Applied and Computational Mathematics Seminar

Multi-agent dynamical systems models are used widely to model the movement of groups of animals. Several species of animal, including penguins and rodent pups, have been observed to huddle for warmth and continuously cycle who is in the center of the huddle. In this talk I present a novel model of individuals that exchange warmth and move based on their own body temperature. This model is built on the physics of heat exchange and well-studied attraction/repulsion dynamics, with some new mathematical results on consequences of parameter choices in attraction/repulsion potentials. I will showcase how the individual dynamics lead to emergent macroscopic movement patterns that mimic observed behavior. Simulations show how different choices of parameters lead to interesting different behaviors that raise questions on fairness and cooperative behavior in animals.

In 2014, the Spotted Lanternfly (SLF) was introduced to Pennsylvania, and with its ability to severely compromise lumber, grape, and crop production, it has become a species of great concern. In this work, we developed a Site-Facilitated spread model motivated by the behavior of the SLF.  A calibrated model prototype is showcased with vineyards in mind, capturing how SLF move between hosts (sites) in the landscapes with different site geometries over the course of a season. We demonstrate the implications of this model on existing vineyards using data for host locations in and around real vineyards. We further look at the macroscopic implications of the Site-Facilitated spread model that has the potential to forecast the spread over large distances and long times while bridging the gap between observed agent behavior and macroscopic population-level spread.

The aggregated and crystalline phases of π-conjugated molecules and polymers continue to receive widespread attention as semiconducting materials for field effect transistors, light emitting diodes and solar cells. However, despite the more than five decades of intensive experimental and theoretical research following Kasha's pioneering work on H- and J-aggregates[1] there remain important questions regarding the nature of the photo-excitations in molecular assemblies and how their spectral signatures are related to crystal packing and morphology. In this talk we explore the absorption and emission of light in molecular aggregates using simplified Hamiltonians which account for electronic coupling between molecules as well as the coupling between electronic and vibrational degrees of freedom. Once a suitable basis set is chosen, such Hamiltonians can be represented as matrices which are readily analyzed numerically using linear algebraic techniques. A range of novel spectral signatures are presented which are directly related to the way molecules pack to form aggregates. Applications are made to a range of p-conjugated chromophores which have been intensively investigated for opto-electronic applications.[2]

[1] M. Kasha, Energy Transfer Mechanisms and the Molecular Exciton Model for Molecular Aggregates, Radiation Research 20 (1963) 55-70.

[2] N.J. Hestand, F.C. Spano, Expanded Theory of H- and J- Molecular Aggregates: The Effects of Vibronic Coupling and Intermolecular Charge Transfer, Chem. Rev. 118 (2018) 7069–7163.

Optimal damping seeks to determine a vector of damping coefficients that maximizes the decay rate of a mechanical system's response. This problem can be formulated as the minimization of the trace of the solution to a Lyapunov equation whose coefficient matrix depends on the damping parameters, which must satisfy nonnegativity constraints for physical relevance. We propose two fast algorithms for solving this resulting nonlinear optimization problem under nonnegativity constraints and demonstrate their computational efficiency.
This is joint work with Professor Qingna Li at the Beijing Institute of Technology.
 

The beautiful displays exhibited by fish schools and bird flocks have long fascinated scientists, but the role of their complex behavior remains largely unknown. In particular, the influence of hydrodynamic interactions on schooling and flocking has been the subject of debate in the scientific literature. I will present a model for flapping wings that interact hydrodynamically in an inviscid fluid, wherein each wing is represented as a plate that executes a prescribed time-periodic kinematics. The model generalizes and extends thin-airfoil theory by assuming that the flapping amplitude is small, and permits consideration of multiple wings through the use of conformal maps and multiply-connected function theory. We find that the model predictions agree well with experimental data on freely-translating, flapping wings in a water tank. The results are then used to motivate a reduced-order model for the temporally nonlocal interactions between schooling wings, which consists of a system of nonlinear delay-differential equations. These equations are used to derive a continuum PDE theory, which we find supports traveling wave solutions. Generally, our results indicate how hydrodynamics may mediate schooling and flocking behavior in biological contexts.

In many areas of scientific computing, statistical analysis, and machine learning, the ability to efficiently and accurately handle large scale and high dimensional data is a rapidly growing necessity. Many powerful techniques have been developed to approach this task, one being Tensor-Train (TT) and variants such as Tensor-Train Cross (TT-Cross). TT-Cross provides a framework suited for working with extremely large datasets while maintaining low memory and computational complexity. In this talk, I will introduce the foundations at the matrix level and the base TT-Cross algorithm. I will then discuss a post-processing oversampling algorithm designed to enhance approximation accuracy. Numerical experiments demonstrate improved computational efficiency and systematic error reduction via oversampling.

We extend results known for the randomized (point and block) Gauss-Seidel and the Gauss-Southwell methods for the case of a Hermitian and positive definite matrix to certain classes of non-Hermitian matrices. We consider cases with overlapping variables (as in Domain Decomposition). We obtain convergence results for a whole range of parameters describing the probabilities in the randomized method or the greedy choice strategy in the Gauss-Southwell-type methods. We identify those choices which make our convergence bounds best possible.
One result is that the best convergence bounds that we obtain for the expected values in the randomized algorithm are as good as the best for the deterministic, but more costly algorithms of Gauss-Southwell type. We use these new results to show a provable convergence rate for asynchronous iterations.

(Joint work with Andreas Frommer, Wuppertal)

Can one learn a partial differential equation (PDE) from only input-output function pairs? If so, how many are needed? We provide theoretical guarantees on the number of input-output training pairs required to learn a 3D uniformly elliptic PDE. In particular, we exploit randomized numerical linear algebra and PDE theory to derive a provably data-efficient algorithm that recovers the corresponding Green's function and achieves an exponential convergence rate of the error with respect to the size of the training dataset with an exceptionally high probability of success. This work provides a theoretical explanation for the observed strong performance of recent deep learning techniques in PDE learning, even when there is limited data availability. We also discuss the importance of access to the adjoint operator in this problem, which relates closely to the role of transpose-matrix-vector products in sketching algorithms.