Mathematics in Counterparty Credit Risk Quantification for Financial Institutions
Counterparty Credit Risk (CCR) faced by modern financial institutions, among many types of risks, is challenging to model, quantify and manage. Dealing
with CCR is a central task of risk management practice lines. In this presentation, we will talk through the typical framework of modeling CCR by major
banks with trading engagements. Meanwhile, the key steps and mathematical challenges are elaborated to the extent of understanding the big picture as
well as the fine details from modeling perspective. This includes but not limited to the stochastic equations to simulate a variouty of risk factor
dynamics and non-arbitrage financial derivative pricing theories. If time permits, we would brief through the AI/ML progress in CCR application.
In addition, we will touch base on relative industrial opportunities and potential job areas.
Department of Mathematics, Colorado State University
Pulmonary imaging with EIT: Reconstructions and results from patient data
Electrical impedance imaging is a non-ionzing dynamic imaging modality that is emerging for clinical use in pulmonary applications. The images are formed by numerically solving the inverse conductivity problem, also known as Calderon's problem, and plotting the reconstructed conductivity distribution. In this talk I will give an introduction to EIT and the associated inverse problem, and I will present two methods of solution: the D-bar method, which is a direct (non-iterative) method of reconstruction, and the classical least-squares approach with linearization. Results will be shown from patient data collected at Children's Hospital Colorado and from Anschutz Hospital.
Department of Mathematics and Statistics, Auburn University
Efficient time-stepping methods for nonlinear evolution problems
Domain decomposition (DD) methods provide a natural computational
framework for multiscale multiphysics problems and a powerful tool for
parallel numerical simulation of large-scale problems. As many physical and
engineering processes are described by evolution partial differential
equations, extensions of DD methods to dynamic systems (i.e., those changing
with time) have been a subject of great interest. Moreover, for applications
in which the time scales vary considerably across the whole domain due to
changes in the physical properties or in the spatial grid sizes, it is
critical and computationally efficient to design DD methods which allow the
use of different time step sizes in different subdomains.
In this talk, we first introduce mathematical concepts of DD methods for
evolution equations, then present DD-based time-stepping methods for the
rotating shallow water equations discretized on spatial meshes with variable
resolutions. Two different approaches will be considered: the first approach
is a fully explicit local time-stepping algorithm based on the Strong
Stability Preserving Runge-Kutta (SSP-RK) schemes, which allows different
time step sizes in different regions of the computational domain. The second
approach is the so-called Localized Exponential Time Differencing (LETD)
method, which makes possible the use of much larger time step sizes compared
to explicit schemes and avoids solving nonlinear systems as required in an
implicit time discretization. Numerical results on various test cases will
be presented to demonstrate the performance of the proposed methods.
Paper Folding: Modeling, Analysis and Simulation
The unfolding of a ladybird's wings, the trapping mechanism used by a flytrap, the design of self-deployable space shades, and the constructions of curved origami are diverse examples where strategically placed material defects are leveraged to generate large and robust deformations. With these applications in mind, we derive plate models incorporating the possibly of curved folds as the limit of thin three-dimensional hyper-elastic materials with defects. This results in a fourth order geometric partial differential equation for the plate deformations further restricted to be isometries. The latter nonconvex constraint encodes the plates inability to undergo shear nor stretch and is critical to justify large deformations.
We explore the rigidity of the folding process by taking advantage of the natural moving frames induced by piecewise isometries along
the creases. We then deduce relations between the crease geodesic curvature, normal curvature, torsion, and folding angle.
Regarding the numerical approximation, we briefly present a locally discontinuous Galerkin method. The second order derivatives present in the energy are replaced by weakly converging discrete reconstructions while the isometry constraint is linearized and incorporated within a gradient flow. We discuss the convergence of the discrete equilibrium deformations towards a minimizer of the exact energy and, in particular, to an isometry. This theory does not require additional smoothness on the deformations besides having a finite energy. The capabilities and efficiency of the proposed algorithm is documented throughout the presentation by illustrating the behavior of the model on relevant examples.
Sample Inverse Problems from Maxar's Earth Imaging Satellites
Maxar Technologies operates a constellation of high-resolution earth imaging satellites with varying spectral capabilities. The imagery is loaded with content, and a variety of techniques are employed to extract information and insight. The nine spectral bands on our WorldView-2 and 3 satellites can be used to determine the water depth in shallow coastal and inland waters, as well as characterize the corresponding benthic habitats at very high resolution. This presentation will introduce the shallow water bathymetry application and describe two inverse problems used in our process - atmospheric compensation and depth retrieval.
Department of Mathematics, University of Nebraska-Lincoln
Incorporating Mass Vaccination into Compartment Models for Infectious Diseases
The standard way of incorporating mass vaccination into a compartment model for an infectious disease is as a spontaneous transition process that applies to the entire population of susceptible individuals. The large degree of COVID-19 vaccine refusal, hesitancy, and ineligibility, and initial limitations of supply and distribution require reconsideration of this standard treatment. We address these issues for models on endemic and epidemic time scales. On an endemic time scale, we partition the susceptible class into prevaccinated and unprotected subclasses. On an epidemic time scale, we develop a vaccination model that addresses limitations of supply and distribution. We then extend this model to a COVID-19 scenario in which the population is divided into two risk classes, with the high-risk class being prioritized for vaccination. In both cases, with and without risk stratification, we see significant differences in epidemiological outcomes between the new vaccination model and naive models.
Learned Spatial Priors for D-bar Reconstructions of 2D EIT Data
The inverse problem for Electrical Impedance Tomography (EIT) is known to be highly ill-posed, leading to poor spatial resolution in reconstructed images. Recent advances in the D-bar reconstruction method for 2D EIT permit the inclusion of spatial priors in the nonlinear Fourier domain, which provide regularizing and stabilizing effects. In previous works, these priors were manually extracted from CT scans or other previous medical images. This methodology is labor intensive, has the potential of introducing human bias, and may not be feasible in real-world clinical scenarios when previous scans do not exist. In this talk we demonstrate that, using a trained convolutional neural network, spatial priors may be computed directly from the nonlinear Fourier data computed from the EIT measurements. This novel approach provides completely automated image stabilization without the use of previous medical images.
Invariant-domain preserving high-order implicit explicit time stepping for nonlinear conservation equations
I consider high-order discretizations of a Cauchy problem where the
evolution operator comprises a hyperbolic part and a parabolic part
with diffusion and stiff relaxation terms. Assuming that this problem
admits non-trivial invariant domains, in the talk I will discuss
approximation techniques in time that preserve these invariant
domains. Before going into the details, I am going to give an
overview of the literature on the topic. Emphasis will be put on
explicit and explicit Runge-Kutta techniques using Butcher's
formalism. Then I am going to describe techniques that make every
implicit-explicit time stepping scheme invariant-domain preserving and
mass conservative. The proposed methodology is agnostic to the space
discretization and allows to optimize the time step restrictions
induced by the hyperbolic sub-step.
Department of Geophysics, Colorado School of Mines
Modeling Earth's and planetary interiors with 3D numerical wave simulations: Seismology with big
& small data and high-performance computing
Seismic waves generated by passive sources such as earthquakes and ambient noise are our primary tools for probing Earth's
deep interior. Improving the resolution of the seismic models of Earth's interior is crucial to understand
the dynamics of the mantle (from ~30 km to 2900 km depth) and the core (from 2900 km to 6371 km depth), which directly control,
for instance, plate tectonics and volcanic activity at the surface, and the generation of Earth's magnetic field, respectively.
Meanwhile, a detailed shallower crustal structure is essential for seismic hazard assessment, better modeling earthquakes
and detecting nuclear explosions, and oil and mineral explorations.
Advances in computational power and the availability of high-quality seismic data from dense seismic networks and emerging instruments offer excellent opportunities to refine our understanding of Earth's structure and dynamics from the surface to the core. We are at a stage where we need to take the full complexity of wave propagation into account and avoid commonly used approximations to the wave equation and corrections in seismic tomography. Imaging Earth's interior globally with full-waveform inversion has been one of the most extreme projects in seismology in terms of computational requirements and available data that can potentially be assimilated in seismic inversions. While we need to tackle computational and "big data" challenges to better harness the available resources on Earth, we have "small data" challenges on other planetary bodies such as Mars, where we now have the first radially symmetric models constrained by seismic waves generated by marsquakes as part of the Mars InSight mission. I will discuss advances in the theory, computations, and data in exploring multi-scale Earth's and Mars' interiors. I will also talk about our recent efforts to address computational and data challenges and discuss future directions in the context of global seismology.