Research Overview.

My research is concerned with the use of variational techniques for analyzing constrained evolutionary equations. Focus is put on the solvability of these problems, their properties and applications, as well as on the construction of approximation schemes for treating their numerical solutions.
I am recently interested in shape analysis applications, PDE on manifolds, intefacial motion, free boundary problems, and image processing.

Interfacial motions (thresholding algorithms).

We have introduced a method for computing interfacial motions governed by curvature dependent acceleration.
Our method is a thresholding algorithm of the MBO-type (a.k.a., BMO) which, instead of utilizing a diffusion process,
thresholds evolution by the wave equation to obtain the desired interfacial dynamics.
We also developed the numerical methods and are investigating related multiphase volume-preserving motions.
Curvature dependent accelerations (oscillating interfaces) contrast mean curvature flow, and it is
natural to search for a thresholding algorithm to approximate their dynamics. The target of this research
is to construct such a method. From the point of view of applications, since interfaces in nature are
often observed to oscillate (e.g., elastic membranes, soap bubbles, and liquid droplets), we remark this class of motions includes interesting physical phenomena.
Click here to view a result of using our method to compute multiphase volume-preserving mean curvature flow. The hyperbolic motions can be compared here). (Note the oscillation of each interface.)

Droplet motion (hyperbolic free boundary problems).

The main target of analysis in this research is a constrained hyperbolic free boundary problem (FBP) for describing the motion of liquid droplets and bubbles. The target problem is a volume-constrained nonlinear wave equation that corresponds to the Euler-Lagrange equation of an appropriate action integral, and we are approaching its analysis and computations by the method of minimizing movements (which is a variational technique that utilizes a discretization of time, see below). In particular, by constructing a sequence of functions in such a way that each element minimizes a prescribed functional, we can show that, upon approximating the minimal surface operator by the Laplacian, this approach allows us to obtain approximate weak solutions and (uniform) energy estimates in arbitrary dimensions. We are also able to show the uniform convergence of the minimizing movement in the case of a one-dimensional domain (which, in turn, yields the existence of a corresponding weak solution). We have developed the corresponding numerical methods for the minimizing movements. See here for a result of its application to the case of droplet motion.

Minimizing movements (MM). Also known as discrete Morse flow（離散勾配流法).

A main topic of my research is the use of MM for both analyzing and computing solutions to differential equations. The target problems are usually nonlinear (constrained) PDE, often of the free boundary type. I have written a short introduction to this subject (it can be downloaded
here).

Image processing.

Here is an application of pde-based image processing for removing noise created by nuclear radiation. The movie is from (here). In the original movie, the noise that we want to remove are the cylinder-like artifacts. A regularized level set equation, which takes values on the unit sphere, was used in processing the images. I have not put much energy into improving the result, but the point is that mathematics allows us to design ways of removing the noise. Moreover, the compuations are really easy to do. A partial differential equation exists for removing artifacts with high curvature (such as the ends of the cylinders), so we create a method for its approximation. One gives the initial condition(s), and then computes.

Undergraduate research at the University of Washington.

(1) I recieved my first Bachelor's of Science in Mathematics, from the University of Washington. ワシントン大学数学科.

(2) I recieved my second Bachelor's of Science in Applied Mathematics, from the University of Washington. ワシントン大学応用数学科.

(3) Under the guidance of Jim Morrow as a student of his honors class, I participated in the world's largest mathematical modeling contest, the COMAP MCM. Our team was selected in the top 10% and won the Meritorious Award in 2004.

(4) I also participated in the following COMAP MCM and our team won an Honorable mention, in 2005.

(5) I also did research with David Catling, Stephen Wood, and Conway Leovy, at the Department of Atmospheric Sciences and the Department of Earth and Space Sciences. I did analysis of NASA's Mars data, which was a great experience and motivation for doing research. This research got me interested in doing applied mathematics through image processing, which is a topic that I am still interested in. 火星気象科学研究グループ.