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Zhiqiang Cai 0001
Person information
- affiliation: Purdue University, Department of Mathematics, West Lafayette, IN, USA
Other persons with the same name
- Zhiqiang Cai — disambiguation page
- Zhiqiang Cai 0002
— University of Wisconsin-Madison, WI, USA (and 1 more)
- Zhiqiang Cai 0003
- Zhiqiang Cai 0004
- Zhiqiang Cai 0005 — Tongji University, School of Electronics and Information, China
- Zhiqiang Cai 0006
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2020 – today
- 2024
- [j51]Zhiqiang Cai
Least-Squares Neural Network (LSNN) Method for Linear Advection-Reaction Equation: Discontinuity Interface. SIAM J. Sci. Comput. 46(4): 448- (2024) - [i16]Zhiqiang Cai, Anastassia Doktorova, Robert D. Falgout, César Herrera
Fast Iterative Solver For Neural Network Method: I. 1D Diffusion Problems. CoRR abs/2404.17750 (2024) - [i15]Zhiqiang Cai, Anastassia Doktorova, Robert D. Falgout, César Herrera
Fast Iterative Solver For Neural Network Method: II. 1D Diffusion-Reaction Problems And Data Fitting. CoRR abs/2407.01496 (2024) - [i14]Zhiqiang Cai, Junpyo Choi, Min Liu:
ReLU neural network approximation to piecewise constant functions. CoRR abs/2410.16506 (2024) - 2023
- [j50]Zhiqiang Cai
Least-squares neural network (LSNN) method for scalar nonlinear hyperbolic conservation laws: Discrete divergence operator. J. Comput. Appl. Math. 433: 115298 (2023) - [i13]Zhiqiang Cai, Junpyo Choi, Min Liu:
Least-Squares Neural Network (LSNN) Method for Linear Advection-Reaction Equation: General Discontinuous Interface. CoRR abs/2301.06156 (2023) - [i12]Zhiqiang Cai, Binghe Chen, Jing Yang:
Adaptive Least-Squares Methods for Convection-Dominated Diffusion-Reaction Problems. CoRR abs/2301.11582 (2023) - [i11]Min Liu, Zhiqiang Cai, Karthik Ramani:
Deep Ritz Method with Adaptive Quadrature for Linear Elasticity. CoRR abs/2305.15257 (2023) - [i10]Zhiqiang Cai, Junpyo Choi, Min Liu:
Least-Squares Neural Network (LSNN) Method For Linear Advection-Reaction Equation: Non-constant Jumps. CoRR abs/2306.07445 (2023) - 2022
- [j49]Min Liu
Adaptive two-layer ReLU neural network: I. Best least-squares approximation. Comput. Math. Appl. 113: 34-44 (2022) - [j48]Min Liu
Adaptive two-layer ReLU neural network: II. Ritz approximation to elliptic PDEs. Comput. Math. Appl. 113: 103-116 (2022) - [j47]Zhiqiang Cai, Jingshuang Chen
Self-adaptive deep neural network: Numerical approximation to functions and PDEs. J. Comput. Phys. 455: 111021 (2022) - 2021
- [j46]Zhiqiang Cai, Cuiyu He
Generalized Prager-Synge identity and robust equilibrated error estimators for discontinuous elements. J. Comput. Appl. Math. 398: 113673 (2021) - [j45]Zhiqiang Cai
Least-squares ReLU neural network (LSNN) method for linear advection-reaction equation. J. Comput. Phys. 443: 110514 (2021) - [j44]Shaohong Du, Zhiqiang Cai:
Adaptive Finite Element Method for Dirichlet Boundary Control of Elliptic Partial Differential Equations. J. Sci. Comput. 89(2): 36 (2021) - [i9]Zhiqiang Cai, Jingshuang Chen, Min Liu:
Least-Squares ReLU Neural Network (LSNN) Method For Scalar Nonlinear Hyperbolic Conservation Law. CoRR abs/2105.11627 (2021) - [i8]Zhiqiang Cai, Jingshuang Chen, Min Liu:
Least-Squares ReLU Neural Network (LSNN) Method For Linear Advection-Reaction Equation. CoRR abs/2105.11632 (2021) - [i7]Difeng Cai, Zhiqiang Cai:
Hybrid A Posteriori Error Estimators for Conforming Finite Element Approximations to Stationary Convection-Diffusion-Reaction equations. CoRR abs/2107.06341 (2021) - [i6]Min Liu, Zhiqiang Cai:
Adaptive Two-Layer ReLU Neural Network: II. Ritz Approximation to Elliptic PDEs. CoRR abs/2107.06459 (2021) - [i5]Min Liu, Zhiqiang Cai, Jingshuang Chen:
Adaptive Two-Layer ReLU Neural Network: I. Best Least-squares Approximation. CoRR abs/2107.08935 (2021) - [i4]Zhiqiang Cai, Jingshuang Chen, Min Liu:
Self-adaptive deep neural network: Numerical approximation to functions and PDEs. CoRR abs/2109.02839 (2021) - [i3]Zhiqiang Cai, Jingshuang Chen, Min Liu:
Finite Volume Least-Squares Neural Network (FV-LSNN) Method for Scalar Nonlinear Hyperbolic Conservation Laws. CoRR abs/2110.10895 (2021) - 2020
- [j43]Zhiqiang Cai
Deep least-squares methods: An unsupervised learning-based numerical method for solving elliptic PDEs. J. Comput. Phys. 420: 109707 (2020) - [j42]Difeng Cai
Robust Equilibrated Error Estimator for Diffusion Problems: Mixed Finite Elements in Two Dimensions. J. Sci. Comput. 83(1): 22 (2020) - [j41]Difeng Cai
Robust equilibrated a posteriori error estimator for higher order finite element approximations to diffusion problems. Numerische Mathematik 144(1): 1-21 (2020) - [j40]Zhiqiang Cai, JaEun Ku
A Dual Finite Element Method for a Singularly Perturbed Reaction-Diffusion Problem. SIAM J. Numer. Anal. 58(3): 1654-1673 (2020) - [i2]Cuiyu He
Generalized Prager-Synge Inequality and Equilibrated Error Estimators for Discontinuous Elements. CoRR abs/2001.09102 (2020)
2010 – 2019
- 2019
- [j39]Fleurianne Bertrand
Least-Squares Methods for Elasticity and Stokes Equations with Weakly Imposed Symmetry. Comput. Methods Appl. Math. 19(3): 415-430 (2019) - [j38]Zhaopeng Hao, Moongyu Park
Finite Element Method for Two-Sided Fractional Differential Equations with Variable Coefficients: Galerkin Approach. J. Sci. Comput. 79(2): 700-717 (2019) - [i1]Zhiqiang Cai, Jingshuang Chen, Min Liu, Xinyu Liu:
Deep least-squares methods: an unsupervised learning-based numerical method for solving elliptic PDEs. CoRR abs/1911.02109 (2019) - 2018
- [j37]Zhiqiang Cai, Seokchan Kim
Error estimate of a finite element method using stress intensity factor. Comput. Math. Appl. 76(10): 2402-2408 (2018) - 2017
- [j36]Zhiqiang Cai, Cuiyu He
Residual-based a posteriori error estimate for interface problems: Nonconforming linear elements. Math. Comput. 86(304): 617-636 (2017) - [j35]Zhiqiang Cai, Cuiyu He
Discontinuous Finite Element Methods for Interface Problems: Robust A Priori and A Posteriori Error Estimates. SIAM J. Numer. Anal. 55(1): 400-418 (2017) - 2015
- [j34]Zhiqiang Cai, Varis Carey, JaEun Ku
Asymptotically exact a posteriori error estimators for first-order div least-squares methods in local and global L2 norm. Comput. Math. Appl. 70(4): 648-659 (2015) - [j33]Zhiqiang Cai, Robert D. Falgout
Div First-Order System LL* (FOSLL*) for Second-Order Elliptic Partial Differential Equations. SIAM J. Numer. Anal. 53(1): 405-420 (2015) - 2012
- [j32]Zhiqiang Cai, Shun Zhang
Mixed methods for stationary Navier-Stokes equations based on pseudostress-pressure-velocity formulation. Math. Comput. 81(280): 1903-1927 (2012) - [j31]Zhiqiang Cai, Shun Zhang
Robust Equilibrated Residual Error Estimator for Diffusion Problems: Conforming Elements. SIAM J. Numer. Anal. 50(1): 151-170 (2012) - 2011
- [j30]Zhiqiang Cai, Xiu Ye, Shun Zhang
Discontinuous Galerkin Finite Element Methods for Interface Problems: A Priori and A Posteriori Error Estimations. SIAM J. Numer. Anal. 49(5): 1761-1787 (2011) - [j29]Zhiqiang Cai, JaEun Ku
Goal-Oriented Local A Posteriori Error Estimators for H(div) Least-Squares Finite Element Methods. SIAM J. Numer. Anal. 49(6): 2564-2575 (2011) - 2010
- [j28]Zhiqiang Cai, JaEun Ku
Optimal Error Estimate for the Div Least-squares Method with Data f∈L2 and Application to Nonlinear Problems. SIAM J. Numer. Anal. 47(6): 4098-4111 (2010) - [j27]Zhiqiang Cai, Shun Zhang
Recovery-Based Error Estimators for Interface Problems: Mixed and Nonconforming Finite Elements. SIAM J. Numer. Anal. 48(1): 30-52 (2010) - [j26]Zhiqiang Cai, Chunbo Wang, Shun Zhang
Mixed Finite Element Methods for Incompressible Flow: Stationary Navier-Stokes Equations. SIAM J. Numer. Anal. 48(1): 79-94 (2010) - [j25]Zhiqiang Cai, Shun Zhang
Flux Recovery and A Posteriori Error Estimators: Conforming Elements for Scalar Elliptic Equations. SIAM J. Numer. Anal. 48(2): 578-602 (2010)
2000 – 2009
- 2009
- [j24]Zhiqiang Cai, Yanqiu Wang:
An error estimate for two-dimensional Stokes driven cavity flow. Math. Comput. 78(266): 771-787 (2009) - [j23]Zhiqiang Cai, Shun Zhang:
Recovery-Based Error Estimator for Interface Problems: Conforming Linear Elements. SIAM J. Numer. Anal. 47(3): 2132-2156 (2009) - [j22]Frank S. Attia, Zhiqiang Cai, Gerhard Starke:
First-Order System Least Squares for the Signorini Contact Problem in Linear Elasticity. SIAM J. Numer. Anal. 47(4): 3027-3043 (2009) - 2008
- [j21]Zhiqiang Cai, Chad R. Westphal:
A Weighted H(div) Least-Squares Method for Second-Order Elliptic Problems. SIAM J. Numer. Anal. 46(3): 1640-1651 (2008) - 2007
- [j20]Zhiqiang Cai, Yanqiu Wang
A Multigrid Method for the Pseudostress Formulation of Stokes Problems. SIAM J. Sci. Comput. 29(5): 2078-2095 (2007) - 2006
- [j19]Zhiqiang Cai, JaEun Ku
The L2 Norm Error Estimates for the Div Least-Squares Method. SIAM J. Numer. Anal. 44(4): 1721-1734 (2006) - 2004
- [j18]Zhiqiang Cai, Gerhard Starke:
Least-Squares Methods for Linear Elasticity. SIAM J. Numer. Anal. 42(2): 826-842 (2004) - [j17]Zhiqiang Cai, Barry Lee, Ping Wang:
Least-Squares Methods for Incompressible Newtonian Fluid Flow: Linear Stationary Problems. SIAM J. Numer. Anal. 42(2): 843-859 (2004) - 2003
- [j16]Zhiqiang Cai, Jim Douglas Jr., Moongyu Park:
Development and Analysis of Higher Order Finite Volume Methods over Rectangles for Elliptic Equations. Adv. Comput. Math. 19(1-3): 3-33 (2003) - [j15]Zhiqiang Cai, Rossen R. Parashkevov, Thomas F. Russell, J. D. Wilson, Xiu Ye:
Domain Decomposition for a Mixed Finite Element Method in Three Dimensions. SIAM J. Numer. Anal. 41(1): 181-194 (2003) - [j14]Zhiqiang Cai, Gerhard Starke:
First-Order System Least Squares for the Stress-Displacement Formulation: Linear Elasticity. SIAM J. Numer. Anal. 41(2): 715-730 (2003) - 2002
- [j13]Zhiqiang Cai, Seokchan Kim, Gyungsoo Woo:
A finite element method using singular functions for the Poisson equation: crack singularities. Numer. Linear Algebra Appl. 9(6-7): 445-455 (2002) - [j12]Zhiqiang Cai, Byeong Chun Shin:
The Discrete First-Order System Least Squares: The Second-Order Elliptic Boundary Value Problem. SIAM J. Numer. Anal. 40(1): 307-318 (2002) - 2001
- [j11]Zhiqiang Cai, Seokchan Kim:
A Finite Element Method Using Singular Functions for the Poisson Equation: Corner Singularities. SIAM J. Numer. Anal. 39(1): 286-299 (2001) - [j10]Zhiqiang Cai, Thomas A. Manteuffel, Stephen F. McCormick, John W. Ruge:
First-Order System \CL\CL* (FOSLL*): Scalar Elliptic Partial Differential Equations. SIAM J. Numer. Anal. 39(4): 1418-1445 (2001) - [j9]Zhiqiang Cai, Seokchan Kim, Byeong Chun Shin:
Solution Methods for the Poisson Equation with Corner Singularities: Numerical Results. SIAM J. Sci. Comput. 23(2): 672-682 (2001) - 2000
- [j8]Zhiqiang Cai:
Least Squares for the Perturbed Stokes Equations and the Reissner-Mindlin Plate. SIAM J. Numer. Anal. 38(5): 1561-1581 (2000) - [j7]Zhiqiang Cai, Chang-Ock Lee, Thomas A. Manteuffel, Stephen F. McCormick:
First-Order System Least Squares For Linear Elasticity: Numerical Results. SIAM J. Sci. Comput. 21(5): 1706-1727 (2000) - [j6]Zhiqiang Cai, Chang-Ock Lee, Thomas A. Manteuffel, Stephen F. McCormick:
First-Order System Least Squares for the Stokes and Linear Elasticity Equations: Further Results. SIAM J. Sci. Comput. 21(5): 1728-1739 (2000)
1990 – 1999
- 1999
- [j5]Thomas Y. Hou, Xiao-hui Wu, Zhiqiang Cai:
Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients. Math. Comput. 68(227): 913-943 (1999) - [j4]Zhiqiang Cai, Xiu Ye, Huilong Zhang:
Least-squares finite element approximations for the Reissner-Mindlin plate. Numer. Linear Algebra Appl. 6(6): 479-496 (1999) - 1996
- [j3]Zhiqiang Cai, Chen-Yao G. Lai:
Convergence Estimates of Multilevel Additive and Multiplicative Algorithms for Non-symmetric and Indefinite Problems. Numer. Linear Algebra Appl. 3(3): 205-220 (1996) - 1993
- [j2]Zhiqiang Cai, Charles I. Goldstein, Joseph E. Pasciak:
Multilevel Iteration for Mixed Finite Element Systems with Penalty. SIAM J. Sci. Comput. 14(5): 1072-1088 (1993)
1980 – 1989
- 1989
- [j1]Zhiqiang Cai, Steve F. McCormick:
Computational Complexity of the Schwarz Alternating Procedure. Int. J. High Speed Comput. 1(1): 1-28 (1989)
Coauthor Index
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