomial approximation for the conservative R-L FDEs while using the polynomial approximation for the conservative Caputo FDEs. We establish the well-posedness of the corresponding weak problems and analyze sufficient conditions for the coercivity of the SPM for different types of fractional boundary value problems. This analysis allows us to estimate the proper values of the penalty parameters at boundary points. We present several numerical examples to verify the theory and demonstrate the high accuracy of SPM, both for stationary and time dependent FDEs. Moreover, we compare the results against a Petrov-Galerkin spectral tau method (PGS-$τ$, an extension of [Z. Mao, G.E. Karniadakis, SIAM J. Numer. Anal., 2018]) and demonstrate the superior accuracy of SPM for all cases considered.

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arXiv2018

Jacobi-Galerkin spectral method for eigenvalue problems of Riesz fractional differential equations

An efficient Jacobi-Galerkin spectral method for calculating eigenvalues of Riesz fractional partial differential equations with homogeneous Dirichlet boundary values is proposed in this paper. In order to retain the symmetry and positive definiteness of the discrete linear system, we introduce some properly defined Sobolev spaces and approximate the eigenvalue problem in a standard Galerkin weak formulation instead of the Petrov-Galerkin one as in literature. Poincaré and inverse inequalities are proved for the proposed Galerkin formulation which finally help us establishing a sharp estimate on the algebraic system's condition number. Rigorous error estimates of the eigenvalues and eigenvectors are then readily obtained by using Babuška and Osborn's approximation theory on self-adjoint and positive-definite eigenvalue problems. Numerical results are presented to demonstrate the accuracy and efficiency, and to validate the asymptotically exponential oder of convergence. Moreover, the Weyl-type asymptotic law $ λ_n=\mathcal{O}(n^{2α})$ for the $n$-th eigenvalue $λ_n$ of the Riesz fractional differential operator of order $2α$, and the condition number $N^{4α}$ of its algebraic system with respect to the polynomial degree $N$ are observed.

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