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Research

1. Mathematical modeling to enhance and extend inter-disciplinary collaboration

Mathematics provides useful tools for representing various phenomena in our life and the universe. Mathematical models help elucidate the phenomena, predict target behaviors, and contribute to optimization/control of the targeted systems. Mathematical models take different forms depending on the targeted problems such as systems of partial differential equations, descriptors for machine learning, statistical models, and graphical network representations. We have been collaborating with scientists and engineers in different scientific fields to construct mathematical models appropriate to their problems.

2. Computational analysis for coupled problems

I am interested in coupled problems represented by different physics in different time/space scales such as fluid-structure interactions, fluid-particle interactions, and fluid-fluid interactions. They are formulated using different equations, but usually with software we already have. Careful treatments, both mathematically and computationally, are necessary to solve such coupled problems, but coupled systems reveal interesting and beautiful characteristics to us.

3. Techniques for numerical simulations, scientific visualization, and machine learning

To obtain practical solutions of mathematical models, numerical simulation techniques have been developed extensively during recent decades. I am also interested in developing high-quality, reliable, robust, and not-so-expensive simulation techniques to tackle real-world problems. The results of computational simulations are huge rows of numerical values. Therefore, visualization techniques have been important tools for use with computational simulations. While applying several visualization techniques in our daily research work, we are finding ways of producing comprehensible explanations of our numerical results for our collaborators and for our own deeper understanding. Machine learning is also a promising tool that has arisen in recent years. We are collaborating with skilled medical doctors and skilled engineers to uncover the algorithms hidden in their experiences.