My research field is numerical solutions for partial differential equations. As matter of fact my major as a undergraduate student was Computational Mathematics, though the first two years we took the same courses with the pure math majors. My focus for graduate study in China was numerical solutions for partial differential equations.
My current research interest focuses on numerical solutions for stochastic partial differential equations and integral equations, with applications to interface problems for flows in porous media, optimal controls for noise radiation problems related aircraft engines, and optimal controls of mathematical biology problems such as vector born diseases.

Current projects:

1. Modeling and numerical simulations for flows in karest aquifers.
2. Numerical solutions for stochastic partial differential equations and nonlinear filter problems. 
3. Numerical methods for flow in poroelastic media

4. Backward doubly stochastic differential equation methods for nonlinear filtering problems

Recent and current collaborators
1. Max Gunzburger, FSU, stochastic partial differential equations, flow in porous media
2. A. J. Meir,  Auburn University, Flow in poroelastic media
3. Song Chen, University of Wisconsin at La Crosse,  Flow in poroelastic media, bioconvenction models
4. Feng Bao, Auburn, Numerical methods for nonlinear filter problems
5. Xiaoying Han, Auburn University, Numerical methods for nonlinear filter problems
6. Yuehseng Xu,   Syracuse University, stochastic integral equations
7. Ying Jiang, Sun Yat-sen University, Stochastic PDES.
8. Xiaoming Wang,  FSU, flow in porous media
9. Hongmei Chi, FAMU, Efficient quasi-Monte Carlo method

10. Xiaoshen Wang,  University of Arkansas at Little Rock, Homotopy method for SDEs.

11. Peng Wang, Jilin University, Numerical methods for SDEs.
12. Dawit Denu, Stochastic population dynamics.


Yanzhao Cao, Ying Jiang, and Yuesheng Xu,  A fast  discrete  spectral  method  for stochastic partial differential equations, accepted by
Adv. Comput. Math.  (2017).

Lijin Wang,  Xiaoying Han, Yanzhao Cao, and Habib Najm,  Computational Singular Perturbation Analysis of Stochastic Chemical Systems with Stiffness, Accepted by JCP (2017).

Yanzhao Cao
, Jialin Hong, Zhihui Liu, Finite element approximations for SDEs driven by fractional Brownian motion noises, accepted by IMA Journal on Numerical Analysis (2017).

Feng Bao, Yanzhao Cao, and Xiaoying Han,  Efficient particle filtering for stochastic Korteweg-de Vries equations, accepted by Stochastics and Dynamics (2017).

Feng Bao, Yanzhao Cao, Clayton Webster, and Guannan Zhang,  An efficient meshfree implicit filter for nonlinear filtering problems,  I. J. UQ, 6(1), 19-33 (2016).

Feng Bao, Yanzhao Cao, A. J. Meir, and Weidong ZhaoA first order scheme for backward doubly stochastic differential equations,   SIAM Journal on Uncertainty Quantification,  4(1), 413-445 (2016).

Yanzhao Cao
and Dawit Denu,  Analysis of stochastic vector-host epidemic model with direct transmission,  DCDS-B,  21(7), 2109-2127, (2016).

F. Bao, Yanzhao  Cao, W. Zhao and B. Hu, Identification of hydraulic conductivity distributions in density dependent flow fields of submarine groundwater discharge modeling using adjoint-state,  SCIENCE CHINA Earth Sciences, doi:10.1007/s11430-015-5236-x, (2016).

Yanzhao Cao,  Song Chen, and Leo Rebholz, Well-posedness and a numerical study of a regularization model with adaptive nonlinear filtering for incompressible fluid flow, Computers & Mathematics with Applications,  71(11), 2192-2205  (2016).

Yanzhao Cao,  Song Chen, and A. J. Meir,  Quasi-linear poroelasticity: analysis and hybrid finite element approximations,  Numer. Meth. PDEs, Vol. 31, no 4, 1174--1189 (2015). 

 Feng Bao, Yanzhao Cao, and Weidong Zhao,  High order numerical algorithms for  forward backward  doubly stochastic differential equations, accepted by Discrete and Continuous Dynamical System-B,  Vol. 20, no 5, 1297--1313 (2015).

Sabit Adanur, Yang Shen, A. J. Meir, and Yanzhao Cao, Finite Element Analysis of Monofilament Woven Fabrics under Uniaxial Tension,  The Journal of the Textile Institute,  Vol. 106, no 1, 90--100 (2015)

Yanzhao Cao
and Zhang, Ran,   A stochastic collocation method for stochastic Volterra equations of the second kind. J. Integral Equations Appl. Vol. 27, no. 1, 1--25 (2015). 

Feng Bao, Yanzhao Cao,  Clayton Webster, and Guannan Zhang,   A hybrid sparse-grid approach for nonlinear filtering problems based on adaptive-domain of the Zakai equation approximations,  accepted by SIAM Journal on Uncertainty Quantification, Vol. 2, No 1, 784-804  (2014)  PDF

Yanzhao Cao,  O. A. IIhan, and S. A. Alimov, On the problems of peridynamics with special convolution kernels,  Journal of Integral Equations and ApplicationsVol. 26, No. 3, 301-321 (2014).

Yanzhao Cao
, Ying Jiang and Yuesheng Xu, A Fast algorithm  for orthogonal polynomial expansions on sparse grids,   Journal of Complexity , Vol. 30, No. 6,  673-682 (2014)  

Yanzhao Cao
,  Gunzburger, Max; He, Xiaoming; Wang, Xiaoming, Parallel non-iterative multi-physics domain decomposition methods for the time-dependent Stokes-Darcy model,   Mathematics of Computation, Vol.  83 no. 288, 1617--1644  (2014).

Yanzhao Cao, Feng Wang and Xiaoshen Wang,  Homotopy continuation methods for stochastic two-point boundary value problems driven by additive noises,  Journal of Computational Mathematics, Vol.32, No.6,  630--642 (2014).

Yanzhao Cao, Feng Bao and Xiaoying Han, An accurate and stable numerical algorithm for nonlinear filtering problems, Commun. Comput. Phys., Volume 16,  382-402 (2014).

Yanzhao Cao,  and  Song Chen,  A computational model of steady bioconvective flow with concentration dependent viscosity,   International Journal of Numerical Analysis and Modeling, Volume  11,  no.  1 86-101,  (2014).

Yanzhao Cao, Song Chen, and A.J. Meir. On steady flow in a poroelastic porous medium,  Mathematical Method in Applied Sciences, Vol. 37-7 1029-1041 (2014). 

Yanzhao Cao, Song Chen, and A.J. Meir. On the equation of quasi static poroelasticity with dilatation dependent hydraulic conductivity: existence uniqueness, and finite element approximation,  Discrete and Continuous Dynamical Systems-Series B, 18-5 July, 1253 - 1273  (2013).

Xiaolong Hu, Xiaoming Wang, Max Gunzburger, Fei Hua and Yanzhao Cao, Experimental and computational validation and verification of the Stokes-Darcy and continuum pipe flow models for karst aquifers with dual porosity structure,  Hydrological Processes, Vol 26, 2031-2040 (2012).

Yanzhao Cao, X. He, M. Gunzburger and X. Wang, Robin Robin domain decomposition methods for the steady-state Stokes-Darcy system with the Beavers Joseph interface condition, Numer. Math. Vol. 117,  601-629, (2011).

Yanzhao Cao and Li Yin, Spectral method for nonlinear stochastic partial differential equations of ellitpic type. Numer. Math. Theor. Meth. Appl. Vol. 4, 38-52, (2011).

Feng Bao,  Yanzhao Cao, and  Weidong Zhao,  Numerical solutions for forward backward doubly stochastic partial differential equations and Zakai equations, International Journal for Uncertainty Quantification, 1 (4): 351--367, (2011).

Yanzhao Cao,  Fei Hua, M. Gunzburger, Xiaoming Wang,  Finite element approximations of pipe flow interface problems.  Numeri. Meth. PDEs.  Vol. 27, 5, 1242-1252,  (2011).

Hongmei Chi and Yanzhao Cao,  Generating parallel random sequences via parameterizing EICGs for heterogeneous computing environments, Lecture Notes in Computer Science Volume 6019,  409-417,  (2010).

Yanzhao Cao, M. Gunzburger, F. Hua and X. Wang, Coupled Stokes-Darcy Model with Beavers-Joseph Interface Boundary Condition, Comm. Math. Sci.,  Volume 8, Number 1, 1-25,  (2010).   PDF

Yanzhao Cao, Z. Chen and Max Gunzburger, Finite element method approximation and error estimates for stochastic Stokes equations,   Advances in Computational Mathe-
, Vol. 33, 215-230, (2010).  

Yanzhao Cao,  Fei Hua, M. Gunzburger, Xiaoming Wang, and Weidong Zhao, Finite Element Approximations for Stokes--Darcy Flow with Beavers--Joseph Interface Conditions. SIAM J. Numer. Anal. 47 4239 (2010). 

Yanzhao Cao,  Bin Wu and Yuesheng Xu,  A fast collocation method for solving stochastic integral equation  SIAM J. Numer. Anal Vol. 47, No. 5, pp. 3744--3767, (2009).

K. Blayneh,
Yanzhao Cao and Hee-Dae Kwon, On asymptotic analysis and numerical optimal control of vector born diseases,  Discrete and Continuous Dynamical System B Volume 11, Number 3  (2009). PDF

Yanzhao Cao, Z. Chen and Max Gunzburger, An ANOVA  analysis for a class of partial differential equations with uncertain boundary conditions, , Int. J. Numer. Anal.& Model.. Volume 6, Number 2, Pages 256--273, (2009).

Yanzhao Cao, H. Chi, C. Milton and  W. Zhao,  Exploitation of sensitivity derivatives on quasi Monte Carlo simulations, Monte Carlo Methods and Applications, Vol. 14, 269-279, (2008) .

Yanzhao Cao, Ran Zhang and Kai Zhang, Finite Element and Discontinuous Galerkin Method for Stochastic Helmholtz Equation in Two- and Three-Dimensions.
J. Comp. Math., 26, 702-715,  (2008).

Yanzhao Cao, K. Zhang and R. Zhang, Finite Element Method and Discontinuous Galerkin Method for Stochastic Scattering Problem of Helmholtz Type, Potential Analysis, Volume 28, 301-319, (2008). PDF

Yanzhao Cao and Li Yin, A spectral finite element method for stochastic wave equations in one dimensional space,   Comm.  Pure & Appl. Anal.  6, 607 - 617, (2007).

Yanzhao Cao, M. Huang and Y. Xu,   A hybrid collocation method for Fredholm integral equations with singular kernels,    Appl. Numer. Math.  57, 549-561 (2007).
Yanzhao Cao,  L. Yin and H. Yang, Finite element methods for  semilinear elliptic stochastic partial differential equations,  Numerische Mathematik, Vol. 106 , 2007 PDF

Yanzhao Cao, G. Cerezo and T. Herdman and J. Turi,  Singularity expansion for a class of neutral delay equations, accepted by    J. Integral Equations Appl.  Vol 19,   13-32,  (2007).

Yanzhao Cao, M. Y. Hussaini and H.  Yang, Numerical optimization of radiated engine noise with uncertain wave numbers,  Int. J. Numer. Anal.& Model. Vol. 4  No. 3-4  392--401, (2007).

Yanzhao Cao, M. Y. Hussaini and Hongtao Yang,  Parameter identification for a noise radiation problem,  Int. J. Numer. Anal.& Model. Vol. 4, Number 1, Pages 116--126, (2007).

Yanzhao Cao,  On the rate of convergence of Wiener-Ito expansion for generalized random variables,  Stochastics, , 78 (3),  179-187, (2006)  PDF 

Yanzhao Cao, M.Y. Hussaini,  T. Zang and A. Zetazalo, A variance  reduction method using high order sensitivity derivatives,  Appl. Numer. Math, 56, 800--813 (2006). 

 S. Wang,
Yanzhao Cao and  P. Rikvold,  First-principles calculations for the adsorption of water molecules on the Cu(100) surface, Physics  Review  B Vol. 70, 205410 (2004).

Yanzhao Cao,  M.Y. Hussaini and T. Zang,  Applications of sensitivity in Monte Carlo simulation.  AIAA Journal, Vol. 42, No. 4,  815-822 (2004). PDF

Yanzhao Cao
,  M. Y. Hussaini and T. Zang, On the exploitation of sensitivity derivatives for accelerating sampling methods for uncertainty analysis,  AIAA paper 2003-1656,  (2003).

Yanzhao Cao,  M. Y. Hussaini and T. Zang, An efficient Monte Carlo method for optimal control problems with uncertainty, Computational Optimization and Applications, Vol. 26,  219-230, (2003). PDF

Yanzhao Cao, T. Herdman and Y. Xu, A hybrid collocation method for Volterra integral equations with weakly singular kernels.  SIAM J. Numer. Anal., Vol. 41, No. 1, 364-381 (2003).

Yanzhao Cao and D. Stanescu, Shape optimization for noise radiation problemsComp. & Math. Appl., Vol. 44, No. 12,  1539-1556,  (2002).PDF

Yanzhao Cao, On convergence of optimal control  problem to exact control problem for systems governed by Navier-Stokes equations (with M. Y. Hussaini),  Proceedings of ICNPPA-2000, Vol 1, 117-126 (2001).

Yanzhao Cao,  Numerical approximations of exact controllability problems using  optimal control problems for parabolic differential equationsAppl. Math. Comp.,  Vol 119, 127-145, (2001).

Yanzhao Cao,  Numerical solution of diffraction problems by a least-square  finite element method (with H. Yang and G. Bao), Math. Meth. Appl. Sci.  Vol. 23, 1073-1092 (2000).

Yanzhao Cao, On exact controllability and convergence of optimal controls to exact controls of parabolic equations, (with M. Gunzburger and and J.Turner)  Optimal Control: Theory,  Algorithms, and Applications, Kluwer, 67-83, (1999).

Yanzhao Cao,  A nonconforming least squares finite element method for interface problems,  (with M. Gunzburger) SIAM J. Numer. Anal. Vol 36, No. 2, 312-322 (1998).

Yanzhao Cao, On conditions of controllability for systems governed by parabolic differential equations, (with M. Gunzburger), Math. Anal. Appl., Vol. 215,  174-189 (1997).

Yanzhao Cao, Singularity preserving Galerkin methods for weakly singular Fredholm integral equations, (with Y. Xu) J. Integral Equations Appl., Vol. 6, 303-334 (1994).

Yanzhao Cao, On finite element approximations for the bifurcation solutions of free convention problems (with M. Huang),  Proceedings on numerical methods for partial differential equations, (refereed),  World Scientific, Singapore, 3-14  (1992).

Feng Bao, Yanzhao Cao,  A. J. Mier,   and Weidong Zhao,  A first order scheme for backward doubly stochastic differential Equations, submitted to SIAM Journal on Uncertainty Quantification (2014) PDF