Bifurcation Theory and the Dynamics of Cardiac Arrhythmias
DateFriday, September 7, 2018 - 3:00pm
AbstractMathematical models of cardiac arrhythmias come in two distinct forms. High-dimensional, biophysically detailed models give us realism; they talk about biophysical quantities that can be experimentally and clinically altered. But there is another critical kind of modeling: Low-dimensional modeling tries to isolate the essential dynamical phenomena responsible for a qualitative type of solution. It gives us deeper insights into causal mechanisms. The most important technique of low-dimensional modeling is bifurcation theory. A bifurcation is a qualitative change in the solution to a differential equation (ODE or PDE), as a key parameter is varied. Identification of these key parameters then becomes the central task, because it is by acting through them that we can produce or prevent the qualitative phenomenon. For example, consider the onset of Ventricular Fibrillation. We will show that VF arises as a series of bifurcations from simple reentry to complex spatio- temporally chaotic behavior, and that manipulations of the key parameters can prevent its onset. A second type of arrhythmia is the onset of the pathological voltage oscillations that are known as early after depolarizations (EADs). EADs are thought to be important triggers of arrhythmia in heart failure, Long QT syndromes and other cardiac conditions. We will show that the onset of these pathological oscillations is a bifurcation at the cellular level, and that manipulation of its key parameters can produce and prevent EADs.
Superadditivity in communication capacity via collective quantum measurements
DateFriday, September 14, 2018 - 3:00pm
AbstractThe overarching theme of this talk will be to explore how to attain the quantum limit of the maximum rate of reliable transmission of (classical) information over a quantum channel. We will consider arguably the simplest setting to develop this problem, that of using two non-orthogonal pure states — of a given inner product — as the symbols of a binary alphabet to encode information, where each “use" of a channel comprises of transmitting one of those two pure states. We will first consider the quantum optimal measurement to distinguish between those two states with the minimum average probability of error, and evaluate the Shannon limit to the capacity (bits per use) when this optimal measurement is used on each symbol by the receiver. We will then evaluate the Holevo limit to the capacity, C_\infty, and quantify the gap between that and the Shannon capacity C_1 of the aforesaid optimal symbol-by-symbol measurement. Next, we will consider an example of a collective (projective) measurement on the two-symbol Hilbert space, whose per-symbol Shannon capacity exceeds C_1. This effect — that a quantum measurement on an n-symbol codeword, one which cannot be expressed as individually measuring each of the n symbols, can attain a higher communication capacity, even though the codeword state on which the measurement is being performed is an n-symbol product state — is known as superadditivity of capacity. Even for this binary pure-state communication problem, C_2, the maximum capacity attainable with a general two-symbol quantum measurement, is not known. We will consider some more explicit examples of superaddivity, and then discuss open problems that underlie our understanding, or lack thereof, of the relationship between the geometry of codeword states and their relationship with optimal collective quantum measurements.
Integrated computational physics and numerical optimization
DateFriday, September 21, 2018 - 3:00pm
AbstractOptimization problems governed by partial differential equations are ubiquitous in modern science, engineering, and mathematics. They play a central role in optimal design and control of multiphysics systems, data assimilation, and inverse problems. However, as the complexity of the underlying PDE increases, efficient and robust methods to accurately compute the objective function and its gradient becomes paramount. To this end, I will present a globally high-order discretization of PDEs and their quantities of interest and the corresponding fully discrete adjoint method for use in a gradient-based PDE-constrained optimization setting. The framework is applied to solve a slew of optimization problems including the design of energetically optimal flapping motions, the design of energy harvesting mechanisms, and data assimilation to dramatically enhance the resolution of magnetic resonance images. In addition, I will demonstrate that the role of optimization in computational physics extends well beyond these traditional design and control problems. I will introduce a new method for the discovery and high-order accurate resolution of shock waves in compressible flows using PDE-constrained optimization techniques. The key feature of this method is an optimization formulation that aims to align discontinuous features of the solution basis with the discontinuities in the solution. The method is demonstrated on a number of one- and two-dimensional transonic and supersonic flow problems. In all cases, the framework tracks the discontinuity closely with curved mesh elements and provides accurate solutions on extremely coarse meshes.