Research Outlines of Bin Han

My research interests concentrate on roughly four major areas:
Book: Bin Han, Framelets and Wavelets: Algorithms, Analysis, and Applications, Applied and Numerical Harmonic Analysis, Birkhauser/Springer, Cham, (2017), 724 pages.
I. Numerical PDEs and computing We are interested in developing numerical methods for effectively solving various linear and nonlinear PDEs using finite difference methods, wavelet-base methods, and Fourier-based methods. Currently, we are working on Helmholtz equations, elliptic interface problems, and Burgers' equations etc.
II. Framelets (=Wavelet frames) with applications Wavelet frames are of particular interest for their applications in signal and image processing. In comparison with classical wavelets, framelets enjoy the desired properties of redundancy for robustness against noise and perturbation, directionality for capturing singularities, flexibility for relatively easy construction, and the near-translation invariance for noise removal. Except image compression and wavelet-based methods for numerical PDEs, one often prefers framelets instead of classical wavelets in many other applications, in particular in image processing and data sciences.
III. Wavelet theory Classical wavelet theory deals with orthonormal wavelets, biorthogonal wavelets, and Riesz wavelets. Such wavelets are of particular interest in image/data compression and in wavelet-methods in numerical PDEs. We are interested in studying various properties of wavelet and framelet systems. Such properties includes characterization of function spaces by wavelets, approximation property of shit-invariant spaces, stability and linear independence of refinable vector functions. We are also interested in constructing various types of wavelets including multivariate biorthogonal wavelets and Riesz wavelets.
IV. Refinable functions and subdivision schemes Wavelets and framelets are often derived from refinable functions and most properties of wavelets and framelets are determined that of a refinable function. The properties of a refinable function are often studied through a subdivision scheme and its associated cascade algorithm in the function setting. A subdivision scheme is a fast local averaging algorithm to generate smooth curves and surface. In particular, a subdivision scheme can be employed in wavelet analysis to compute and analyze refinable functions and in the fast wavelet/framelet transform to reconstruct the data. We are interested in the convergence and smoothness of various subdivision schemes
V. Other research topics We also worked on compressed sensing, shearlets, sampling theory etc,