- Wavelet Analysis
- Signal Processing
- Wavelets Methods for Solving Partial Differential Equations
- Tensor Computation

- Project Title: A dynamically adaptive wavelet algorithm for solution of evolution equations with localized structures

PI: Ratikanta Behera

Status: 2018-2021. - Project Title: Tensor-valued wavelet analysis and application to solution of differential equations

PI:Ratikanta Behera

Status: 2018-2021.

- Wavelets on Signal Processing My research is concerned with the development of adaptive wavelet based method for multicomponent signals analysis. Developing a truly adaptive method for signal analysis is important for the understanding of many natural phenomena. Traditional signal analysis methods, such as the Fourier transform, use pre-determined basis. They provide an effective tool to process linear and stationary data. However, there are still some limitations in applying these methods to analyze nonlinear and nonstationary data. The recently developed wavelet based synchrosqueezing technique have opened a new path for time-frequency analysis. Synchrosqueezing by definition is a highly nonlinear operator along with special properties: (1) it is adaptive to the given signal; (2) the signal can be reconstructed from the reallocated coefficients. This represents a significant improvement over the traditional signal analysis method.
- Wavelets on PDEs My research is concerned with the development of adaptive wavelet based methods for the solution of Partial Differential Equations (PDEs) and its applications on physical problems. Since this method provide a simple, efficient and automatic way to adapt computational refinements to local demands of the solution, i.e., fine resolution only in areas where it is needed and coarser resolution elsewhere. Which is necessary for those problems having localized structures or sharp transitions occur intermittently anywhere on the computational domain or change their locations and scales in space and time. The numerical solution of such problems on uniform grids is computational expensive, since high-resolution computations are required only in regions where sharp transitions occur. Hence the goal is ultimately to improve the computational efficiency and the computational grid should adapt dynamically in time to reflect local changes in the solution.
- Tensor Computation The generalized inverse of a tensor is important in analysis because it provides an extension of the concept of an inverse which applies to all tensors. Further, it is a great tool in solving linearly dependent and unbalanced system of linear equations.