About

Xiangmin Jiao is an Associate Professor in the Department of Applied Mathematics and Statistics at Stony Brook University, with additional affiliations in the Computer Science Department and the Institute for Advanced Computational Science. He received his B.S. in 1995 from Peking University, China, his M.S. in 1997 from the University of California Santa Barbara, and his Ph.D. in computer science in 2001 from the University of Illinois at Urbana-Champaign. After working as a Research Scientist at the Center for Simulation of Advanced Rockets at UIUC and as a Visiting Assistant Professor at Georgia Institute of Technology, he joined Stony Brook University in Fall 2007. Dr. Jiao's research interests are in high-performance geometric and numerical computing in science and engineering. His work focuses on developing efficient and robust algorithms and high-performance software implementations for dynamic surfaces, mesh optimization, applied computational and differential geometry, and multiphysics coupling, with applications in computational fluid dynamics, biomedical engineering, climate modeling, and geometric modeling.

Research topics

• Computer Science
• Mathematics
• Algorithm
• Thermodynamics
• Parallel computing
• Geology
• Physics

Selected publications

• Preserving superconvergence of spectral elements in three-dimensional curved domains via ApSEM

  Engineering With Computers · 2026-04-28

  articleSenior author
  PublisherDOI

• Unified SMW-Like Identities of Low-Rank Updates for Generalized Inverses and Pseudoinverses

  SIAM Journal on Matrix Analysis and Applications · 2025-08-18

  article1st authorCorresponding
  PublisherDOI

• P-order: Unified Convergence Analysis for Nonlinear Iterative Methods

  ArXiv.org · 2025-03-12

  preprintOpen access1st authorCorresponding

  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.

  PublisherOA PDFDOI

• Accelerating correlated wave function calculations with hierarchical matrix compression of the two-electron integrals

  The Journal of Chemical Physics · 2025-10-01

  article

  Leveraging matrix sparsity has proven to be a fruitful strategy for accelerating quantum chemical calculations. Here, we present the hierarchical SOS-MP2 algorithm, which uses hierarchical matrix (H2) 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 H2 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 O(N2⁡log⁡N) and a space complexity of O(N2⁡log⁡N), where N denotes the number of basis functions. Numerical tests show asymptotic time and space complexities better than O(N2) for both linear alkanes and three-dimensional water clusters.

  PublisherDOI

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

  Mathematics · 2025-07-29

  articleOpen accessSenior authorCorresponding

  The accurate reconstruction of piecewise continuous functions on meshes is challenging due to potential spurious oscillations—namely the Gibbs phenomenon—especially for high-order methods. This paper introduces the Robust Discontinuity Indicators (RDI) method, a novel technique for constructing discontinuity indicators. These indicators can effectively identify both C0 and C1 discontinuities in a single pass using a new comprehensive theoretical analysis combined with cell-based overshoot–undershoot indicators and node-based oscillation indicators. In addition to detecting discontinuities, these indicator values can also facilitate the construction of adaptive weighting schemes to mitigate the Gibbs phenomenon. Due to its flexibility, RDI can accommodate complex geometries and applies to nonuniform unstructured meshes and general surfaces, broadening its utility. Through experiments, we show that RDI can accurately capture discontinuities while producing fewer false positives than two-pass methods. By providing a more rigorous method for discontinuity detection, RDI has the potential to offer significant improvements in computational simulations and data remapping.

  PublisherDOI

• Efficient and scalable wave function compression using corner hierarchical matrices

  The Journal of Chemical Physics · 2024-11-25 · 2 citations

  article

  The exponential scaling of complete active space 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 block-wise low-rank decomposition. By application to dodecacene, a strongly correlated molecule, we demonstrate that CH matrix compression provides superior compression compared to 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.

  PublisherDOI

• Efficient and Scalable Wave Function Compression Using Corner Hierarchical Matrices

  arXiv (Cornell University) · 2024-07-30

  preprintOpen access

  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.

  PublisherOA PDFDOI

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

  arXiv (Cornell University) · 2023-04-26

  preprintOpen accessSenior author

  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.

  PublisherOA PDFDOI

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

  arXiv (Cornell University) · 2023-03-07

  preprintOpen access1st authorCorresponding

  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.

  PublisherOA PDFDOI

• Preserving superconvergence of spectral elements for curved domains via h- and p-geometric refinement

  Engineering With Computers · 2023-09-30 · 1 citations

  articleSenior author
  PublisherDOI

Recent grants

• Optimal Cross Parameterizations of Surfaces: Computational Methods and Scientific Applications

  NSF · $226k · 2007–2011

Frequent coauthors

• Michael T. Heath

  University of Illinois Urbana-Champaign

  13 shared

• Daniel R. Einstein

  Saint Martin's University

  13 shared

• Jiang Long

  12 shared

• Shikui Chen

  Stony Brook University

  12 shared

• Rebecca Conley

  10 shared

• Navamita Ray

  Los Alamos National Laboratory

  10 shared

• Tristan J. Delaney

  Stony Brook University

  9 shared

• Byung Moon Kim

  Yonsei University

  8 shared

Education

• Ph.D., Computer Science

  University of California, Los Angeles

  1995

• M.S., Computer Science

  University of California, Los Angeles

  1992

• B.S., Computer Science

  University of Science and Technology of China

  1988