The focus of my research is on nonlinear scientific computing and applications in modeling biomedical diseases. In particular, I develop numerical methods based on homotopy continuation setup for solving nonlinear problems and study problems in cardiovascular disease, fibrosis, tumor growth, and Alzheimer’s disease.
Computing multiple solutions of nonlinear PDEs
Nonlinear differential equations may have many (even infinitely many) solutions with complex structures. Computing multiple solutions and their structure has attracted the attention of many mathematicians, physicists, and engineers. So far, the understanding of multiple solutions is still quite limited: analytic solutions are too difficult to obtain. Due to their strong nonlinearity, multiplicity, and unstable nature, such solutions are extremely difficult to compute and very elusive to traditional numerical methods. Thus the development of efficient and reliable numerical methods to compute the multiple solutions of the nonlinear problem is very interesting to both research and applications. I have developed several numerical methods for computing multiple solutions of nonlinear PDEs:bootstrapping method coupled domain decomposition algorithm with homotopy method;
two-level spectral method;
a homotopy method with adaptive basis selection.
I have applied these methods to solve spatial pattern formation in biology for both one and two dimensional cases.
Numerical methods for solving data-driven nonlinear systems
Nonlinear systems are omnipresent and play crucial roles in data-driven modeling and machine learning. My research is to develop computational methods, with efficient numerical algorithms and analytical guidance, to solve nonlinear systems that include ill-conditioned or parameterized systems arising from real-world applications. I have developed several efficient numerical methods for solving data-driven nonlinear systems such as
an Equation-By-Equation method for solving the multidimensional moment constrained Maximum Entropy problem;
a homotopy method for parameter estimation;
a homotopy training algorithm for fully connected neural networks,
and gradient descent method for solving nonlinear systems.
Nonlinear modeling for public health
I have applied efficient computational tools to clinical intervention such as identifying effective clinical biomarkers to reflect the progression of atherosclerotic aneurysms,
providing long-term prediction on aneurysms growth, etc. I also initiate a computational approach to predict personalized Alzheimer's disease progressions based on the Alzheimer's disease
pathophysiology and patients' clinical data. Moreover, machine learning techniques are also incorporated to predict the clinical outcomes and to simulate the nonlinear models.