Research 

I am interested in applying algorithm into different areas.  Here is a list of projects that I am working on.


We've developed new algorithms for option pricing based on the binomial, trinomial, and Black-Scholes-Merton models. These algorithms are designed for parallel computing, resulting in substantial performance and energy savings. In practical terms, using our method in a binomial model scenario on a multi-core system, our approach is significantly faster than existing methods. This speed advantage becomes more pronounced with larger datasets. Remarkably, our algorithms operate more efficiently, consuming less time and energy than traditional methods. Beyond its application in finance, our approach, which employs the Fast Fourier Transform technique, can be beneficial in various other fields, highlighting its versatility and efficiency.


Polynomial zonotopes, a non-convex set representation, have a wide range of applications from real-time motion planning and control in robotics, to reachability analysis of nonlinear systems and safety shielding in reinforcement learning. Despite this widespread use, a frequently overlooked difficulty associated with polynomial zonotopes is intersection checking. Determining whether the reachable set, represented as a polynomial zonotope, intersects an unsafe set is not straightforward. In fact, we show that this fundamental operation is NP-hard, even for a simple class of polynomial zonotopes.

The standard method for intersection checking with polynomial zonotopes is a two-part algorithm that overapproximates a polynomial zonotope with a regular zonotope and then, if the overapproximation error is deemed too large, splits the set and recursively tries again. Beyond the possible need for a large number of splits, we identify two sources of concern related to this algorithm: (1) overapproximating a polynomial zonotope with a zonotope has unbounded error, and (2) after splitting a polynomial zonotope, the overapproximation error can actually increase. Taken together, this implies there may be a possibility that the algorithm does not always terminate. We perform a rigorous analysis of the method and detail sufficient conditions for the union of overapproximations to provably converge to the original polynomial zonotope.


We investigate a family of approximate multi-step proximal point methods, accelerated by implicit linear discretizations of gradient flow. The resulting methods are multi-step proximal point methods, with  similar computational cost in each update as the proximal point method. We explore several optimization methods where applying an approximate  multistep  proximal points method results in improved convergence behavior. We argue that this is the result of the lowering of truncation error in approximating gradient flow.