Research Statement

Maksym Pryporov

My research area is Applied Mathematics with research interests in analysis of PDEs. I am working on asymptotic methods for recovery of the high frequency waves governed by linear partial differential equations such as semi-classical Schrödinger equations with periodic potentials and symmetric hyperbolic systems.  Those types of problems arise in solid state physics, quantum chemistry, acoustic waves, seismology and other areas. I am dealing with equations with highly oscillatory initial data which produce high frequency waves. The direct computational methods for such problems are prohibitively costly because of the high frequency parameter.  In order to deal with this challenge, several asymptotic approaches were proposed in the past.  The main focus of my research is on the error estimates for the asymptotic solution based on the Gaussian beam superposition method.  Gaussian beams are asymptotic solutions concentrated on trajectories for Hamiltonian and they remain valid beyond “caustics”. At present, there is considerable interest in using superposition of beams to compute high frequency waves. The accuracy of Gaussian beam superposition was an open problem for several decades. In the past few years some progress on estimates of the error was done for hyperbolic and Schrödinger equations.  One of the first results was obtained by Tanushev for the initial error in 2008. Liu and Ralston gave rigorous convergence rates in terms of small wave length for the acoustic wave equation in the scaled energy norm and for the Schrödinger equations in the L^2 norm in 2009-2010. In 2012, Liu, Runborg and Tanushev further obtained sharp error estimates for a large class of linear wave equations.  My research projects are built upon those advances.

My current research is focused on two projects:

1.    Error estimates of the Bloch band-based Gaussian beam superposition for the Schrödinger equation;

2.    The Gaussian beam method for linear symmetric hyperbolic systems.

As for the Schrödinger equation, the main challenge is the band structure of the solution, which leads to the theory of Bloch waves. The problem is studied in the case of the separated energy bands, which allows constructing smooth asymptotic solutions. During this work, the following results were obtained:

-       Gaussian beam superposition in the two-scale formulation;

-       Initial error estimate;

-       Evolution error estimate;

-       Validation of the two-scale results for the original problem.

Limitation:

-       In our model, we consider the finite number of energy bands.

The two-scale approach is a powerful tool in dealing with high frequency problems. It is a simplest case of a problem with many high frequency scales. Thus, it would be interesting for me to investigate more general high frequency problems.  The initial error and the evolution error estimates give a total error estimate. The time is considered to be finite. Since Bloch functions form an orthonormal basis in L^2 space, the initial data can be represented by this basis and it can be approximated well by the finite number of terms, which makes our results applicable to practical problems.  As for the infinite representation of the initial data, a deeper understanding of Bloch bands and energy bands is needed. In particular, whether there exist uniform bounds for their derivatives.

In the second project, I study the symmetric hyperbolic systems with highly oscillatory initial data. For this project, there are the following results:

-       Asymptotic solution via Gaussian beams;

-       Initial error estimate;

-       Evolution error estimate;

-       Found a fix for the stationary phase case.

The construction of the asymptotic approach using Gaussian beams for hyperbolic systems is a new result itself. The existence of the orthogonal basis formed by eigenvectors of the symmetric hyperbolic system is a key condition which makes this construction possible. Several ideas developed in the previous project are used for the proofs of the error bounds.

I am eager to work on more research projects in the next few months and years.  The most preferred goal for me right now is to develop a deeper theory for the hyperbolic systems. So far I considered a relatively simple setting and my feeling is that there is a lot of room to make it more general. It can be a system with non-constant coefficients, with non-zero right hand side, high frequency coefficients and other. The more advanced goal is to understand more realistic models, for example, Maxwell equations; so far the divergence free condition seems to be an obstacle. I am also familiar with the “frozen” Gaussian beam method and I hope to make contribution to this theory as well.

The second priority for my research is to continue to study Bloch bands and energy bands. This topic is important for applications in the quantum chemistry and the study of solids with periodic structure. For instance, in experiments, the energy bands are crossing or touching each other, which in theory affects analyticity properties of band functions. Since then, there is a need to study this topic in greater depth in order to make the Gaussian beam theory more applicable to real processes. Problems involving almost periodic and quasi periodic structures or materials with mixed structure are also important and are even more practical, this is another possible extension for my research in future.  

Although my research was on analysis so far, it is not restricted to it. I took several graduate level courses in computational mathematics and I would like to work in the numerical study of Gaussian beams as well. Another possible direction of my future research is to study the high frequency boundary value problems instead of initial value problems as I did so far, since this has a lot of applications in geophysics. In particular, the decomposition of the multi-scale high frequency data from the boundary into Gaussian beams, which are used to construct an asymptotic solution.

In the long run, I am interested in the study of nonlinear high frequency problems, for example, nonlinear Schrödinger equations. Since the possibility to construct the superposition of Gaussian beams is based on the linearity of the equation, some new ideas need to be developed.

References

1.      H. Liu and J. Ralston. Recovery of high frequency wave fields for the acoustic wave equation. Multiscale Model. Simul., 8(2) (2009), 428-444.

2.      H. Liu and J. Ralston. Recovery of high frequency wave fields from phase space based measurements. Multiscale Model. Simul., 8(2) (2010) 622-644.

3.      H. Liu, O. Runborg and N. Tanushev. Error estimates for Gaussian beam superpositions. Math. Comp. In press. (2012)

4.      N.M. Tanushev. Superpositions and higher order Gaussian beams. Comm. In Math. Sci., 6(2) (2008), 449-475.