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Jinchao Xu

Publications and source records attributed to Jinchao Xu.

At least 109 records · Page 6Linked to original sources
arXiv2014

A Nearly Optimal Multigrid Method for General Unstructured Grids

In this paper, we develop a multigrid method on unstructured shape-regular grids. For a general shape-regular unstructured grid of ${\cal O}(N)$ elements, we present a construction of an auxiliary coarse grid hierarchy on which a geometric multigrid method can be applied together with a smoothing on the original grid by using the auxiliary space preconditioning technique. Such a construction is realized by a cluster tree which can be obtained in ${\cal O}(N\log N)$ operations for a grid of $N$ elements. This tree structure in turn is used for the definition of the grid hierarchy from coarse to fine. For the constructed grid hierarchy we prove that the convergence rate of the multigrid preconditioned CG for an elliptic PDE is $1 - {\cal O}({1}/{\log N})$. Numerical experiments confirm the theoretical bounds and show that the total complexity is in ${\cal O}(N\log N)$.

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arXiv2014

Stable Finite Element Methods Preserving $\nabla \cdot \boldsymbol{B} = 0$ Exactly for MHD Models

This paper is devoted to the design and analysis of some structure-preserving finite element schemes for the magnetohydrodynamics (MHD) system. The main feature of the method is that it naturally preserves the important Gauss law, namely $\nabla\cdot\boldsymbol{B}=0$. In contrast to most existing approaches that eliminate the electrical field variable $\boldsymbol{E}$ and give a direct discretization of the magnetic field, our new approach discretizes the electric field $\boldsymbol{E}$ by Nédélec type edge elements for $H(\mathrm{curl})$, while the magnetic field $\boldsymbol{B}$ by Raviart-Thomas type face elements for $H(\mathrm{div})$. As a result, the divergence-free condition on the magnetic field holds exactly on the discrete level. For this new finite element method, an energy stability estimate can be naturally established in an analogous way as in the continuous case. Furthermore, well-posedness is rigorously established in the paper for both the Picard and Newton linearization of the fully nonlinear systems by using the Brezzi theory for both the continuous and discrete cases. This well-posedness naturally leads to robust (and optimal) preconditioners for the linearized systems.

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arXiv2014

Well-posedness and Robust Preconditioners for the Discretized Fluid-Structure Interaction Systems

In this paper we develop a family of preconditioners for the linear algebraic systems arising from the arbitrary Lagrangian-Eulerian discretization of some fluid-structure interaction models. After the time discretization, we formulate the fluid-structure interaction equations as saddle point problems and prove the uniform well-posedness. Then we discretize the space dimension by finite element methods and prove their uniform well-posedness by two different approaches under appropriate assumptions. The uniform well-posedness makes it possible to design robust preconditioners for the discretized fluid-structure interaction systems. Numerical examples are presented to show the robustness and efficiency of these preconditioners.

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arXiv2013

Convergence and optimality of the adaptive Morley element method

This paper is devoted to the convergence and optimality analysis of the adaptive Morley element method for the fourth order elliptic problem. A new technique is developed to establish a quasi-orthogonality which is crucial for the convergence analysis of the adaptive nonconforming method. By introducing a new parameter-dependent error estimator and further establishing a discrete reliability property, sharp convergence and optimality estimates are then fully proved for the fourth order elliptic problem.

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arXiv2013

Convergence and optimality of the adaptive nonconforming linear element method for the Stokes problem

In this paper, we analyze the convergence and optimality of a standard adaptive nonconforming linear element method for the Stokes problem. After establishing a special quasi--orthogonality property for both the velocity and the pressure in this saddle point problem, we introduce a new prolongation operator to carry through the discrete reliability analysis for the error estimator. We then use a specially defined interpolation operator to prove that, up to oscillation, the error can be bounded by the approximation error within a properly defined nonlinear approximate class. Finally, by introducing a new parameter-dependent error estimator, we prove the convergence and optimality estimates.

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arXiv2013

An Error-Resilient Redundant Subspace Correction Method

As we stride toward the exascale era, due to increasing complexity of supercomputers, hard and soft errors are causing more and more problems in high-performance scientific and engineering computation. In order to improve reliability (increase the mean time to failure) of computing systems, a lot of efforts have been devoted to developing techniques to forecast, prevent, and recover from errors at different levels, including architecture, application, and algorithm. In this paper, we focus on algorithmic error resilient iterative linear solvers and introduce a redundant subspace correction method. Using a general framework of redundant subspace corrections, we construct iterative methods, which have the following properties: (1) Maintain convergence when error occurs assuming it is detectable; (2) Introduce low computational overhead when no error occurs; (3) Require only small amount of local (point-to-point) communication compared to traditional methods and maintain good load balance; (4) Improve the mean time to failure. With the proposed method, we can improve reliability of many scientific and engineering applications. Preliminary numerical experiments demonstrate the efficiency and effectiveness of the new subspace correction method.

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arXiv2013

Block Triangular Preconditioning for Stochastic Galerkin Method

In this paper we study fast iterative solvers for the large sparse linear systems resulting from the stochastic Galerkin discretization of stochastic partial differential equations. A block triangular preconditioner is introduced and applied to the Krylov subspace methods, including the generalized minimum residual method and the generalized preconditioned conjugate gradient method. This preconditioner utilizes the special structures of the stochastic Galerkin matrices to achieve high efficiency. Spectral bounds for the preconditioned matrix are provided for convergence analysis. The preconditioner system can be solved approximately by geometric multigrid V-cycle. Numerical results indicate that the block triangular preconditioner has better performance than the traditional block diagonal preconditioner for stochastic problems with large variance.

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arXiv2013

Comparative Convergence Analysis of Nonlinear AMLI-cycle Multigrid

The main purpose of this paper is to provide a comprehensive convergence analysis of nonlinear AMLI-cycle multigrid method for symmetric positive definite problems. Based on classical assumptions for approximation and smoothing properties, we show that the nonlinear AMLI-cycle MG method is uniformly convergent. Furthermore, under only the assumption that the smoother is convergent, we show that the nonlinear AMLI-cycle method is always better (or not worse) than the respective V-cycle MG method. Finally, numerical experiments are presented to illustrate the theoretical results.

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arXiv2013

Combined Preconditioning with Applications in Reservoir Simulation

We develop a simple algorithmic framework to solve large-scale symmetric positive definite linear systems. At its core, the framework relies on two components: (1) a norm-convergent iterative method (i.e. smoother) and (2) a preconditioner. The resulting preconditioner, which we refer to as a combined preconditioner, is much more robust and efficient than the iterative method and preconditioner when used in Krylov subspace methods. We prove that the combined preconditioner is positive definite and show estimates on the condition number of the preconditioned system. We combine an algebraic multigrid method and an incomplete factorization preconditioner to test the proposed framework on problems in petroleum reservoir simulation. Our numerical experiments demonstrate noticeable speed-up when we compare our combined method with the standalone algebraic multigrid method or the incomplete factorization preconditioner.

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arXiv2013

Numerical Study of Geometric Multigrid Methods on CPU--GPU Heterogeneous Computers

The geometric multigrid method (GMG) is one of the most efficient solving techniques for discrete algebraic systems arising from elliptic partial differential equations. GMG utilizes a hierarchy of grids or discretizations and reduces the error at a number of frequencies simultaneously. Graphics processing units (GPUs) have recently burst onto the scientific computing scene as a technology that has yielded substantial performance and energy-efficiency improvements. A central challenge in implementing GMG on GPUs, though, is that computational work on coarse levels cannot fully utilize the capacity of a GPU. In this work, we perform numerical studies of GMG on CPU--GPU heterogeneous computers. Furthermore, we compare our implementation with an efficient CPU implementation of GMG and with the most popular fast Poisson solver, Fast Fourier Transform, in the cuFFT library developed by NVIDIA.

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arXiv2012

A Parallel Auxiliary Grid AMG Method for GPU

In this paper, we develop a new parallel auxiliary grid algebraic multigrid (AMG) method to leverage the power of graphic processing units (GPUs). In the construction of the hierarchical coarse grid, we use a simple and fixed coarsening procedure based on a region quadtree generated from an auxiliary grid. This allows us to explicitly control the sparsity patterns and operator complexities of the AMG solver. This feature provides (nearly) optimal load balancing and predictable communication patterns, which makes our new algorithm suitable for parallel computing, especially on GPU. We also design a parallel smoother based on the special coloring of the quadtree to accelerate the convergence rate and improve the parallel performance of this solver. Based on the CUDA toolkit [40], we implemented our new parallel auxiliary grid AMG method on GPU and the numerical results of this implementation demonstrate the efficiency of our new method. The results achieve an average speedup of over 4 on quasi-uniform grids and 2 on shape regular grids when compared to the AMG implementation in CUSP.

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arXiv2012

On Adaptive Eulerian-Lagrangian Method for Linear Convection-Diffusion Problems

In this paper, we consider the adaptive Eulerian--Lagrangian method (ELM) for linear convection-diffusion problems. Unlike the classical a posteriori error estimations, we estimate the temporal error along the characteristics and derive a new a posteriori error bound for ELM semi-discretization. With the help of this proposed error bound, we are able to show the optimal convergence rate of ELM for solutions with minimal regularity. Furthermore, by combining this error bound with a standard residual-type estimator for the spatial error, we obtain a posteriori error estimators for a fully discrete scheme. We present numerical tests to demonstrate the efficiency and robustness of our adaptive algorithm.

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arXiv2012

Optimal solvers for fourth-order PDEs discretized on unstructured grids

This paper provides the first provable $\mathcal{O}(N \log N)$ algorithms for the linear system arising from the direct finite element discretization of the fourth-order equation with different boundary conditions on unstructured grids of size $N$ on an arbitrary polygoanl domain. Several preconditioners are presented, and the conjugate gradient methods applied with these preconditioners are proven to converge uniformly with respect to the size of the preconditioned linear system. One main ingredient of the optimal preconditioners is a mixed-form discretization of the fourth-order problem. Such a mixed-form discretization leads to a non-desirable ---either non-optimal or non-convergent--- approximation of the original solution, but it provides optimal preconditioners for the direct finite element problem. It is further shown that the implementation of the preconditioners can be reduced to the solution of several discrete Poisson equations. Therefore, any existing optimal or nearly optimal solver, such as geometric or algebraic multigrid methods, for Poisson equations would lead to a nearly optimal solver for the discrete fourth-order system. A number of nonstandard Sobolev spaces and their discretizations defined on the boundary of polygonal domains are carefully studied and used for the analysis of those preconditioners.

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arXiv2012

Local Multilevel Preconditioners for Elliptic Equations with Jump Coefficients on Bisection Grids

The goal of this paper is to design optimal multilevel solvers for the finite element approximation of second order linear elliptic problems with piecewise constant coefficients on bisection grids. Local multigrid and BPX preconditioners are constructed based on local smoothing only at the newest vertices and their immediate neighbors. The analysis of eigenvalue distributions for these local multilevel preconditioned systems shows that there are only a fixed number of eigenvalues which are deteriorated by the large jump. The remaining eigenvalues are bounded uniformly with respect to the coefficients and the meshsize. Therefore, the resulting preconditioned conjugate gradient algorithm will converge with an asymptotic rate independent of the coefficients and logarithmically with respect to the meshsize. As a result, the overall computational complexity is nearly optimal.

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arXiv2011

Lower Bounds of the Discretization for Piecewise Polynomials

Assume that $V_h$ is a space of piecewise polynomials of degree less than $r\geq 1$ on a family of quasi-uniform triangulation of size $h$. Then the following well-known upper bound holds for a sufficiently smooth function $u$ and $p\in [1, \infty]$ $$ \inf_{v_h\in V_h}\|u-v_h\|_{j,p,Ω,h} \le C h^{r-j} |u|_{r,p,Ω},\quad 0\le j\le r. $$ In this paper, we prove that, roughly speaking, if $u\not\in V_h$, the above estimate is sharp. Namely, $$ \inf_{v_h\in V_h}\|u-v_h\|_{j,p,Ω,h} \ge c h^{r-j},\quad 0\le j\le r, \ \ 1\leq p\leq \infty, $$ for some $c>0$. The above result is further extended to various situations including more general Sobolev space norms, general shape regular grids and many different types of finite element spaces. As an application, the sharpness of finite element approximation of elliptic problems and the corresponding eigenvalue problems is established.

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arXiv2011

Analysis of two-level method for anisotropic diffusion equations on aligned and non-aligned grids

This paper is devoted to the multigrid convergence analysis for the linear systems arising from the conforming linear finite element discretization of the second order elliptic equations with anisotropic diffusion. The multigrid convergence behavior is known to strongly depend on whether the discretization grid is aligned or non-aligned with the anisotropic direction and analyses in the paper will be mainly focused on two-level algorithms. For an aligned grid case, a lower bound is given for point-wise smoother which shows deterioration of convergence rate. In both aligned and non-aligned cases we show that for a specially designed block smoother the convergence is uniform with respect to both anisotropy ratio and mesh size in the energy norm. The analysis is complemented with numerical experiments which confirm the theoretical results

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