Anderson localization in nonlocal models; Analysis Seminar; Department of Mathematics and Statistics, Texas Tech University; Lubbock, Texas (February 2019) (invited).
Abstract: It is well known that many physical systems will exhibit localized energy states in the presence of certain environmental disturbances. Anderson localization has attracted attention from the physics, mathematical physics, numerical analysis, and pure analysis communities, but in this talk we will provide a more operator theoretical approach. In this talk, we will provide two new directions of study of the Anderson localization problem. First, we will extend the problem to consider nonlocal operators on discrete graphs.
Next, we will develop a novel method of studying the localization properties of these nonlocal operators via the consideration of the spectrum of the operators. This approach allows for the development of surprising results that allow for the improvement of many existing results. This series of talks will include a review of the pertinent concepts from analysis, making the talk accessible to all graduate students (even those who do not study analysis).
Operator splitting methods for approximating singular nonlinear differential equations; Numerical Analysis Seminar; Department of Mathematical Sciences, University of Delaware; Newark, Delaware (November 2018) (invited).
Abstract: Operator splitting techniques were originally introduced in an effort to save computational costs in numerical simulations. Classically, such methods were restricted to dimensional splitting of evolution operators. However, these methods have since been extended to allow for splitting of problems involving nonlinear operators which evolve on vastly different time scales. In this talk I will introduce the notion of nonlinear operator splitting and rigorously justify the approach by considering some techniques from Lie group theory. This is a nonstandard presentation that should also be accessible to graduate students. The second half of the talk will provide results concerning two very interesting applications of operator splitting techniques: nonlinear stochastic problems and singular combustion problems. The former problems have traditionally been plagued with low-order techniques with restrictive regularity conditions, while the latter have the need for strongly adaptive methods which recover important qualitative properties. We will discuss in detail how operator splitting provides solutions to these issues, while also being straightforward to implement.
Operator splitting methods for approximating singular nonlinear differential equations; Department Colloquium; Department of Mathematics, Baylor University; Waco, Texas (November 2018) (invited).
Abstract: In this talk we introduce the notion of operator splitting for nonlinear equations. We formulate the approach in the language of Magnus expansions in abstract spaces, allowing us to combine the language of semigroups with nonlinear operators. The focus of the talk will be extending these techniques to approximating solutions of stochastic differential equations in Hilbert spaces. These approximation techniques allow for the development of numerical methods which are of arbitrary order, yet have lower regularity conditions when compared to many existing methods. Moreover, the methods may easily be generalized to differential problems posed on smooth manifolds. If time permits, we will discus how operator splitting methods may be employed to construct approximations which respect the underlying Lie group structure of the problem at hand. There will be a thorough introduction to the considered methods and the talk will be accessible to interested graduate students.
Numerical integration techniques on manifolds and their Hopf algebraic structure; Geometry Seminar; Department of Mathematics and Statistics, Texas Tech University; Lubbock, Texas (October 2018) (invited).
Abstract: Lie group integrators are a class of numerical integration methods which approximate the solution to differential equations which preserve the underlying geometric structure of the true solution. In this talk, we consider a commutative graded Hopf algebraic structure arising in the order theory and backward error analysis of such Lie group methods. We will consider recursive and direct formulae for the coproduct and antipode, while emphasizing the connection to the Hopf algebra of classical Butcher theory and the Hopf algebra structure of the shuffle algebra. The talk will provide the necessary background to make it accessible to graduate students.
Analysis of exponential-type integration method for nonlocal diffusion problems; SIAM Annual Meeting; Special Session; Eugene, Oregon (June 2018) (invited).
Abstract: In recent years numerous physically relevant phenomena have been shown to demonstrate a non-standard diffusive process known as anomalous diffusion. Such models are mathematically interesting due to the non-local nature of the involved operators, such as the fractional Laplacian. Despite the growing interest in such problems, the existing numerical methods are still plagued by reduced convergence rates and inefficient implementations. This talk will focus on approximating an abstract Bessel-type equation which is an extension of the non-local problem of interest. From this extension problem, an efficient method with desirable convergence properties will be developed and analyzed. Numerical examples will be provided to demonstrate the results.
Lie-Butcher series from an algebraic geometry point of view; Geometry Seminar; Department of Mathematics and Statistics, Texas Tech University; Lubbock, Texas (April 2018) (invited).
Approximating the fractional Laplace equation via operator theoretical methods; West Texas Applied Math Symposium; Department of Mathematics and Statistics, Texas Tech University; Lubbock, Texas (April 2018) (invited).
Abstract: In recent years numerous physically relevant phenomena have
been shown to demonstrate a non-standard diffusive process known
as anomalous diffusion. Such models are mathematically interesting
due to the non-local nature of the involved operators, such as the
fractional Laplacian. Despite the growing interest in such problems,
the existing numerical methods are still plagued by reduced conver-
gence rates and inefficient implementations. This talk will focus on
approximating an abstract extension problem which is equivalent
to the non-local problem of interest. From this extension problem,
an efficient method with desirable convergence properties will be
developed and analyzed. Numerical examples will be provided to
demonstrate the results
An introduction to geometric numerical integration; Geometry Seminar; Department of Mathematics and Statistics, Texas Tech University; Lubbock, Texas (March 2018) (invited).
Operator splitting methods for approximating differential equations; Junior Scholar Symposium; Department of Mathematics and Statistics, Texas Tech University; Lubbock, Texas (February 2018) (invited).
An operator theoretical approach to nonlocal differential equations; Analysis Seminar; Department of Mathematics and Statistics, Texas Tech University; Lubbock, Texas (November 2017) (invited).
Abstract: Nonlocal differential equations are receiving increasing attention due to their ability to accurately model many physically relevant phenomena such as anomalous diffusion and non-Fickian transport. The resulting models exhibit difficulties not seen in standard local models and require careful treatment and attention. In an effort to study problems more efficiently, we consider an operator theoretical approach to solving such problems. This method attempts to mirror the classical approach of solving differential equations through semigroup theory, however, nonlocal problems will have solutions generated by generalized Mittag-Leffler functions. Properties of these operators will be discussed and compared with classical semigroup operators. The theory will be well-motivated through an explicit example of interest. Proposed future considerations will also be briefly mentioned in order to give an idea of research potential.
Operator splitting and Lie group methods for geometric integration; Seminar in Applied Mathematics; Department of Mathematics and Statistics, Texas Tech University; Lubbock, Texas (November 2017) (invited).
Abstract: Geometric integration is the discipline concerned with the discretization of differential equations while conserving exactly their invariants. The motivation for developing structure-preserving algorithms for certain classes of differential equations originates from diverse areas such as astronomy, molecular dynamics, mechanics, control theory, theoretical physics, and numerical analysis. If the equations of interest evolve on Lie groups, then geometric integration methods will guarantee that all approximations remain in the appropriate Lie group. In this talk we will consider Lie group methods for designing structure-preserving schemes with special emphasis on operator splitting methods. Operator splitting in a Lie group setting is primarily concerned with effectively approximating the exponential map and guaranteeing that such approximations map elements from the appropriate Lie algebra to the corresponding Lie group. The talk will consider concrete examples throughout, removing the need for a deep understanding of Lie group theory, and will conclude with several examples of how such methods may be applied to problems of interest.
An exploration of quenching-combustion via globalized fractional models; SIAM Annual Meeting, Special Session; Pittsburgh, Pennsylvania (July 2017) (invited).
Solving degenerate stochastic Kawarada equations via adaptive operator splitting methods; University of Central Arkansas; Conway, Arkansas (January 2017) (invited).
An approach to the numerical solution of multidimensional stochastic Kawarada equations via adaptive operator splitting; Joint Mathematics Meeting; Atlanta, Georgia (January 2017).
Abstract: This talk concerns the numerical solution of multidimensional nonlinear Kawarada equations. The stochastically influenced degenerate reaction-diffusion equations exhibit strong singularities and play an important role in numerous industrial applications. Moving mesh strategies and operator splitting are utilized throughout the approach to yield favorable adaptive grids in both space and time. Highly efficient and effective nonuniform difference schemes are developed. It is shown that the numerical solution acquired not only approximates the theoretical solution satisfactorily, but also preserves the required positivity, monotonicity and stability of the solution when proper constraints are satisfied. The
latter is particularly crucial to quenching-combustion simulations. Numerical experiments are given to illustrate and demonstrate our conclusions.
Using Matlab to solve nonlinear PDE; AMS Student Meeting; Baylor University; Waco, Texas (October 2016).
Using an adaptive Crank-Nicolson scheme to solve the degenerate stochastic Kawarada equation on nonuniform grids; SIAM Central States Section Meeting, Special Session; Little Rock, Arkansas (September 2016) (invited).
Positive and monotone solutions to quenching differential equations; Differential Equations Seminar; Baylor University; Waco, Texas (April 2016, 6 lectures).
A semi-adaptive LOD method for solving three-dimensional degenerate Kawarada equations; AMS Spring Southeastern Sectional Meeting; Athens, Georgia (March 2016).
A novel LOD method for solving degenerate Kawarada equations; CASPER Seminar; Waco, Texas (February (2016) (invited).
An exploration of exponential splitting; Joint Mathematics Meeting, Special Session; San Antonio, Texas (January 2015) (invited).
Abstract: Exponential splitting methods have been widely utilized for computing numerical solutions of partial differential equations. Different types of error estimates for the splitting procedures have been introduced and studied. In this talk, we will
present an improved, new global error analysis for key exponential splitting formulations based on the commutativity of matrix exponentials resulting from different exponential splitting formulas. Computational examples will be provided to illustrate our theoretical results and expectations.