Xuanzhao Gao (高煊钊), Ph.D.

I'm a Flatiron Research Fellow in the Center for Computational Mathematics at the Flatiron Institute interested in developing fast numerical methods for a variety of problems in computational mathematics. I am working with Alex Barnett and Jason Kaye on developing boundary integral solvers for system with complex geometries.

I received my Ph.D. in Hong Kong University of Science and Technology, where I was advised by Zecheng Gan, co-advised by Yang Xiang and Jinguo Liu, with my thesis on Confined Quasi-2D Coulomb Systems: Theory, Algorithms, and Applications.

Download my CV (updated on 2025-10-15).

Research

Boundary Integral Equations

I am currently working on developing BIE solvers for problems with edge and corner singularities.

Boundary Integral Equations

Fast Summation Algorithms

I develop fast summation algorithms for electrostatics in Molecular Dynamics simulations and Monte Carlo simulations, including stochastic and spectral methods. I do analysis on the accuracy and efficiency of the algorithms, and providing efficient implementations in Julia and C++.

Fast Summation Algorithms

Tensor Network Methods

I work on tensor network methods for solving combinatorial optimization problems, especially the maximum independent set problem. We provide a set of tools for constructing and solving tensor network models in Julia.

Tensor Network Methods

Papers

  1. X. Gao, Q. Zhou, Z. Gan, and J. Liang, Accurate error estimates and optimal parameter selection in Ewald summation for dielectrically confined Coulomb systems, J. Chem. Theory Comput. 21, 5890 (2025)
  2. (alphabetical order) Z. Gan, X. Gao, J. Liang, and Z. Xu, Random batch Ewald method for dielectrically confined Coulomb systems, SIAM J. Sci. Comput. 47, B846 (2025)
  3. X. Gao, X. Li, and J. Liu, Programming guide for solving constraint satisfaction problems with tensor networks, Chinese Physics B 34, 50201 (2025)
  4. (alphabetical order) Z. Gan, X. Gao, J. Liang, and Z. Xu, Fast algorithm for quasi-2D Coulomb systems, J. Comput. Phys. 524, 113733 (2025)
  5. M. Roa-Villescas, X. Gao, S. Stuijk, H. Corporaal, and J.-G. Liu, Probabilistic inference in the era of tensor networks and differential programming, Phys. Rev. Res. 6, 33261 (2024)
  6. X. Gao and Z. Gan, Broken symmetries in quasi-2D charged systems via negative dielectric confinement, J. Chem. Phys. 161, (2024)
  7. (co-first author) Z. Nie, X. Gao, Y. Ren, S. Xia, Y. Wang, Y. Shi, J. Zhao, and Y. Wang, Harnessing hot phonon bottleneck in metal halide perovskite nanocrystals via interfacial electron-phonon coupling, Nano Lett. 20, 4610 (2020)

Preprints

  1. (alphabetical order) X. Gao, S. Jiang, J. Liang, Z. Xu, and Q. Zhou, A fast spectral sum-of-Gaussians method for electrostatic summation in quasi-2D systems, Arxiv Preprint Arxiv:2412.04595 (2024)
  2. X. Gao, Y.-J. Wang, P. Zhang, and J.-G. Liu, Automated discovery of branching rules with optimal complexity for the maximum independent set problem, Arxiv Preprint Arxiv:2412.07685 (2024)

Talks

  1. A Fast Spectral Sum-of-Gaussians Method for Electrostatic Summation in Quasi-2D Systems, The 14th CSCM Annual Conference, Invited Talk, August 17-21, 2025
  2. TreeWidthSolver.jl: From Treewidth to Tensor Network Contraction Order, JuliaCN Meetup 2024, Invited Talk, Nov 2-3, 2024
  3. Fast Algorithm for Quasi-2D Coulomb Systems, SciCADE 2024, Contributed Talk, July 15-19, 2024
  4. How to Implement Generic Matrix-Mul with Generic Element Types on GPU?, JuliaCN Meetup 2023, Contributed Talk, Dec 9, 2023
  5. Random Batch Quasi-Ewald Method for the Simulations of Charged Particles under Dielectric Confinement, ICIAM 2023, Poster, August 20-25, 2023

Blogs

Below are some blogs on my personal website (link).

  1. How to implement generic matrix multiplication (GEMM) with generic element types on GPU?

    This blog is a technical note for the Open Source Promotion Plan 2023 project "TropicalGEMM on GPU" released by JuliaCN, where I developed a Julia package CuTropicalGemm.jl, to calculate Generic Matrix Multiplication (GEMM) of Tropical Numbers on Nvidia GPUs.

  2. Tensor Network Contraction Order Optimization with Exact Tree Width Solver

    This blog is a technical note for the Google Summer of Code 2024 project "Tensor network contraction order optimization and visualization" released by The Julia Language, where I implemented an optimizer for tensor network contraction order based on tree decomposition in the Julia package OMEinsumContractionOrders.jl.

  3. Finding the Optimal Tree Decomposition with Minimal Treewidth

    This blog is a supplementary for the note Tensor Network Contraction Order Optimization with Exact Tree Width Solver, where I detailed introduce the algorithm to find the optimal tree decomposition with minimal treewidth of a given simple graph, and how it is implemented in Julia package TreeWidthSolver.jl.

And some other technical notes.

  1. How to install slurm on Ubuntu 22.04

    This blog is a technical note for the installation of slurm on Ubuntu 22.04 with NIS and apt tools.

Software

I contribute to a number of open source software projects.

Name Description
EwaldSummations.jl A comprehensive implementation of the Ewald summation method for electrostatic interactions in both triply and doubly periodic systems with and without dielectric mismatches.
TropicalNumbers.jl A refined implementation of the tropical semiring.
CuTropicalGEMM.jl A GPU-accelerated implementation of the tropical matrix multiplication, supported by OSPP 2023.
TreeWidthSolver.jl A collection of tools for calculating the exact tree width and tree decomposition of a given graph, supported by GSOC 2024.
OptimalBranching.jl A framework for automated discovery of optimal branching rules for the branch-and-bound algorithm.

Contact