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Zexuan Liu

Biography[CV]

I am currently a Research Assistant in University of California, Berkeley, majoring in Mathematics, under the guidance of Professor Paul Grigas. I will (suppose to) receive the B.S degree from Wuhan University (WHU), China, in 2021.

Publications

• Z. Liu, Z. Sun, J. Yang, "A Numerical Study of Superconvergence of the Discontinuous Galerkin Method by Patch Reconstruction", Submitted to the Electronic Research Archive. [pdf]

Research

Interests

My research interests are at the interface between high performance computing and optimization. More speifically, I am interested in fast convergence theory in numerical PDEs, highly efficient optimization algorithms and their connections in machine learning.

Non-convex Parametric Optimization via Differential Equations

Used ordinary differential equation to develop different second order algorithms for computing anapproximately optimal solution path of a parameterized non-convex problem.

• Derived approximate solution path by Euler discretization method

• Developed the error bound of the algorithm.

• Modified the algorithm by solving a sub-problem to minimize the upper bound (MINIUPPER).

• Implement MINIUPPER algorithm and prove it has the optimal convergence rate.

Superconvergence of the Discontinuous Galerkin Method

Developed a new Galerkin method by patch reconstruction requiring much less free order than the traditional methods to solve elliptic PDEs and explored the superconvergence property.

• Developed symmetry element patch picking rule and defined only one degree of freedom (DOF) per element.

• Constructed the global stiffness matrix with fix size regardless of the approximation order which may vary.

• Implemented the discontinuous Galerkin method by patch reconstruction (DGPR) in MATLAB and C++.

• Extended the DGPR method in MATLAB into the 6th polynomials while traditional method only use 1st order or 2nd order polynomials.

• Found three different patterns of superconvergence from 1 to 3 dimensions with our DGPR method in elliptic problem when the mesh has geometric symmetry.

High Performance Machine Learning Library

Developed a high efficiency machine learning library library for an ARMv8-a architecture server and implemented a scikit-learn like C machine learning toolkit optimized for the specific architecture.

• Developed linear regression models (Ridge regression, Lasso regression and Elastic Net methods) based on OpenBLAS, which are 140% times faster than scikit-learn functions on the ARMv8 sever.

• Implemented the Word2Vec function whose training is 171% times faster than scikit-learn functions on the ARMv8 sever.

• Developed givens transformation operator via TVM to provide cross platform library.

Teaching experience

Teaching Assistant

• Fall 2019

Numerical Analysis

Direct solution of linear systems, including error bounds, iteration methods, least square approximation, eigenvalues and eigenvectors of matrices. I taught students with theoretical and practical results of iteration methods and eigen problems.

Seminar

• Fall 2018 & Spring 2019

Seminar of advanced analysis

Covering the first three books of Princeton Lectures in Analysis written by Elias M. Stein and Rami Shakarchi. I delivered the most part of the Real Analysis and several chapters of Fourier Analysis.