Georgia Institute of Technology College of Sciences

We approximate a function defined over a $d$-dimensional manifold $M ⊂R^n$ utilizing only noisy function values at noisy locations on the manifold. To produce the approximation we do not require any knowledge regarding the manifold other than its dimension $d$. The approximation scheme is based upon the Manifold Moving Least-Squares (MMLS) and is therefore resistant to noise in the domain $M$ as well. Furthermore, the approximant is shown to be smooth and of approximation order of $O(h^{m+1})$ for non-noisy data, where $h$ is the mesh size w.r.t $M,$ and $m$ is the degree of the local polynomial approximation. In addition, the proposed algorithm is linear in time with respect to the ambient space dimension $n$, making it useful for cases where d is much less than n. This assumption, that the high dimensional data is situated on (or near) a significantly lower dimensional manifold, is prevalent in many high dimensional problems. Thus, we put our algorithm to numerical tests against state-of-the-art algorithms for regression over manifolds and show its dominance and potential.

When we get to the point of including the huge and relevant experience of finite element fluid modeling collected in over 25 years of experience in the treatment of cardiovascular diseases, the risk of getting “lost in translation” is real. The most important issues are the reliability that we need to guarantee to provide a trustworthy decision support to clinicians; the efficiency we need to guarantee to fit into the demand coming from a large volume of patients in Computer Aided Clinical Trials as well as short timelines required by special circumstances (emergency) in Surgical Planning. In this talk, we will report on some recent activities taken at Emory to make this transition possible. Reliability requirements call for an appropriate integration of measurements and numerical models, as well as for uncertainty quantification. In particular, image and data processing are critical to feeding mathematical models. However, there are several challenges still open, e.g. in simulating blood flow in patient-specific arteries after stent deployment; or in assessing the correct boundary data set to be prescribed in complex vascular districts. The gap between theory, in this case, is apparent and good simulation and assimilation practices in finite elements for clinical hemodynamics need to be drawn. The talk will cover these topics. For computational efficiency, we will cover some numerical techniques currently in use for coronary blood flow, like the Hierarchical Model Reduction or efficient methods for coping with turbulence in aortic flows. As Clinical Trials are currently one of the most important sources of information for medical research and practice, we envision that the suitable achievement of reliability and efficiency requirements will make Computer Aided Clinical Trials (specifically with a strong Finite-Elements-in-Fluids component) an important source of information with a significant impact on the quality of healthcare. This is a joint work with the scholars and students of the Emory Center for Mathematics and Computing in Medicine (E(CM)2), the Emory Biomech Core Lab (Don Giddens and Habib Samady), the Beta-Lab at the University of Pavia (F. Auricchio ). This work is supported by the US National Science Foundation, Projects DMS 1419060, 1412963 1620406, Fondazione Cariplo, Abbott Vascular Inc., and the XSEDE Consortium.

In this talk, i shall provide some optimal PIR bounds, which confirmed a conjecture on optimal RIP bound. Furtheremore, i shall also investigate some results on signals recovery with redundant dictionaries, which are also related to statistics and sparse representation.

In this talk we discuss how to find probabilities of extreme events in stochastic differential equations. One approach to calculation would be to perform a large number of simulations and gather statistics, but an efficient alternative is to minimize Freidlin-Wentzell action. As a consequence of the analysis one also determines the most likely trajectory that gave rise to the extreme event. We apply this approach to stochastic systems whose deterministic behavior exhibit chaos (Lorenz and Kuramoto-Sivashinsky equations), comment on the observed behavior, and discuss.