Pierre A. Gremaud - US grants

Fundamental and Applied Problems in Granular Flow Michael Shearer/David Schaeffer This project focuses on three aspects of the flow of granular materials: an investigation of fluctuations, flows and stresses in industrial silos, and liquefaction of soils. The various issues will be addressed using an interdisciplinary approach involving analysis, computation and experiment. The latter two parts will also involve input from industrial and geotechnical experts. Details of each subproject follow. Subproject 1, fluctuations in sheared granular materials: Recent work has shown that fluctuations of forces and to some extent velocities can be very large for moderate scale systems. The experimental part of this project will provide additional quantitative characterizations of these fluctuations for modest scale slowly sheared systems. In addition, new experiments will be constructed of a Couette type that will probe force fluctuations on larger length and time scales. These experiments will be integrated into ongoing work to model force fluctuations by lattice type models, and computations using novel hybrid molecular dynamics and finite element codes. Subproject 2, flow in industrial silos: In collaboration with engineers at the firm Jenike and Johanson, Inc. the co-PI's of this project will analyze flows in a spatial region that corresponds to the shape of a typical hopper. This analysis will be based both on Coulomb materials and on critical state soil mechanics (CSSM). Some of the aspects under study will include an investigation of the relationship between CSSM and Coulomb models, shock and rarefaction wave solutions, boundary value problems for hopper flow, and stability of such solutions. An important application is the design of flow corrective devices. Subproject 3, liquefaction of soils: This phenomenon corresponds to the abrupt loss of load-bearing capacity of a loose, water-saturated soil, possibly leading to a massive landslide. Real world s oil failure/liquefaction will be investigated in collaboration with G. Gudehus and his associates. This project will combine experiments, mathematical analysis, computer simulation and industrial/geotechnical expertise to better understand the flow of granular materials. The area of study is of considerable importance to technical processes involving all types of granular materials, including but by no means limited to chemical process industries, and to the handling of coal, ores, food grains, and pharmaceuticals. Many aspects of the above processes are not fully understood, leading in some cases to enormous financial losses. Also under consideration are geotechnical issues such as the stability of embankments, as well as the stability of soils under earthquake conditions. The project will involve the application of existing theories for granular materials to such fundamental problems as flows in hoppers and stability of soils in landfills. New models will be developed in order to take into account some important aspects of granular flows such as fluctuations of forces. Recent experiments in this lab have shown that fluctuations, which are not accounted for in existing models, can be very strong and may well be necessary to provide safe and reliable design criteria for industrial devices involving granular flows. The models will be tightly linked to the experimental data, on the one hand, and, on the other hand, will be the basis of computer solutions for relevant technical problems.

The 2000 NCSU Industrial Mathematics Modeling Workshop for graduate students (IMMW2000) is designed to expose students in mathematics, statistics, and engineering to problems from industry and government laboratories, and introduce them to a team approach to problem solving. The workshop is scheduled for the time period July 24,2000 to August 1, 2000 and is organized by Pierre Gremaud, Zhilin Li, Ralph Smith, and Hien Tran. The workshop accommodates approximately 36 graduate students (for a total of 6 teams)
from national and international institutions. Scientists from industry or government laboratories will be invited to present current research problems and lead teams of 5-6 students through model formulation and at least partial solution of the problems. This enriches the traditional graduate experience and provides valuable training for both students considering academic careers and those students preparing for nonacademic careers. For students preparing for an academic career, the workshop provides experience which will significantly broaden their perspective in the classroom and may provide a catalyst for later research. The experience is even more significant for students pursuing nonacademic careers since it provides them with an exposure to important ``real life'' problems and gives them some initial experience at addressing such problems.

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Gremaud

This award will support the Southeast Conference on Applied
Mathematics (SECAM). The conference will focus on four overlapping
themes: i) Mathematical Methods for Multiscale Problems, ii) Industrial
Mathematics, iii) Mathematical and Computational Biology, and iv)
Material Science. Multiscaled models are found in a wide range of
applications and pose difficulties for both mathematical and numerical
analysis. What is, for instance, the effect of the fine scale behavior on the
solutions? How well should the fine scales be resolved to obtain accurate
information? etc. The problems to be discussed in Industrial Mathematics
come from areas such as microelectronic, chemical manufacturing and
material processing, and will focus on the process of technology transfer.
The extreme complexity of the problems studied in Mathematical Biology
precludes a traditional "pencil and paper" approach and calls rather for an
integrated approach between mathematical modeling and computer
analyses. This trend has been accelerated by the development of
productive partnerships between experimentalists, theoreticians and
mathematicians with similar interests but very different areas of expertise.
Successful examples will be discussed. The study of emerging
applications in the Material Sciences such as granular flows, thin films or
smart materials has led to the discovery of new kind of instabilities and
interface propagation problems. This session will aim at facilitating the
migration of the most successful analysis and computational techniques
from specific areas of Material Science to other problems.

The conference aims at accomplishing three main goals: Interaction,
Showcase, and New Activities. Interaction: The conference will provide a
forum and gathering place for applied mathematicians in the Southeast. It
will reach out to several groups that are underrepresented in traditional
meetings. First, many successful collaborations have been established
between applied mathematicians and scientists from local companies and
governmental laboratories in technological parks such as Research
Triangle Park, NC. Several industrial partners have been invited to give
presentations. Second, there are emerging centers of excellence within the
many local colleges and historically black institutions. Speakers and
students from such institution have been invited to contribute to the
success of the conference. Showcase: The high level of activity in Applied
and Industrial Mathematics is not always accurately represented through
the usual channels of communication. Indeed, by its very nature, a sizable
part of the work done in Industrial Mathematics gets to be published in
specialized engineering type journals, reaching only a small portion of the
Applied Mathematics community. Conferences such as this one can play a
fundamental role in unifying and improving approaches that may be
similar in many aspects but are used in very distinct applications. Special
care has been taken to provide younger scientists and graduate students
with the opportunity to present their work and take an active part in the
conference. New activities: The audience is expected to span a whole
spectrum of expertise, experience and background, from recognized
researchers in academia to engineers from the private sector and national
research laboratories to students and people with limited research
experience in Applied Mathematics. It is expected that not only will the
conference give an accurate idea of significant ongoing and future trends
in Applied Mathematics, but also will lead to new collaborations and
applications.

This research effort addresses a spectrum of fundamental and applied problems in the slow flow of granular materials. It is organized into four projects, chosen partly because of their importance in the field of granular materials, but also because they raise intriguing mathematical and scientific issues of broader significance. The first project attacks a fundamental physical problem: How to include micromechanical effects in a continuum description of granular flow, especially the effect of velocity fluctuations. The second project concerns the mathematics surrounding multidimensional continuum models for granular flow, specifically the issue of extracting mathematically rigorous information from ill-posed partial differential equations. The third project proposes to extend Jenike's radial solution for flows in axisymmetric hoppers to conical hoppers with a general cross section. The fourth project deals with flows of fine granular materials, where the interstitial gas significantly affects the flow. The research program involves coordinated efforts in modeling, analysis, numerical simulations, and experiment.

At the heart of this research project is a basic question concerning the flow of granular materials: "What behaviors of slowly flowing granular material can be understood in terms of a continuum formulation?" This question may be viewed as an attempt to reconcile continuum models, used in industrial design and engineering problem solving, with discrete models, introduced to understand the results of small- to medium-scale physical experiments. Continuum models are highly desirable since they are much more tractable analytically and computationally than particle dynamics simulations, which treat the discreteness of the flow directly. An important issue that arises from this basic question, and which is addressed in this project, is the form that a continuum description should take. This issue has been the subject of debate in the engineering literature ever since Janssen in 1895 demonstrated that stresses in a column of granular material do not increase indefinitely with depth, but approach asymptotically to a constant. The research program has significance well beyond the context of granular materials in mathematics as well as physics. The project is supported by a long-standing industrial collaboration.

The 2002 NCSU Industrial Mathematics Modeling Workshop for graduate students (IMMW2002) is designed to expose students in mathematics, statistics and engineering to problems from industry and government laboratories, and introduce them to a team approach to problem solving. The workshop is scheduled for the time period July 22, 2002 to July 30, 2002 and is organized by Pierre Gremaud, Zhilin Li, Ralph Smith, and Hien Tran. The workshop accommodates approximately 36 graduate students (for a total of 6 teams)
from national and international institutions. Scientists from industry or government laboratories will be invited to present current research problems and lead teams of 6-7 students through model formulation and at least partial solution of the problems. This enriches the traditional graduate experience and provides valuable training for both students considering academic careers and those students preparing for noncademic careers. For students preparing for an academic career, the workshop provides experience which will significantly broaden their perspective in the classroom and may provide a catalyst for later research. The experience is even more significant for students pursuing nonacademic careers since it provides them with
an exposure to important ``real life'' problems and gives them some initial experience at addressing such problems.

Proposal: DMS-0244488
PI: Michael Shearer
Institution: North Carolina State University
Title: FRG: Collaborative Research: Physical, Mathematical and Engineering Problems in Slow Granular Flows

ABSTRACT

This proposal addresses a spectrum of fundamental and applied problems in the slow flow of granular materials. It includes coordinated efforts in experiment, mathematical analysis, modeling, and numerical simulation, supported by a long-standing industrial collaboration. Continuum models of granular materials are desirable from several points of view, but the associated partial differential equations are typically ill-posed, a mathematical difficulty that has severely hampered progress, especially for multidimensional flow. This ill-posedness reflects a real instability, the tendency of the deformation in granular flow to localize into shear bands. Fundamental issues surrounding the continuum description will be addressed through inter-related projects that investigate: (i) the role of discreteness and nonlinearity in regularizing the ill-posed continuum description, (ii) the formulation from micromechanical considerations, based on experiments and MD simulations, of a continuum model that accounts for microscopic velocity fluctuations, (iii) mathematical predictions and experimental tests of multidimensional steady-state hopper flows, and (iv) the settling of powders, an industrially significant prototype two-phase flow problem.

The research program attacks different aspects of a basic question: What behavior of slowly flowing granular material can be understood by modeling the material as a continuum? This is particularly an issue for industrial settings such as the flow of agricultural grains in a converging hopper: When a silo is discharged, the granular material flows somewhat like a continuous fluid, but the forces on the hopper walls are unlike those exerted by a fluid due to the solid-like properties of the material and its discreteness. Scientifically, we wish to understand these properties, especially the role that the discreteness of the material plays in the models. Among other contributions, this research is expected to improve predictive capabilities in the materials-handling industry and thereby increase efficiency in manufacturing. The group will create a research environment that integrates research with the training of students and post-docs, through weekly group meetings, participation in conferences and a new course on granular materials. Collaboration across fields will be facilitated by the proximity of the mathematics and physics departments at Duke University, and by the existing collaborations between PIs at NC State and Duke Universities.

The mechanical behavior of granular materials is still only partially
undertsood even though many continuum models have been proposed. The
goal of this project is to build, study, implement and test efficient
and reliable numerical methods allowing for a quantitative study and
comparison of those models. Specific problems to be studied include wave
propagation in bulk materials, static granular piles, multiphase models
for fine powders and multidimensional granular flows. In all those
problems, dry friction plays a central role. Depending on the problem
and its formulation, the presence of frictional effects manifests itself
mathematically in various ways: presence of a graph, stiff source term
in systems of balance laws, algebraic constraints. Each of those
difficulties leads to new numerical challenges. The wide variety of
mathematical problems resulting from the modelization of the above
phenomena (differential inclusion, systems of conservation laws, balance
laws, Hamilton-Jacobi equations, elliptic and parabolic problems,
non-local free boundary problems) is a testimony to the incredible
richness of this field.

In countless industries, materials have to be processed, stored and
retrieved in granular form. Surprisingly many problems occur during
those various phases. Difficulties range from the complete structural
collapse of silos during discharge to poor performance and
unpredictability of the manufacturing process. This comes at a very high
financial cost to solid processing plants. It is proposed to design and
implement, in consultation with engineers from both Academia and
Industry, numerical methods that allow for the calculations of granular
flows and related problems in general geometries. The current methods in
use date back to the 1950's and have limited predicitive capabilities.

Cerebral autoregulation is one of the most critical control systems in the body, as a constant tissue perfusion is necessary for proper functioning of the brain. As a response to changes in blood pressure, this control system modulates cardiovascular parameters to maintain a constant cerebral blood flow. Transcranial Doppler ultrasound measurements are routinely used to measure blood flow velocity in the middle cerebral arteries, one of the largest suppliers of blood to the brain. These measurements are then used to estimate blood flow and assess efficacy of cerebral autoregulation. However, these measurements do not currently provide reliable indicators for early diagnosis of potential impairments in the cerebral arteries, as they lack the necessary accuracy. One problem from basing the estimates derived from measurements is the questionable assumption that regulation only influences the diameter of microvasculature, while the diameter of larger vessels, such as the middle cerebral artery, remains constant. It is now clear that the large arteries are compliant suggesting that the diameter of the middle cerebral artery can indeed change in response to variations in pulsatility. In addition, estimates derived from measurements do not account for topological variations in network of cerebral arteries, such as the main distribution system, the circle of Willis. These questions will be studied using a new one-dimensional fluid dynamic model of the circle of Willis. Geometric data for this model will be obtained from magnetic resonance angiographs. To solve these equations, new numerical methods will be used. Viscoelastic equations describing the compliance of the vascular wall will be introduced and the effects of including non-Newtonian flow will be studied. Additionally, the effects of curvature of the vessel topology will be estimated. In particular, the internal carotid artery, curves about 180 degrees from when it enters the scull to it is attached to the circle of Willis. To validate this model, computed results will be compared with measurements of cerebral blood flow and network topology. The model will be used to predict effects of changes in the topology as well as changes in outflow boundary conditions. For example, plan to study the effects on distribution of blood flow in response to changes in boundary conditions and compare this with changes in diameters of the proximal vessels. Furthermore, we plan to study changes between healthy subjects and in elderly DM patients. Mathematical models have long been used to study fluid dynamic properties of arteries, however no studies have used this approach to design patient specific models to predict CBF and cerebral autoregulation.

Cerebral autoregulation is a critical control system in the body, as constant tissue perfusion is necessary for proper functioning of the brain. In response to changes in blood pressure, this control system modulates cardiovascular parameters to maintain a constant cerebral blood flow. Impairments in cerebral autoregulation have been observed in patients with type II diabetes and are associated with an increased risk of stroke. Ultrasound measurements are routinely used to measure blood flow velocity in the cerebral arteries, the largest suppliers of blood to the brain. These measurements are then used to estimate cerebral blood flow and assess efficacy of cerebral autoregulation. One problem is the questionable assumption that regulation only influences the diameter of the microvasculature, while the diameter of larger vessels, such as the middle cerebral artery, remains constant. It is now clear that the large arteries are compliant suggesting that the diameter of middle cerebral artery can indeed change in response to variations in pulsatility. In addition, estimates derived from measurements do not account for topological variations in network of cerebral arteries, such as the main distribution system, the circle of Willis. These facts suggest that there is a need for development of more advanced methods to estimate cerebral blood flow. In this study we propose to combine physiological data analysis with mathematical fluid dynamic modeling to predict cerebral blood in healthy and diabetic patients. Mathematical models have long been used to study fluid dynamic properties of arteries, however no studies have used this approach to design patient specific models to predict cerebral blood flow. Modeling detailed hemodynamics allows us to develop hypotheses that can predict mechanisms that underlie regulatory failure. The proposed model and new numerical methods for fluid dynamics models with time dependent boundary conditions will be a considerable contribution to applied mathematics and biological sciences applications. Students, who will be doing research in this area, will have skills and knowledge in applied and computational mathematics, and physiology. Such professionals are in great demand.

The past few years have seen spectacular successes in the handling of ever larger- and higher-dimensional data sets. Those advances are based on multiscale techniques and are applied, for instance, in the new FBI fingerprint database and in JPEG2000, the new standard for image compression.

However, recent advances in mathematics have shown these techniques to be far from optimal: there is ample room for improvement. In other words, a new generation of methods can be built that will significantly expand their range of applications. This answers a growing need in many sensitive applications for more powerful tools to represent data and extract relevant information in an efficient and accurate way. Potential applications include remote sensing, medical diagnosis, data transmission and classification, video surveillance, and data storage.

It is proposed to develop the shearlet representation, which combines the power of multiscale methods with a unique ability to capture the geometry of multidimensional data. This approach opens the door to a new generation of methods that are optimally efficient (no room for improvement) for handling multidimensional data and are fast to compute.

The investigators propose a program of research that involves the development of both mathematical and numerical aspects of the shearlet representation. Not only will this lead to improved algorithms for data compression and analysis, it is also a very promising approach for the numerical solution of partial differential equations. Both avenues of investigation will be pursued in the project. Specific applications to biomedical and geological data will also be considered.

The program "Mathematics of Materials" provides a wide range of interdisciplinary research and training opportunities for undergraduates, graduate students, and postdocs in the mathematical sciences. The research activities are organized around five topics that play a fundamental role in emerging technologies: multifunctional materials, polymers and composites including carbon nanotubes, orthopaedic biomaterials, dynamics of thin material layers, and material behavior of laser welding. Within each topic, investigations focus on fundamental model development, mathematical solution, numerical simulation, and control design. Each project involves substantial collaboration with experimental colleagues that provides interdisciplinary training opportunities to mathematics students and postdocs. The training component is designed to prepare students and postdocs for the varied roles of interdisciplinary research mathematicians. This includes training modules that introduce participants to research and career topics not typically covered in coursework, targeted courses on topics pertaining to research areas and national research agendas, and participation in summer internships and national conferences.

"Mathematics of Materials" is an interdisciplinary research training group program designed for undergraduates, graduate students and postdocs in the mathematical sciences. The research activities are organized around five topics that play a fundamental role in emerging
technologies: multifunctional materials, polymers and composites including carbon nanotubes, orthopaedic biomaterials, dynamics of thin material layers, and material behavior of laser welding. The training component involves a coordinated set of activities designed to prepare students and postdocs for the varied roles of interdisciplinary research mathematicians.
This includes training modules that introduce participants to research and career topics not covered in traditional coursework, new courses pertaining to the five research topics and areas of national need, and participation in summer internships and national conferences spanning multiple disciplines. The objective of the program is to attract and train highly qualified students and postdocs for academic and nonacademic careers at the interface between applied mathematics, materials science, engineering, physics, and advanced technology.

This SGER grant aims at developing a pilot system to achieve synergistic and stable deep penetration welding by combining a fiber laser and a plasma jet. The success of this pilot system heavily relies on a real-time plasma sensor and the integration of the fiber laser and the plasma jet. To develop this real-time plasma sensor, a high speed photography system will be integrated with a high performance computing system for image processing at a frame rate up to 200,000 fps. This high speed photography system allows the PIs to observe the oscillation of keyhole and plasma ejection in order to validate and refine the design of the new plasma sensor array. The observation also allows the validation of rigorous mathematical modeling of the laser/plasma-jet process currently under development.

The key technology resides in synergistic energy deposition of a laser beam with moderate power and a plasma torch with a special nozzle design. The laser, with its much higher power density, will create the keyhole while a finely focused plasma jet from the plasma torch will provide additional energy deposition. The keyhole will be stabilized through active regulation of the plasma jet and the laser-induced plasma. The successful development of closed-loop deep penetration laser welding processes will be a breakthrough in this area. The request SGER grant will allow the PIs to prove the synergistic energy deposition concept as well as the inherent stability of the process, thus, establishing the foundation for a closed-loop welding system. Such a closed-loop system has not been attempted because the lack of real-time feedback sensors.

The technology will render laser welding economically viable for a much larger range of industries. The US industry is in need of this kind of advanced technology to boost competitiveness in order to turn back the tide of off-shoring jobs by many major corporations. Further, US laser job shops (over 10,000) will see their service capability significantly increased by the availability of such a technology. Computationally, the project uses novel combination of numerical strategies. The work is being performed in an interdisciplinary group environment that integrates research with undergraduate and graduate students through both courses and projects.

The objective of this research is to develop efficient numerical methods for the simulation of networked systems of hyperbolic balance laws. In such networks, each edge is a quasi one-dimensional domain interacting with the rest of the system through junctions at each of its ends. The character of those interactions depends on the applications at hand; ideally, they are
modeled to mitigate the effects of dimension reduction. Mathematically, the presence of junctions complicates the selection process of proper solutions. The naive use of existing numerical methods in the present context may be inefficient, unstable and lead to nonphysical solutions. Numerical methods specifically optimized for network problems will be designed, analyzed and implemented. This involves not only discretization issues but also and more importantly the construction of new solvers. Those solvers will be designed by building on recent progress in both numerical methods for differential algebraic equations and in domain decomposition methods. Some phenomena are
essentially one-dimensional in most of the computational domain and only "locally multidimensional". Being able to reliably switch to one-dimensional approximations represents significant savings; how to do this efficiently will be investigated. Transport phenomena in trees, which play an essential role in many organisms (breathing, blood circulation,etc...), lead to other types of couplings for which new numerical approaches are also proposed. Two applications are considered as test beds for various aspects of the research. They respectively involve blood flows in arteries and gas flows.

Networks of roads, pipelines or arteries play a fundamental role in many aspects of our lives. They allow the efficient transport and distribution of, for instance, cars, raw sewage, gas or blood in respectively cities, countries, organisms, etc... Related practical problems range from business
(optimization of natural gas pipeline networks) and public safety (emergency evacuation schedules in specific geographic areas) to health (likelihood of stroke based on patients' vasculature). While the tools of scientific computing have been applied very successfully to many types of transport
phenomena such as problems in aerodynamics, the numerical simulation of transport on networks faces several specific challenges that have yet to resolved. Three main issues will be studied. (i) Efficiency: the methods have to be nimble enough to allow the simulation of entire networks as opposed to only some of their parts. (ii) Accuracy: flows are more involved near junctions or crossroads than they are away from them. Different models may have to be used at different locations of a same network. The project will study efficient implementation of such multi-physics models for network flows. (iii) Finally, the various theoretical and numerical aspects of the project will be testedon two specific applications, arterial blood flow and gas flow in rigid pipes.

Transport and distribution networks come in a number of forms, from animal cardiovascular and respiratory systems to communication and industrial infrastructures. Practical issues abound: prediction of local spikes, estimation of perfusion, and impact of structural changes such as vessel occlusion. The complexity of such phenomena can be illustrated by the well-known Braess' paradox: adding links to a transportation network might not improve the operation of the system! In spite of recent successes, our understanding of network flows is usually limited to small deterministic problems, while most applications correspond to large uncertain ones. The goal of this project is to enable improved predictions in biological and technological transport and diffusion networks. For instance, can one predict how cerebral blood flow will be affected if one of the carotids becomes narrow or blocked? Will the vasculature allow for re-routing? If so, with what probability and how fast?

The main challenge in this research project is the presence of uncertainties. For many applications, only partial information about the systems is available. For instance the size or even the presence of a specific vessel might be uncertain or the status of a router unknown. The analysis of such problems requires the creation of novel mathematical tools and numerical methods to describe how uncertainties propagate through vast and complex networks. The computational tools to be constructed will provide information, usually probabilistic in nature, regarding phenomena that are difficult, expensive, or impossible to measure.

Syncope or, more colloquially, fainting, is a surprisingly common and little understood condition. The long term goal of the proposed line of work is the identification of the root causes of syncope. Fainting can result from the failure of one or more internal control mechanisms. There are currently no clear causal links between these controls and the observed symptoms. In order to understand the involved mechanisms, this project will start by analyzing clinical data to determine the number and characterization of the different types of syncope. A better understanding hinges on the analysis of clinical data, here time series, and the ability to infer from these, patient classification. Various scenarios will be tested through mathematical modeling to confirm both the soundness of the obtained classification and the nature and source of the pathology for each identified class. The ability to identify subjects as members of a class or group also makes it possible to leverage information about the other members of that group for individual diagnosis purposes. The methodology developed here will contribute to the implementation of this approach, sometimes referred to as "bringing cohort studies to the bedside". This approach will also be applicable to the study of other diseases where similar clinical data are being collected such as epilepsy.

Time series data are ubiquitous. In fact, the development of an ever increasing number of applications depends on their analysis, from stock market and economics to weather predictions. Practical issues include signal matching, classification, pattern detection and early prediction. Signals of interest are often high dimensional and noisy and the underlying dynamics are generally unknown. This award supports initiation of a collaborative research project that addresses all the above issues in the context of medical times series data and, more specifically, for syncope. To do so, a novel paradigm combining non-parametric statistics, machine learning, and applied mathematics is proposed. The first objective is to design and compute features in multivariate time series that are well adapted to classification and clustering. The proposed approach relies on recent advances in the evaluation of variable importance for scattered data. The second objective is to facilitate calibration of mathematical models by extending machine-learning concepts to this new realm. The goal is to determine how much data is necessary to "learn" biomathematics models and increasing their predictive power. This award is supported by the National Institutes of Health Big Data to Knowledge (BD2K) Initiative in partnership with the National Science Foundation Division of Mathematical Sciences.