Abstract:

The classical phase retrieval problem arises in contexts ranging from speech recognition to x-ray crystallography and quantum state tomography. The generalization to matrix frames that are U(r) phase retrievable is natural in the sense that it corresponds to quantum tomography of impure states. The U(r) phase retrieval problem is also essentially equivalent to low rank matrix recovery, and in particular to the Euclidean Distance Geometry problem. We provide computable global stability bounds for the quasi-linear analysis map β and a path forward for understanding related problems in terms of the differential geometry of the semi-algebraic variety of positive semi-definite matrices. In particular, we manifest a Whitney stratification of the PSD matrices of low rank, which allows us to “stratify” the computation of the global stability bound. We show that in contrast to the rank 1 case, for the impure state case no such global stability bounds can be obtained for the non-linear analysis map α with respect to certain natural distance metrics. Finally, our computation of the global lower Lipschitz constant for the β analysis map provides novel conditions for a matrix frame to be U(r) phase retrievable.  

Abstract: In this talk, I will discuss some key ideas and methods of two disparate fields of mathematical research, frame theory and optimal transport, using the methods of the second to answer questions posed in the first. In particular, I will discuss construction of gradient flows in the Wasserstein space for a new potential, the tightness potential, which is a modification of the probabilistic frame potential. The potential is suited for the application of a gradient descent scheme from optimal transport that can be used as the basis of an algorithm to evolve a given frame toward a tight probabilistic frame.

Abstract:

This talk will review models and structural properties of the
equations used to model geophysical flows which involve multiple
components undergoing phase transitions. Simulations of these problems
only model the gross properties of these flows since a precise
description of the physical system is neither available nor
computationally tractable. In this context mathematics is essential
if phenomenology, physical intuition, mechanics, thermodynamics, and
thought experiments, are to be integrated into well posed models.
Numerical schemes which inherit the essential structural and physical
properties of the underlying models can then be developed.

Gap junctions are channels that connect cell membranes allowing electric ions to pass directly between cells. They connect cells throughout the body, including heart myocytes, neurons, and astrocytes. Voltage propagation mediated by gap junctions can be passive or active. In passive propagation, the voltage of one cell affects the voltage of neighboring cells without triggering action potentials (APs). In active propagation, an AP in one cell triggers APs in neighboring cells; this occurs in cardiac tissue and throughout the nervous system. It is known experimentally that there is an ideal gap junction conductance for AP propagation — weaker or stronger conductance can block propagation. We present a theory explaining this phenomenon by analyzing an idealized model that focuses exclusively on gap junctional and spike-generating currents. We also find a novel type of behavior that we call semi-active propagation, where cells in the network are so strongly connected that they are not excitable at rest, but still propagate action potentials.

Abstract:

Cellular networks are ubiquitous in nature. Most technologically useful
materials arise as polycrystalline microstructures, composed of a myriad
of small monocrystalline cells or grains, separated by interfaces, or
grain boundaries of crystallites with different lattice orientations. A
central problem in materials science is to develop technologies capable of
producing an arrangement of grains that provides for a desired set of
material properties. One method by which the grain structure can be
engineered is through grain growth (also termed coarsening) of a starting
structure.

The evolution of grain boundaries and associated grain growth is a very
complex multiscale process. It involves, for example, dynamics of grain
boundaries, triple junctions, and the dynamics of lattice misorientations.
Grain growth can be viewed as the evolution of a large metastable network,
and can be mathematically modeled by a set of deterministic local
evolution laws for the growth of an individual grain combined with
stochastic models to describe the interaction between them. In this talk,
we will discuss recent progress in mathematical modeling, simulation and
analysis of the evolution of the grain boundary network in polycrystalline
materials.

Abstract:

The placenta is the essential organ of maternal-fetal interactions, where nutrient, oxygen, and waste exchange occur. In recent studies, differences in the morphology of the placental chorionic surface vascular network (PCSVN) have been associated with developmental disorders such as autism. This suggests that the PCSVN could potentially serve as a biomarker for the early diagnosis and treatment of autism. Studying PCSVN features in large cohorts requires a reliable and automated mechanism to extract the vascular networks. In this talk, we present a method for PCSVN extraction. Our algorithm builds upon a directional multiscale mathematical framework based on a combination of shearlets and Laplacian eigenmaps and can isolate vessels with high success in high-contrast images such as those produced in CT scans.

Abstract:

In computer graphic applications or design problems, a three-dimensional object is often represented by a triangular mesh. In many instances, individual or groups of triangles in such representation need to satisfy spatial constraints that are imposed either by observation from the real world, or by concrete design specifications of the object. In this talk, we model various geometric constraints as convex sets in Euclidean spaces. We then apply splitting algorithms in these constructions to find optimal solutions.

Abstract:

The Riesz maps of the spaces H^1, H(curl) and H(div) frequently arise as
subproblems in the construction of fast preconditioners for more
complicated problems.  In this work we present multigrid solvers for
high-order finite element discretizations of these Riesz maps with
optimal complexity in polynomial degree, i.e.~with the same time and
space complexity as sum-factorized operator application.  The key idea
of our approach is to build new finite elements for each space in the de
Rham complex with orthogonality properties in both the $L^2$ and
$H(\mathrm{d})$ inner products ($\mathrm{d} \in \{\mathrm{grad},
\mathrm{curl}, \mathrm{div}\})$ on the reference hexahedron.  The
resulting sparsity enables the fast solution of the patch problems
arising in the Arnold–Falk–Winther and Hiptmair space
decompositions, in the separable case.  In the non-separable case, the
method can be applied to a spectrally-equivalent auxiliary operator.
With exact Cholesky factorizations of the sparse patch problems, the
application complexity is optimal but the setup costs and storage are
not. We overcome this with the use of incomplete Cholesky factorizations
with carefully specified sparsity patterns arising from static
condensation. This yields multigrid relaxations with computational
complexity and storage that are both optimal in the polynomial degree.

Abstract:
In this talk I will discuss various approaches to modeling. I will start with introducing Energetic variational approach , discuss modeling and numerical simulations of multi-component fluid flow, sintering process and flow through poroelastic medium. Then I will discuss modeling carcinogenesis and immune response in cancer patients using chemical reaction networks and mass-action law. Lastly I will discuss opinion dynamics model.
The models presented were made in collaborations with Chun Liu, James Brannick, Qingcheng Yang, James Adler, Xiaozhe Hu, Yiwei Wang, Leili Shahriyari, Bruce Boghosian, Christoph Borgers, Natasa Dragovic, Anna Haensch.

Abstract:

Among infinitely many factorizations A=VV* of a psd matrix A, we seek the factor V that has the smallest (1,2) norm. In this talk we review the origin of this problem as well as existing results regarding the optimal value. We prove two new results describing two properties of the optimal factorization: full continuity over the set of psd matrices (previously, the continuity was shown over strictly positive matrices), and equivalence to a linear program over the space of Borel measures. We discuss also the conjecture that the squared (1,2) norm of V is equivalent to the (1,1) norm of A (a.k.a. the projective norm).

Due to recent technological advances, multi-modal data have become abundant as many data-acquisition systems record information through multiple sensors simultaneously. Coupled with an unknown system model and high data dimensions, such data poses a significant challenge for common analysis techniques. We approach this problem from a geometric perspective, using manifold learning to recover the underlying data structures, yet, comparing and combining such representations is non-trivial. I will discuss two recent approaches to data fusion, with known and unknown dataset alignment, in which we exploit the Riemannian geometry of symmetric positive definite matrices to compare diffusion operators, a manifold learning tool.