# Applied/ACMS/absF18

## Contents

# ACMS Abstracts: Fall 2018

### Ting Zhou (Northeastern University)

*Nonparaxial near-nondiffracting accelerating optical beams*

We show that new families of accelerating and almost nondiffracting beams (solutions) for Maxwell’s equations can be constructed. These are complex geometrical optics (CGO) solutions to Maxwell’s equations with nonlinear limiting Carleman weights. They have the form of wave packets that propagate along circular trajectories while almost preserving a transverse intensity profile. We also show similar waves constructed using the approach combining CGO solutions and the Kelvin transform.

### Daniel Sanz-Alonso (University of Chicago)

*Discrete and Continuous Learning in Information and Geophysical Sciences*

The formulation of Bayesian inverse problems in function space has led to new theoretical and computational developments, providing improved understanding on regularization techniques and suggesting new scalable algorithms. The approach has found numerous applications throughout the geophysical and medical sciences, where interest often lies in recovering an unknown field defined on a physical domain. Learning problems in the information sciences, in contrast, typically seek to recover functions defined on discrete point clouds. My talk will have two parts. In the first one, I will prove that in certain large data limit, discrete learning problems converge to a continuous one, thus allowing to transfer scalable Markov chain Monte Carlo methodology developed in the geophysical sciences to novel applications in the information sciences. In the second part I will introduce a fully Bayesian, data-driven methodology to discretize complex forward models with the specific goal of solving inverse problems. This methodology has the potential of producing cheap surrogates that still allow for satisfactory input reconstruction.

### Matthew Dixon (Illinois Institute of Technology)

*"Quantum Equilibrium-Disequilibrium”: Asset Price Dynamics, Symmetry Breaking and Defaults as Dissipative Instantons*

We propose a simple non-equilibrium model of a financial market as an open system with a possible exchange of money with an outside world and market frictions (trade impacts) incorporated into asset price dynamics via a feedback mechanism. Using a linear market impact model, this produces a non-linear two-parametric extension of the classical Geometric Brownian Motion (GBM) model, that we call the ”Quantum Equilibrium-Disequilibrium” model. Our model gives rise to non-linear mean-reverting dynamics, broken scale invariance, and corporate defaults. In the simplest one-stock (1D) formulation, our parsimonious model has only one degree of freedom, yet calibrates to both equity returns and credit default swap spreads. Defaults and market crashes are associated with dissipative tunneling events, and correspond to instanton (saddle-point) solutions of the model. When market frictions and inflows/outflows of money are neglected altogether, ”classical” GBM scale-invariant dynamics with an exponential asset growth and without defaults are formally recovered from our model. However, we argue that this is only a formal mathematical limit, and in reality the GBM limit is non- analytic due to non-linear effects that produce both defaults and divergence of perturbation theory in a small market friction parameter.