The focus of my research is on scientific computing and mathematical models' study. In particular, I study problems in tumor growth, blood coagulation, cell cycle control using both stochastic and deterministic approaches. The mathematical tools that I use include ODEs, PDEs, numerical algebraic geometry, uncertainty quantification, sensitivity analysis, bifurcation analysis, and computational methods.

Mathematical Models

1. Tumor growth models

Tumor growth is often modeled by PDEs with a free boundary where the changing shape of the tumor is of prime importance. Specifically, the models depend on values of the controlling parameter, called the tumor-aggressiveness factor, where bifurcations occur are nontrivial to compute. We have been using bifurcation analysis to prove that there exists a sequence of symmetric-breaking bifurcation branches of stationary solutions and using numerical algebraic geometry methods to compute the steady states of tumor growth models such as solid tumor, tumor with a necrotic core as well as fluid-like tumor.

In tumor models, it is interesting to find out whether it is possible for the tumor to grow into other shapes besides spherical shape. Homotopy continuation method combined with adaptive path tracking as well as multi-precision is used to compute nonradial solutions. Such procedure automatically and systematically handles the singular system when the parameter approaches the bifurcation points. Numerical examples indicate that the homotopy continuation method is efficient to manipulate parametric problems with bifurcation. Moreover, discretization of the free boundary by floating grids yields a polynomial system and reaches high order accuracy on the unconstructive grids.

Related references:
1. W. Hao, J. D. Hauenstein, B. Hu and A. J. Sommese, A three-dimensional steady-state tumor system, Applied Mathematics and Computation, Volume 218, Issue 6, pp. 2661-2669, (2011). [pdf]
2. W. Hao, J. D. Hauenstein, B. Hu Y. Liu, A. J. Sommese and Y.-T. Zhang, Bifurcation for a free boundary problem modeling the growth of a tumor with a necrotic core, Nonlinear Analysis Series B: Real World Applications, Volume 13, Issue 2, pp. 694-709, (2012). [pdf]
3. W. Hao, J. D. Hauenstein, B. Hu Y. Liu, A. J. Sommese and Y.-T. Zhang, Continuation along bifurcation branches for a tumor model with a necrotic core, Journal of Scientific Computing, to appear , (2012). [pdf]
4. W. Hao, J. D. Hauenstein, B. Hu T. McCoy and A. J. Sommese, Computing steady-state solutions for a free boundary problem modeling tumor growth by Stokes equation, Journal of Computational and Applied Mathematics, to appear, (2012). [pdf]
2. Blood coagulation model

The structural integrity of a thrombus has important medical consequences, as fragments washed away from an unstable clot in a peripheral vein can embolize to the lungs with sometimes fatal results. The stability of the thrombus is influenced by the structural heterogeneity of the thrombus as the boundaries between discreet domains with different mechano-elastic properties are susceptible to fracture. We develop algorithms based on numerical algebraic geometry to compute steady state solutions for different initial conditions. Because of rank-deficiency of the steady state system, a numerical algebraic geometry algorithm is introduced to explore the system and obtain the steady state solutions. Steady states which can not be obtained by time marching method were discovered. We then perform both variance decomposition based sensitivity analysis and Morris design method to rank the significance of the 16 uncertain reaction rates with respect to the total thrombin production. Good agreement on the sensitivity ranking of the reaction rates is observed using both methods. Sparse grid probabilistic collocation method (SGPCM) is employed to quantify how uncertainties of these 16 reaction rates influence total thrombin production. We also used the importance ranking of the corresponding reaction rates obtained through sensitivity analysis study to identify the most effective drug therapy and employ the model to predict the performance of the most effective drug therapy with confidence interval of corresponding thrombin production using SGPCM.
The effect of platelet on blood coagulation is studied by developing a distributed-Lagrange-multiplier/fictitious-domain(DLM/FD). Using this method, the three-dimensional motion of a viscoelastic platelet in a shear blood flow is simulated and compared with experiments on tracking platelets in a blood chamber. The blood flow field is modeled using an incompressible Navier-Stokes fluid solver; the platelet is decoupled by Lagrange multiplier. We formulate potential energy to model the forces of platelet, i.e., worm-like force, bending force, surface force and volume force to allow accurate representations of ''flipping" dynamics of platelets.
Related reference:
1. W. Hao, G. Lin, Z. Xu, A.J. Sommese and M. Alber, Analyzing stoichiometric regulation of blood coagulation using numerical algebraic geometry method and sensitivity analysis, Submitted, (2012). [pdf]

Numerical analysis/Numerical PDE

1. Bootstrapping method for computing multiple solutions of nonlinear PDEs

Discretization of systems of differential equations often leads to systems of polynomials. Though using modern numerical codes for the complete solution of polynomial systems can yield new solutions, the systems of polynomials (arising even from very sparse grids) are usually much too large for direct solution by these codes. The realization underlying this article is that domain decomposition gives excellent guidance on how to bootstrap'' from the solutions of many small systems of polynomials to often large numbers of solutions of a system of polynomials arising from a discretization with a realistic grid. We introduces a new domain decomposition algorithm which is called the bootstrapping method. This method computes multiple solutions of nonlinear PDE systems and is naturally parallelizable. Numerical examples for both one and two dimensional cases show the feasibility of algorithm.
Related references:
1. W. Hao, J. D. Hauenstein, B. Hu and A. J. Sommese, A domain decomposition algorithm for computing multiple steady states of differential equations, Submitted, (2011). [pdf]
2. Homotopy method based on WENO scheme for solving steady state problems of hyperbolic conservation laws

Homotopy continuation is an efficient tool originally designed for solving polynomial systems via numerical algebraic geometry. Its efficiency relies on utilizing adaptive stepsize and adaptive precision path tracking, and endgames. In this paper, we apply homotopy continuation to solve steady state problems of hyperbolic conservation laws. The algorithm is based on a finite difference discretization using the Lax-Friedrich numerical fluxes evaluated based on the WENO scheme. Extensive numerical examples in both scalar and system test problems in one and two dimensions demonstrate the efficiency. Finite volume and finite difference methods are widely used in hyperbolic conservation laws $\mathbf{u}_t+\nabla\cdot F(\mathbf{u})=g(\mathbf{u}).$ In both types of schemes, time marching approach is expensive to compute steady states due to small stepsize on time space. We are interested in developing homotopy method which is based on WENO scheme and have a faster convergence than the regular time marching approaches. Our effort is restricted to steady state problems. We introduce a homotopy function $H(u,\epsilon)=\big(\nabla\cdot F(\mathbf{u})-g(\mathbf{u})-\epsilon \Delta \mathbf{u}\big)(1-\epsilon)-\epsilon(\mathbf{u}-\mathbf{u}^0)\equiv0,$ where $$\mathbf{u}^0$$ is the initial condition, $$\epsilon$$ is a parameter between 0 and 1. In particular, the initial condition automatically satisfies the homotopy function when $$\epsilon=1$$, and it becomes the steady state problem by setting $$\epsilon=0$$. Our homotopy method are tested on both one and two dimensional examples and shows faster convergence than the time marching approach. The figure below shows the result of one dimensional Burger's equation.
Related references:
1. W. Hao, J. D. Hauenstein, A.J. Sommese, C.W. Shu, Z. Xu and Y. Zhang.Homotopy method for steady state problems on hyperbolic conservation laws, Submitted, (2012) [pdf]
3. A package for computing multiplicity structures of nonlinear systems

Solving nonlinear systems of equations is one of the most basic problems in applied mathematics. It is well known that a solution of the system is multiple or singular when the Jacobian is rank-deficient at the solution. In other words, this particular solution is a converging point of a few, a few dozens or even hundreds of solutions. Finding the characteristics of the solution such as multiplicity is a natural component of nonlinear system solving. However, the basic concept of multiplicity of the solution and its identification are largely unknown to numerical analysts and scientific computing practitioners. None of the numerical analysis textbooks in all levels we have seen elaborates the multiplicity beyond the single univariate equations. We implemented a Matlab software, MULTIPLICITY, of a numerical algorithm for computing the multiplicity structure of a nonlinear system at an isolated zero is presented. The software incorporates a newly developed equation-by-equation strategy that significantly improves the efficiency of the closedness subspace algorithm and substantially reduces the storage requirement. As a result, the algorithm and software can handle much larger nonlinear systems and higher multiplicities than their predecessors.
Related references:
1. W. Hao, Andrew J. Sommese and Zhonggang Zeng, An algorithm and software for computing multiplicity structures at zeros of nonlinear systems, ACM Transactions on Mathematical Software, to appear, (2012). [pdf]
2. W. Hao, Andrew J. Sommese and Zhonggang Zeng, Matlab software for computing multiplicity structure of nonlinear systems, Available at here.
4. Parallel schemes for parabolic equations

Domain decomposition is a powerful tool for devising parallel methods to solve time-dependent partial differential equations. There is rich literature on domain decomposition finite difference methods for solving parabolic equations on parallel computers. For the non-overlapping domain decomposition methods, the explicit nature of the calculation at the interface of sub-domain leads some domain decomposition schemes to be conditionally stable, which implies that they have to suffer from temporal step-size restrictions. We developed parallel iterative method with unconditional stability and parallel finite difference scheme with high order accuracy as well as unconditional stability for solving parabolic equation. These schemes are based on domain decomposition method. The numerical stability and convergence are derived in the $$H^1$$ norm in one dimensional case. Numerical results of two and three dimensions examine the stability, accuracy, and parallelism of the procedure.
Related references:
1. W. Hao and S. Zhu, A domain decomposition finite difference scheme with third-order accuracy and unconditional stability, submitted, (2012). [pdf]
2. W. Hao and S. Zhu, Parallel iterative methods for parabolic equations, International Journal of Computer Mathematics, Volume 86, Issue 3, pp. 431-440, (2009). [pdf]

Other interdisciplinary areas

I am also interested other interdisciplinary areas which numerical methods can be applied such as, numerical optimization on factorial design in Statistics; application of numerical algebraic geometry in the dorsal-ventral patterning of a zebrafish; mathematical modeling in gait control of the biped robot.

Gait control with data computed from a mathematical model we built up

Related references:
1. F. Sun, M.-Q. Liu and W. Hao, An algorithmic approach to finding factorial designs with generalized minimum aberration, Journal of Complexity, Volume 25, Issue 1, pp. 75-84, (2009). [pdf]
2. W. Hao, J. D. Hauenstein, B. Hu, Y. Liu, A. J. Sommese and Y.-T. Zhang, Multiple stable steady states of a reaction-diffusion model on zebrafish dorsal-ventral patterning, Discrete and Continuous Dynamical Systems - Series S, Volume 4, Number 6, pp. 1413-1428, (2011). [pdf]
3. W. Hao and C. Liu, A high degree freedom mathematical model for the biped robot with the gait control, Mechanic in Engineering, Volume 28, No.6, pp. 69-72, (2006).