**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.