Xiangmin Jiao

Xiangmin Jiao, Hongyuan Zha

Differential quantities, including normals, curvatures, principal directions, and associated matrices, play a fundamental role in geometric processing and physics-based modeling. Computing these differential quantities consistently on surface meshes is important and challenging, and some existing methods often produce inconsistent results and require ad hoc fixes. In this paper, we show that the computation of the gradient and Hessian of a height function provides the foundation for consistently computing the differential quantities. We derive simple, explicit formulas for the transformations between the first- and second-order differential quantities (i.e., normal vector and principal curvature tensor) of a smooth surface and the first- and second-order derivatives (i.e., gradient and Hessian) of its corresponding height function. We then investigate a general, flexible numerical framework to estimate the derivatives of the height function based on local polynomial fittings formulated as weighted least squares approximations. We also propose an iterative fitting scheme to improve accuracy. This framework generalizes polynomial fitting and addresses some of its accuracy and stability issues, as demonstrated by our theoretical analysis as well as experimental results.

Aditi Ghai, Cao Lu, Xiangmin Jiao

Preconditioned Krylov subspace (KSP) methods are widely used for solving large-scale sparse linear systems arising from numerical solutions of partial differential equations (PDEs). These linear systems are often nonsymmetric due to the nature of the PDEs, boundary or jump conditions, or discretization methods. While implementations of preconditioned KSP methods are usually readily available, it is unclear to users which methods are the best for different classes of problems. In this work, we present a comparison of some KSP methods, including GMRES, TFQMR, BiCGSTAB, and QMRCGSTAB, coupled with three classes of preconditioners, namely Gauss-Seidel, incomplete LU factorization (including ILUT, ILUTP, and multilevel ILU), and algebraic multigrid (including BoomerAMG and ML). Theoretically, we compare the mathematical formulations and operation counts of these methods. Empirically, we compare the convergence and serial performance for a range of benchmark problems from numerical PDEs in 2D and 3D with up to millions of unknowns and also assess the asymptotic complexity of the methods as the number of unknowns increases. Our results show that GMRES tends to deliver better performance when coupled with an effective multigrid preconditioner, but it is less competitive with an ineffective preconditioner due to restarts. BoomerAMG with proper choice of coarsening and interpolation techniques typically converges faster than ML, but both may fail for ill-conditioned or saddle-point problems while multilevel ILU tends to succeed. We also show that right preconditioning is more desirable. This study helps establish some practical guidelines for choosing preconditioned KSP methods and motivates the development of more effective preconditioners.

Hongxu Liu, Xiangmin Jiao

ENO (Essentially Non-Oscillatory) and WENO (Weighted Essentially Non-Oscillatory) schemes are widely used high-order schemes for solving partial differential equations (PDEs), especially hyperbolic conservation laws with piecewise smooth solutions. For structured meshes, these techniques can achieve high order accuracy for smooth functions while being non-oscillatory near discontinuities. For unstructured meshes, which are needed for complex geometries, similar schemes are required but they are much more challenging. We propose a new family of non-oscillatory schemes, called WLS-ENO, in the context of solving hyperbolic conservation laws using finite-volume methods over unstructured meshes. WLS-ENO is derived based on Taylor series expansion and solved using a weighted least squares formulation. Unlike other non-oscillatory schemes, the WLS-ENO does not require constructing sub-stencils, and hence it provides more flexible framework and is less sensitive to mesh quality. We present rigorous analysis of the accuracy and stability of WLS-ENO, and present numerical results in 1-D, 2-D, and 3-D for a number of benchmark problems, and also report some comparisons against WENO.

Xingyu Wang, Roman Samulyak, Xiangmin Jiao et al.

We propose a new adaptive Particle-in-Cloud (AP-Cloud) method for obtaining optimal numerical solutions to the Vlasov-Poisson equation. Unlike the traditional particle-in-cell (PIC) method, which is commonly used for solving this problem, the AP-Cloud adaptively selects computational nodes or particles to deliver higher accuracy and efficiency when the particle distribution is highly non-uniform. Unlike other adaptive techniques for PIC, our method balances the errors in PDE discretization and Monte Carlo integration, and discretizes the differential operators using a generalized finite difference (GFD) method based on a weighted least square formulation. As a result, AP-Cloud is independent of the geometric shapes of computational domains and is free of artificial parameters. Efficient and robust implementation is achieved through an octree data structure with 2:1 balance. We analyze the accuracy and convergence order of AP-Cloud theoretically, and verify the method using an electrostatic problem of a particle beam with halo. Simulation results show that the AP-Cloud method is substantially more accurate and faster than the traditional PIC, and it is free of artificial forces that are typical for some adaptive PIC techniques.

Rebecca Conley, Tristan J. Delaney, Xiangmin Jiao

The finite element methods (FEM) are important techniques in engineering for solving partial differential equations, but they depend heavily on element shape quality for stability and good performance. In this paper, we introduce the Adaptive Extended Stencil Finite Element Method (AES-FEM) as a means for overcoming this dependence on element shape quality. Our method replaces the traditional basis functions with a set of generalized Lagrange polynomial (GLP) basis functions, which we construct using local weighted least-squares approximations. The method preserves the theoretical framework of FEM, and allows imposing essential boundary conditions and integrating the stiffness matrix in the same way as the classical FEM. In addition, AES-FEM can use higher-degree polynomial basis functions than the classical FEM, while virtually preserving the sparsity pattern of the stiffness matrix. We describe the formulation and implementation of AES-FEM, and analyze its consistency and stability. We present numerical experiments in both 2D and 3D for the Poison equation and a time-independent convection-diffusion equation. The numerical results demonstrate that AES-FEM is more accurate than linear FEM, is also more efficient than linear FEM in terms of error versus runtime, and enables much better stability and faster convergence of iterative solvers than linear FEM over poor-quality meshes

Cao Lu, Tristan Delaney, Xiangmin Jiao

Saddle-point systems appear in many scientific and engineering applications. The systems can be sparse, symmetric or nonsymmetric, and possibly singular. In many of these applications, the number of constraints is relatively small compared to the number of unknowns. The traditional null-space method is inefficient for these systems, because it is expensive to find the null space explicitly. Some alternatives, notably constraint-preconditioned or projected Krylov methods, are relatively efficient, but they can suffer from numerical instability and even nonconvergence. In addition, most existing methods require the system to be nonsingular or be reducible to a nonsingular system. In this paper, we propose a new method, called OPINS, for singular and nonsingular saddle-point systems. OPINS is equivalent to the null-space method with an orthogonal projector, without forming the orthogonal basis of the null space explicitly. OPINS can not only solve for the unique solution for nonsingular saddle-point problems, but also find the minimum-norm solution in terms of the solution variables for singular systems. The method is efficient and easy to implement using existing Krylov solvers for singular systems. At the same time, it is more stable than the other alternatives, such as projected Krylov methods. We present some preconditioners to accelerate the convergence of OPINS for nonsingular systems, and compare OPINS against some present state-of-the-art methods for various types of singular and nonsingular systems.