arXiv subjects

Xiangmin Jiao

Publications and source records attributed to Xiangmin Jiao.

At least 19 recordsLinked to original sources

arXiv2025

Accelerating Correlated Wave Function Calculations with Hierarchical Matrix Compression of the Two-Electron Integrals

Leveraging matrix sparsity has proven a fruitful strategy for accelerating quantum chemical calculations. Here we present the hierarchical SOS-MP2 algorithm, which uses hierarchical matrix ($\mathcal{H}^{2}$) compression of the electron repulsion integral (ERI) tensor to reduce both time and space complexity. This approach is based on the atomic orbital Laplace transform MP2 calculations, leveraging the data sparsity of the ERI tensor and the element-wise sparsity of the energy-weighted density matrices. The $\mathcal{H}^{2}$ representation approximates the ERI tensor in a block low-rank form, taking advantage of the inherent low-rank nature of the repulsion integrals between distant sets of atoms. The resulting algorithm enables the calculation of the Coulomb-like term of the MP2 energy with a theoretical time complexity of $\mathcal{O}(N^{2}\log N)$ and a space complexity of $\mathcal{O}(N^{2}\log N)$, where $N$ denotes the number of basis functions. Numerical tests show asymptotic time and space complexities better than $\mathcal{O}(N^{2})$ for both linear alkanes and three-dimensional water clusters.

physics.chem-ph↗

arXiv2025

P-order: Unified Convergence Analysis for Nonlinear Iterative Methods

Measuring how quickly iterative methods converge is essential in computational mathematics, but current approaches have significant limitations. Q-order analysis requires strict smoothness conditions, while R-order analysis lacks precision and creates ambiguity, especially when analyzing convergence rates close to linear. We introduce P-order, a new framework that overcomes these limitations by using a power function $ψ(k)$ combined with asymptotic notation ($Θ, o, ω$). Our approach offers two key advantages: it works independently of the chosen norm while providing the precision needed to classify diverse convergence behaviors, including previously hard-to-characterize rates like fractional-power and linearithmic convergence. P-order also systematically accommodates weaker continuity conditions by naturally connecting mathematical assumptions to appropriate Taylor approximation forms. To enhance practical analysis, we develop two important subclasses, QUP-order and UP-order, which work effectively under different smoothness conditions. We demonstrate P-order's practical value through three applications: (1) refining fixed-point iteration analysis with minimal smoothness requirements (mere differentiability suffices where classical analysis required stronger conditions), (2) identifying previously unreported convergence rates for Newton's method and gradient descent algorithms, and (3) providing a unified analysis of $K$-point methods under $C^{K-1,ν}$ (i.e., with Hölder continuous $(K-1)$th derivatives), yielding a new characteristic rate $q_{K}(ν)$. Our P-order framework provides researchers and practitioners with a sharper, more comprehensive toolbox for convergence analysis, particularly valuable when classical assumptions fail or when analyzing complex convergence behaviors in modern computational applications.

math.NA↗

arXiv2024

Efficient and Scalable Wave Function Compression Using Corner Hierarchical Matrices

The exponential scaling of complete active space (CAS) and full configuration interaction (CI) calculations limits the ability of quantum chemists to simulate the electronic structures of strongly correlated systems. Herein, we present corner hierarchically approximated CI (CHACI), an approach to wave function compression based on corner hierarchical matrices (CH-matrices) -- a new variant of hierarchical matrices based on a block-wise low-rank decomposition. By application to dodecacene, a strongly correlated molecule, we demonstrate that CH matrix compression provides superior compression compared to a truncated global singular value decomposition. The compression ratio is shown to improve with increasing active space size. By comparison of several alternative schemes, we demonstrate that superior compression is achieved by a) using a blocking approach that emphasizes the upper-left corner of the CI vector, b) sorting the CI vector prior to compression, and c) optimizing the rank of each block to maximize information density.

physics.chem-ph↗

arXiv2023

Robust Discontinuity Indicators for High-Order Reconstruction of Piecewise Smooth Functions

In many applications, piecewise continuous functions are commonly interpolated over meshes. However, accurate high-order manipulations of such functions can be challenging due to potential spurious oscillations known as the Gibbs phenomena. To address this challenge, we propose a novel approach, Robust Discontinuity Indicators (RDI), which can efficiently and reliably detect both C^{0} and C^{1} discontinuities for node-based and cell-averaged values. We present a detailed analysis focusing on its derivation and the dual-thresholding strategy. A key advantage of RDI is its ability to handle potential inaccuracies associated with detecting discontinuities on non-uniform meshes, thanks to its innovative discontinuity indicators. We also extend the applicability of RDI to handle general surfaces with boundaries, features, and ridge points, thereby enhancing its versatility and usefulness in various scenarios. To demonstrate the robustness of RDI, we conduct a series of experiments on non-uniform meshes and general surfaces, and compare its performance with some alternative methods. By addressing the challenges posed by the Gibbs phenomena and providing reliable detection of discontinuities, RDI opens up possibilities for improved approximation and analysis of piecewise continuous functions, such as in data remap.

math.NA↗

arXiv2023

Preserving Superconvergence of Spectral Elements for Curved Domains via $h$ and $p$-Geometric Refinement

Spectral element methods (SEM), which are extensions of finite element methods (FEM), are important emerging techniques for solving partial differential equations in physics and engineering. SEM can potentially deliver better accuracy due to the potential superconvergence for well-shaped tensor-product elements. However, for complex geometries, the accuracy of SEM often degrades due to a combination of geometric inaccuracies near curved boundaries and the loss of superconvergence with simplicial or non-tensor-product elements. We propose to overcome the first issue by using $h$- and $p$-geometric refinement, to refine the mesh near high-curvature regions and increase the degree of geometric basis functions, respectively. We show that when using mixed-meshes with tensor-product elements in the interior of the domain, curvature-based geometric refinement near boundaries can improve the accuracy of the interior elements by reducing pollution errors and preserving the superconvergence. To overcome the second issue, we apply a post-processing technique to recover the accuracy near the curved boundaries by using the adaptive extended stencil finite element method (AES-FEM). The combination of curvature-based geometric refinement and accurate post-processing delivers an effective and easier-to-implement alternative to other methods based on exact geometries. We demonstrate our techniques by solving the convection-diffusion equation in 2D and show one to two orders of magnitude of improvement in the solution accuracy, even when the elements are poorly shaped near boundaries.

math.NA↗

arXiv2023

Optimal Solutions of Well-Posed Linear Systems via Low-Precision Right-Preconditioned GMRES with Forward and Backward Stabilization

Linear systems in applications are typically well-posed, and yet the coefficient matrices may be nearly singular in that the condition number $κ(\boldsymbol{A})$ may be close to $1/\varepsilon_{w}$, where $\varepsilon_{w}$ denotes the unit roundoff of the working precision. It is well known that iterative refinement (IR) can make the forward error independent of $κ(\boldsymbol{A})$ if $κ(\boldsymbol{A})$ is sufficiently smaller than $1/\varepsilon_{w}$ and the residual is computed in higher precision. We propose a new iterative method, called Forward-and-Backward Stabilized Minimal Residual or FBSMR, by conceptually hybridizing right-preconditioned GMRES (RP-GMRES) with quasi-minimization. We develop FBSMR based on a new theoretical framework of essential-forward-and-backward stability (EFBS), which extends the backward error analysis to consider the intrinsic condition number of a well-posed problem. We stabilize the forward and backward errors in RP-GMRES to achieve EFBS by evaluating a small portion of the algorithm in higher precision while evaluating the preconditioner in lower precision. FBSMR can achieve optimal accuracy in terms of both forward and backward errors for well-posed problems with unpolluted matrices, independently of $κ(\boldsymbol{A})$. With low-precision preconditioning, FBSMR can reduce the computational, memory, and energy requirements over direct methods with or without IR. FBSMR can also leverage parallelization-friendly classical Gram-Schmidt in Arnoldi iterations without compromising EFBS. We demonstrate the effectiveness of FBSMR using both random and realistic linear systems.

math.NA↗

arXiv2022

ARPIST: Provably Accurate and Stable Numerical Integration over Spherical Triangles

Numerical integration on spheres, including the computation of the areas of spherical triangles, is a core computation in geomathematics. The commonly used techniques sometimes suffer from instabilities and significant loss of accuracy. We describe a new algorithm, called ARPIST, for accurate and stable integration of functions on spherical triangles. ARPIST is based on an easy-to-implement transformation to the spherical triangle from its corresponding linear triangle via radial projection to achieve high accuracy and efficiency. More importantly, ARPIST overcomes potential instabilities in computing the Jacobian of the transformation, even for poorly shaped triangles that may occur at poles in regular longitude-latitude meshes, by avoiding potential catastrophic rounding errors. We compare our proposed technique with L'Huilier's Theorem for computing the area of spherical triangles, and also compare it with the recently developed LSQST method (J. Beckmann, H.N. Mhaskar, and J. Prestin, GEM - Int. J. Geomath., 5:143-162, 2014) and a radial-basis-function-based technique (J. A. Reeger and B. Fornberg, Stud. Appl. Math., 137:174-188, 2015) for integration of smooth functions on spherical triangulations. Our results show that ARPIST enables superior accuracy and stability over previous methods while being orders of magnitude faster and significantly easier to implement.

math.NA↗

arXiv2021

Approximate Generalized Inverses with Iterative Refinement for $ε$-Accurate Preconditioning of Singular Systems

We introduce a new class of preconditioners to enable flexible GMRES to find a least-squares solution, and potentially the pseudoinverse solution, of large-scale sparse, asymmetric, singular, and potentially inconsistent systems. We develop the preconditioners based on a new observation that generalized inverses (i.e., $\boldsymbol{A}^{g}\in\{\boldsymbol{G}\mid\boldsymbol{A}\boldsymbol{G}\boldsymbol{A}=\boldsymbol{A}\}$) enable the preconditioned Krylov subspaces to converge in a single step. We then compute an approximate generalized inverse (AGI) efficiently using a hybrid incomplete factorization (HIF), which combines multilevel incomplete LU with rank-revealing QR on its final Schur complement. We define the criteria of $ε$-accuracy and stability of AGI to guarantee the convergence of preconditioned GMRES for consistent systems. For inconsistent systems, we fortify HIF with iterative refinement to obtain HIFIR, which allows accurate computations of the null-space vectors. By combining the two techniques, we then obtain a new solver, called PIPIT, for obtaining the pseudoinverse solutions for systems with low-dimensional null spaces. We demonstrate the robustness of HIF and HIFIR and show that they improve both accuracy and efficiency of the prior state of the art by orders of magnitude for systems with up to a million unknowns.

math.NA↗

arXiv2021

HIFIR: Hybrid Incomplete Factorization with Iterative Refinement for Preconditioning Ill-conditioned and Singular Systems

We introduce a software package called HIFIR for preconditioning sparse, unsymmetric, ill-conditioned, and potentially singular systems. HIFIR computes a hybrid incomplete factorization, which combines multilevel incomplete LU factorization with a truncated, rank-revealing QR factorization on the final Schur complement. This novel hybridization is based on the new theory of approximate generalized inverse and $ε$-accuracy. It enables near-optimal preconditioners for consistent systems and enables flexible GMRES to solve inconsistent systems when coupled with iterative refinement. In this paper, we focus on some practical algorithmic and software issues of HIFIR. In particular, we introduce a new inverse-based rook pivoting into ILU, which improves the robustness and the overall efficiency for some ill-conditioned systems by significantly reducing the size of the final Schur complement for some systems. We also describe the software design of HIFIR in terms of its efficient data structures for supporting rook pivoting in a multilevel setting, its template-based generic programming interfaces for mixed-precision real and complex values in C++, and its user-friendly high-level interfaces in MATLAB and Python. We demonstrate the effectiveness of HIFIR for ill-conditioned or singular systems arising from several applications, including the Helmholtz equation, linear elasticity, stationary incompressible Navier--Stokes equations, and time-dependent advection-diffusion equation.

math.NA↗

arXiv2021

Robust and Efficient Multilevel-ILU Preconditioning of Hybrid Newton-GMRES for Incompressible Navier-Stokes Equations

We introduce a robust and efficient preconditioner for a hybrid Newton-GMRES method for solving the nonlinear systems arising from incompressible Navier-Stokes equations. When the Reynolds number is relatively high, these systems often involve millions of degrees of freedom (DOFs), and the nonlinear systems are difficult to converge, partially due to the strong asymmetry of the system and the saddle-point structure. In this work, we propose to alleviate these issues by leveraging a multilevel ILU preconditioner called HILUCSI, which is particularly effective for saddle-point problems and can enable robust and rapid convergence of the inner iterations in Newton-GMRES. We further use Picard iterations with the Oseen systems to hot-start Newton-GMRES to achieve global convergence, also preconditioned using HILUCSI. To further improve efficiency and robustness, we use the Oseen operators as physics-based sparsifiers when building preconditioners for Newton iterations and introduce adaptive refactorization and iterative refinement in HILUCSI. We refer to the resulting preconditioned hybrid Newton-GMRES as HILUNG. We demonstrate the effectiveness of HILUNG by solving the standard 2D driven-cavity problem with Re 5000 and a 3D flow-over-cylinder problem with low viscosity. We compare HILUNG with some state-of-the-art customized preconditioners for INS, including two variants of augmented Lagrangian preconditioners and two physics-based preconditioners, as well as some general-purpose approximate-factorization techniques. Our comparison shows that HILUNG is much more robust for solving high-Re problems and it is also more efficient in both memory and runtime for moderate-Re problems.

cs.CE↗

arXiv2021

Optimal and Low-Memory Near-Optimal Preconditioning of Fully Implicit Runge-Kutta Schemes for Parabolic PDEs

Runge-Kutta (RK) schemes, especially Gauss-Legendre and some other fully implicit RK (FIRK) schemes, are desirable for the time integration of parabolic partial differential equations due to their A-stability and high-order accuracy. However, it is significantly more challenging to construct optimal preconditioners for them compared to diagonally implicit RK (or DIRK) schemes. To address this challenge, we first introduce mathematically optimal preconditioners called block complex Schur decomposition (BCSD), block real Schur decomposition (BRSD), and block Jordan form (BJF), motivated by block-circulant preconditioners and Jordan form solution techniques for IRK. We then derive an efficient, near-optimal singly-diagonal approximate BRSD (SABRSD) by approximating the quasi-triangular matrix in real Schur decomposition using an optimized upper-triangular matrix with a single diagonal value. A desirable feature of SABRSD is that it has comparable memory requirements and factorization (or setup) cost as singly DIRK (SDIRK). We approximate the diagonal blocks in these preconditioning techniques using an incomplete factorization with (near) linear complexity, such as multilevel ILU, ILU(0), or a multigrid method with an ILU-based smoother. We apply the block preconditioners in right-preconditioned GMRES to solve the advection-diffusion equation in 3D using finite element and finite difference methods. We show that BCSD, BRSD, and BJF significantly outperform other preconditioners in terms of GMRES iterations, and SABRSD is competitive with them and the prior state of the art in terms of computational cost while requiring the least amount of memory.

math.NA↗

arXiv2021

HILUCSI: Simple, Robust, and Fast Multilevel ILU for Large-Scale Saddle-Point Problems from PDEs

Incomplete factorization is a widely used preconditioning technique for Krylov subspace methods for solving large-scale sparse linear systems. Its multilevel variants, such as ILUPACK, are more robust for many symmetric or unsymmetric linear systems than the traditional, single-level incomplete LU (or ILU) techniques. However, the previous multilevel ILU techniques still lacked robustness and efficiency for some large-scale saddle-point problems, which often arise from systems of partial differential equations (PDEs). We introduce HILUCSI, or Hierarchical Incomplete LU-Crout with Scalability-oriented and Inverse-based dropping. As a multilevel preconditioner, HILUCSI statically and dynamically permutes individual rows and columns to the next level for deferred factorization. Unlike ILUPACK, HILUCSI applies symmetric preprocessing techniques at the top levels but always uses unsymmetric preprocessing and unsymmetric factorization at the coarser levels. The deferring combined with mixed preprocessing enabled a unified treatment for nearly or partially symmetric systems, and simplified the implementation by avoiding mixed $1\times 1$ and $2\times 2$ pivots for symmetric indefinite systems. We show that this combination improves robustness for indefinite systems without compromising efficiency. Furthermore, to enable superior efficiency for large-scale systems with millions or more unknowns, HILUCSI introduces a scalability-oriented dropping in conjunction with a variant of inverse-based dropping. We demonstrate the effectiveness of HILUCSI for dozens of benchmark problems, including those from the mixed formulation of the Poisson equation, Stokes equations, and Navier-Stokes equations. We also compare its performance with ILUPACK, the supernodal ILUTP in SuperLU, and multithreaded direct solvers in PARDISO and MUMPS.

math.NA↗

arXiv2020

A Hybrid Method and Unified Analysis of Generalized Finite Differences and Lagrange Finite Elements

Finite differences, finite elements, and their generalizations are widely used for solving partial differential equations, and their high-order variants have respective advantages and disadvantages. Traditionally, these methods are treated as different (strong vs. weak) formulations and are analyzed using different techniques (Fourier analysis or Green's functions vs. functional analysis), except for some special cases on regular grids. Recently, the authors introduced a hybrid method, called Adaptive Extended Stencil FEM or AES-FEM (Int. J. Num. Meth. Engrg., 2016, DOI:10.1002/nme.5246), which combines features of generalized finite differences and Lagrange finite elements to achieve second-order accuracy over unstructured meshes. However, its analysis was incomplete due to the lack of existing mathematical theory that unifies the formulations and analysis of these different methods. In this work, we introduce the framework of generalized weighted residuals to unify the formulation of finite differences, finite elements, and AES-FEM. In addition, we propose a unified analysis of the well-posedness, convergence, and mesh-quality dependency of these different methods. We also report numerical results with AES-FEM to verify our analysis. We show that AES-FEM improves the accuracy of generalized finite differences while reducing the mesh-quality dependency and simplifying the implementation of high-order finite elements.

math.NA↗

arXiv2019

WLS-ENO Remap: Superconvergent and Non-Oscillatory Weighted Least Squares Data Transfer on Surfaces

Data remap between non-matching meshes is a critical step in multiphysics coupling using a partitioned approach. The data fields being transferred often have jumps in function values or derivatives. It is important but very challenging to avoid spurious oscillations (a.k.a. the Gibbs Phenomenon) near discontinuities and at the same time to achieve high-order accuracy away from discontinuities. In this work, we introduce a new approach, called WLS-ENOR, or Weighted-Least-Squares-based Essentially Non-Oscillatory Remap, to address this challenge. Based on the WLS-ENO reconstruction technique proposed by Liu and Jiao (J. Comput. Phys. vol 314, pp 749--773, 2016), WLS-ENOR differs from WLS-ENO and other WENO schemes in that it resolves not only the O(1) oscillations due to C 0 discontinuities, but also the accumulated effect of O(h) oscillations due to C 1 discontinuities. To this end, WLS-ENOR introduces a robust detector of discontinuities and a new weighting scheme for WLS-ENO near discontinuities. We also optimize the weights at smooth regions to achieve superconvergence. As a result, WLS-ENOR is more than fifth-order accurate and highly conservative in smooth regions, while being non-oscillatory and minimally diffusive near discontinuities. We also compare WLS-ENOR with some commonly used methods based on L 2 projection, moving least squares, and radial basis functions.

math.NA↗

arXiv2019

Compact Feature-Aware Hermite-Style High-Order Surface Reconstruction

High-order surface reconstruction is an important technique for CAD-free, mesh-based geometric and physical modeling, and for high-order numerical methods for solving partial differential equations (PDEs) in engineering applications. In this paper, we introduce a novel method for accurate and robust reconstructions of piecewise smooth surfaces from a triangulated surface. Our proposed method extends the Continuous Moving Frames (CMF) and the Weighted Averaging of Local Fittings (WALF) methods (Engrg. Comput. 28 (2012)) in two main aspects. First, it utilizes a Hermite-style least squares approximation to achieve fourth and higher-order accuracy with compact support, even if the input mesh is relatively coarse. Second, it introduces an iterative feature-aware parameterization to ensure high-order accurate, G0 continuous reconstructions near sharp features. We present the theoretical framework of the method and compare it against CMF and WALF in terms of accuracy and stability. We also demonstrate that the use of the Hermite-style reconstruction in the solutions of PDEs using finite element methods (FEM), and show that quartic and sextic FEMs using the high-order reconstructed surfaces produce nearly identical results as using exact geometry while providing additional flexibility.

math.NA↗

arXiv2019

Robust Optimal-Complexity Multilevel ILU for Predominantly Symmetric Systems

Incomplete factorization is a powerful preconditioner for Krylov subspace methods for solving large-scale sparse linear systems. Existing incomplete factorization techniques, including incomplete Cholesky and incomplete LU factorizations, are typically designed for symmetric or nonsymmetric matrices. For some numerical discretizations of partial differential equations, the linear systems are often nonsymmetric but predominantly symmetric, in that they have a large symmetric block. In this work, we propose a multilevel incomplete LU factorization technique, called PS-MILU, which can take advantage of predominant symmetry to reduce the factorization time by up to half. PS-MILU delivers robustness for ill-conditioned linear systems by utilizing diagonal pivoting and deferred factorization. We take special care in its data structures and its updating and pivoting steps to ensure optimal time complexity in input size under some reasonable assumptions. We present numerical results with PS-MILU as a preconditioner for GMRES for a collection of predominantly symmetric linear systems from numerical PDEs with unstructured and structured meshes in 2D and 3D, and show that PS-MILU can speed up factorization by about a factor of 1.6 for most systems. In addition, we compare PS-MILU against the multilevel ILU in ILUPACK and the supernodal ILU in SuperLU to demonstrate its robustness and lower time complexity.

math.NA↗

arXiv2018

A Comparison of Preconditioned Krylov Subspace Methods for Large-Scale Nonsymmetric Linear Systems

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.

math.NA↗

arXiv2016

AP-Cloud: Adaptive Particle-in-Cloud Method for Optimal Solutions to Vlasov-Poisson Equation

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.

math.NA↗