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Colloquium April 14

Date: Friday April 14

Time: 16:00 to 17:00

Place: 455 McBryde (Commons Room)

Speaker: Dongwoo Sheen of Seoul National University

Title: Method of Laplace transformation to solve backward parabolic problems and its applications

Abstract

In this talk we will discuss several mathematical issues in the application of Laplace transformation to solve backward parabolic equations. Backward parabolic equations are well-known ill-posed problems in the sense a small perturbation in the initial data may result in a very large change in the solution at a later time. Interesting application areas of backward parabolic problems include image processing, mathematical finance, and several physics problems related with heat conduction and diffusion. Due to its ill-posedness devising stable numerical methods is very important.

In the first part of this talk, we will recall the nature of ill-posedness in backward parabolic problems and some numerical methods to resolve this difficulty. We will then introduce a parallel method for time-discretization of backward parabolic problems. The problem is reformulated to a set of Helmholtz-type problems with a parameter on a suitably chosen contour in the complex plane. After solving the resulting elliptic equations, which can be solved in parallel, we obtain a regularized solution with high frequency terms cut off by the inverse Laplace transforms without requiring the knowledge of the eigenfunctions of the differential operator. Since the regularized solution is obtained without artificial perturbation and high frequency components of the noise are suppressed, the quality of the solution is improved significantly compared to those obtained by the other methods. Two different numerical inversions of Laplace transforms, with arbitrary high order of accuracy, and the spectral accuracy, respectively, are used. Lastly we will present some numerical examples, especially from image processing.

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