# Grants and Awards

## Wu, Z.

### Optimal tests for weak, sparse, and complex signals with application to genetic association studies

National Science Foundation

2013-2016

Detection of sparse and weak signals is a key for analyzing big data in many fields. Recent statistical research has made celebrated theoretical progress in revealing the detectability boundaries under the Gaussian means model and an idealized linear regression model. Detectability boundary illustrates the border in the two-dimensional phase space of signal sparsity and weakness, below which the signals are asymptotically too weak and sparse to be detectable by any statistical methods. Certain statistics are optimal for these models in the sense that they reach the boundary (i.e., the least requirements) for reliable signal detection. However, there are significant gaps between these theoretical models and practical meaningful models. In this project, the investigators extend statistical theory to handle weak, sparse, correlated, and interactive signals under the framework of generalized linear models. The investigators develop optimal testing procedures to address the realistic data features in genome-wide association studies and next-generation sequence studies. Learn more about the grant and Professor Wu.

## Larsen, C.

### New Variational Methods for Quasi-static and Dynamic Material Defect Evolution

National Science Foundation

2013 - 2016

While defects in materials play a fundamental role in material failure, their analysis remains a major challenge in applied mathematics. This is partly due to the difficulty of formulating precise mathematical models, and partly due to the difficulty of analyzing the free surfaces and singularities involved. The investigator extends recent successes in the analysis of globally minimizing and locally minimizing quasi-static evolutions to both locally stable quasi-static evolutions and dynamic evolutions. One goal is to develop and study new models for cohesive fracture and plasticity with softening, based on local stability rather than global minimality (which is mathematically problematic). The investigator also studies existence and analyzes fundamental properties of dynamic fracture solutions, based on models he formulated previously. The failure of materials rests on the nucleation and evolution of defects such as cracks, plastic regions, and damage. The ability to accurately predict failure depends on the quality of the underlying mathematical models of these defects, as well as on understanding fundamental properties of solutions. Substantial challenges remain in these areas, both in formulating sound models and in the analysis of qualitative behavior of solutions. The investigator seeks to make fundamental progress on these fronts, by developing new models that are both mathematically well-posed and significantly more physically realistic than existing models, and performing the mathematical analysis necessary to assess their accuracy. Learn more about the grant and Professor Larsen.

## Wu, Z.

### Analysis of Deep Sequencing Data to Identify Genes Causative for Neurodegenerative Diseases

UMass Medical School

This project will analyze the exome-sequencing data for amyotrophic lateral sclerosis (ALS, often referred to as "Lou Gehrig's Disease") patients, which were generated at the University of Massachusetts School of Medicine and collaborators. We will apply and assess existing methods as well as develop new statistical genetic association tests. The purpose is to develop more effective statistical methods for hunting new ALS genes, and to obtain a set of putative loci that have high potentials for follow-up biological studies. Learn more about Professor Wu.

## Fehribach, J., Co-PI: Tilley, B., Weekes, S.

### Expanding Links with Industry through Collaborative Research and Education in Applied Mathematics

National Science Foundation

2013-2016

The project is a collaborative program of research, education and training based on the Mathematical Problems in Industry (MPI) Workshop and the Graduate Student Mathematical Modeling (GSMM) Camp. The project is part of an ongoing effort organized by the principal investigators at Rensselaer Polytechnic Institute, University of Delaware, Worcester Polytechnic Institute and New Jersey Institute of Technology. These annual meetings, held during successive weeks in June, attract mathematicians (graduate students, postdoctoral fellows and faculty), scientists, and engineers from academic institutions and from industry. The focus of the MPI Workshop is a set of problems brought by contributing participants from industry. These problems span a wide range of areas of applications, often in fluid and solid mechanics but also in mathematical biology, data analysis, and mathematical finance, among others. The scientific objective of the activity generated by the Workshop and its intellectual merit is the study of mathematical problems of significant interest for industrial applications. The GSMM Camp is held during the week prior to the Workshop, and graduate students attending the Camp also attend the Workshop. The main objective of the Camp is graduate student education and training. The two meetings complement each other and form a comprehensive program of interdisciplinary research, education and training that is unique amongst universities in the United States. Learn more about the grant, Professor Fehribach, Professor Tilley, and Professor Weekes.

## Weekes, S., Co-PI: Tilley, B.

### REU in Industrial Mathematics and Statistics

National Science Foundation

2013 - 2015

The WPI REU program in Industrial Mathematics and Statistics provides a unique educational experience for students of the mathematical sciences by introducing them to research in an industrial environment. It aims to expose students to mathematics research that occurs outside of the university setting. It aims to provide challenges to students not faced in standard undergraduate programs and to develop skills not always developed in traditional educational programs, such as (a) communication at several levels, including reading, writing, speaking, and listening; (b) problem formulation as an interactive, evolutionary process; and (c) the ability to work with a diverse team. It provides a glimpse of the many career possibilities which are open to students with a strong mathematical background. The students work in teams on problems that come directly from local business and industry. Many projects require some combination of classical analysis, modeling, statistics, optimization, and computer programming. The results from the projects completed in the REU program are of direct importance to the industrial partners and impact research and development at these companies. Learn more about the grant, Professor Weekes or Professor Tilley.

## Fehribach, J., Co-PI: Tilley, B.

### The MPI Workshop and the GSMM Camp 2012-2013

The project is a collaborative program of research, education and training based on the Mathematical Problems in Industry (MPI) Workshop and the Graduate Student Mathematical Modeling (GSMM) Camp. The project is an ongoing effort organized by the principal investigators at Rensselaer Polytechnic Institute, University of Delaware, Worcester Polytechnic Institute and New Jersey Institute of Technology. These annual meetings, held during successive weeks in June, attract mathematicians (graduate students, postdoctoral fellows and faculty), scientists, and engineers from academic institutions and from industry. The focus of the MPI Workshop is a set of problems brought by contributing participants from industry. These problems span a wide range of areas of applications, often in fluid and solid mechanics but also in mathematical biology, data analysis, and mathematical finance, among others. The scientific objective of the activity generated by the Workshop and its intellectual merit is the study of mathematical problems of significant interest for industrial applications. The GSMM Camp is held during the week prior to the Workshop, and graduate students attending the Camp also attend the Workshop. The main objective of the Camp is graduate student education and training. The two meetings complement each other and form a comprehensive program of interdisciplinary research, education and training that is unique amongst universities in the United States.

The first year of the grant the MPI will take place at the University of Delaware and the camp at RPI; in 2013 the MPI will take place at WPI. Learn more about Professor Fehribach or Professor Tilley.

## Martin, W.

Some Problems on Association Schemes*2012-2014*

In this effort, we investigate problems in algebraic combinatorics with applications in digital communications and quantum information theory. Polynomial association schemes can be viewed as highly structured matrix algebras built from regular graphs and are finite analogues of two-point homogeneous spaces. Here we employ a local approach motivated by Lie theory to investigate both metric schemes (distance-regular graphs) using discrete homotopy theory and cometric schemes using ideas from algebraic geometry, as well as additive error-correcting codes via their coset graphs. In addition to error-correcting codes, applications include combinatorial designs and mutually unbiased bases. Learn more about Professor Martin.

## Olson, S.

### Mechanisms of marine invertebrate sperm chemotaxis: from cellular signaling to flagellar swimming

National Science Foundation

*2011 - 2014 *

In order to fully understand how marine invertebrate sperm reach the egg, we need to have an understanding of how the sperm is able to modify its waveform in response to the surrounding environment. Since sperm motility is an emergent property of a complex system, in this project, we will develop multiscale models that account for chemical signaling, mechanics, and hydrodynamics. For marine invertebrate sperm, a chemoattractant is able to guide sperm to the egg by initiating a signaling pathway that results in an increase in calcium. This increase in calcium modifies the waveform, and ultimately the trajectory. This project will develop a hierarchy of models that will be used to investigate the relevant biochemistry of chemotaxis, how and where calcium is acting to modify the flagellar waveform, and how this couples to trajectories that allow the sperm to reach the egg. Multiscale models will be developed to couple the relevant biochemistry at the cellular level to trajectories and flagellar bending at the macroscale. In order to accurately account for nonplanar bending of the flagellum in a 3-dimensional fluid governed by the Stokes equations, we will develop a new regularized formulation of the generalized immersed boundary method. This method will provide a framework to numerically solve a coupled system that includes an immersed structure in a viscous fluid, where the force and torque that the sperm flagellum exerts on the fluid can depend on the biochemistry. Although the main focus of this project is on studying aspects of invertebrate sperm motility, the new numerical method that will be developed is also be applicable to study DNA supercoiling and aspects of motility in other microorganisms such as E. coli. Learn more about the grant and Professor Olson.

## Mosco, U., Co-PI: Vernescu, B.

### Fractal Fibers and Singular Homogenization

National Science Foundation

*2011 - 2014 *

The proposed research aims at further extending the scope and applicability of boundary value problems in domains with fractal boundaries, along three main lines of investigation: Boundary value and transmission problems, Singular homogenization and Variational fractals. The investigators will extend the range of applicability of their theory of singular boundary value and transmission problems, by considering wider classes of 3-dimensional domains, with possibly irregularly and/or randomly scaled fractal boundaries and layers. They propose to give fractal models more plausible physical ground, by approximating fractals with thin fibers and fractal equations with partial differential operators, bringing dynamical fractal theory into the more familiar realm of singular homogenization. Finally, the investigators will enhance connections with classical effective properties of elliptic and sub-elliptic operators, by developing a theory of variational fractals, that replaces similarity with quasi-metric scaling and provides a rigorous foundation to the power-law formalism, widely used in physics and applications in the study of phenomena with no characteristic length. The proposed research, focused on constructive and metric methods, opens new perspectives in applied boundary value problems in domains with complicated and interacting boundaries and in homogenization theory with fractal components. Problems with large surfaces and interfaces confined in small volumes occur in many applications, in material sciences (e.g. porous films and dendritic structures grown by diffusion limited aggregation), physics (e.g., highly fragmented electrical conductors, high voltage electric discharges, electrolytic deposition and diffusion-limited-aggregation), chemistry (e.g., catalytic converters), biology (e.g., cells membranes, biological tissues, bones), engineering (e.g., hydraulic fracturing in oil wells), just to mention a few. Learn more about the grant, Professor Mosco or Professor Vernescu.

## Tilley, B.

### Thermal transport models for layered materials

Air Force Office of Scientific Research

*2011 - 2014 *

This new award investigates how the microscale geometry and compositional and conductivity gradients in coolants affect net heat transport in liquid-cooled electronics applications. Liquid cooling has shown promise in improving the rates of heat transport in these devices. Although the experimental capability to design and fabricate specific microstructures in materials has advanced significantly in the past few decades, a description of their effective thermal behavior on the microscale has not kept pace. Current modeling of these multi-physics systems has focused on direct computational approaches, but they are limited due to the need to resolve the finest length-scales (microns) over the length scale of the full application (centimeters). We shall employ an asymptotic approach to formulate effective transport equations that capture the dominant net fine-scale physical effects on the application length scale. This modeling approach provides an efficient means to determine how competition of different microscale effects can change macroscale behavior. Learn more about Professor Tilley.

## Sunar, B., Co-PI: Martin, W.

### Homomorphic Encryption for Cloud Privacy

National Science Foundation

*2011-2014 *

Cloud computing functionalities constitute poignant and current examples of the need for encryption schemes that allow untrusted parties to perform computations on encrypted data. An encryption function e(.) taking inputs from some ring R is "homomorphic" if it allows someone to efficiently compute encryptions e(x+y) and e(xy) only knowing encryptions e(x) and e(y) of plaintexts x and y from the ring and yet makes computation of x and y themselves intractable. Solving a 30-year-old open problem, Gentry introduced the first homomorphic encryption schemes in 2009/2010. Since any set of ring operations is Turing complete, this advance opens the door to a wide array of applications such as secure search, data aggregation, and server-blind cloud computation. Unfortunately, the existing schemes are far from efficient and are therefore impractical. The goal of this project is to explore efficient implementations, optimizations, tailored arithmetic units, and new homomorphic schemes all in an effort to bolster secure computing in cloud contexts. Learn more about the grant and Professor Martin.

## Yakovlev, V.

### High Temperature Microwave Processing of Clay-Type Materials – Modeling-Based Feasibility Study

Spheric Technologies

*2010*

This program is designed as a continuation of a recently completed project sponsored by Spheric Technologies. The new study will be dedicated to the further development of a multiphysics modeling technique allowing for computing 3D temperature fields at the arbitrary instances of high temperature processing of powder materials. The effort will be focused on making the modeling technique more adequate to physics by incorporating the algorithms for handling convective and radiant thermal boundary conditions. The developed technique will be used to help design laboratory and industrial microwave equipment for high temperature processing of clay-type materials. Learn more about Professor Yakovlev.

## Walker, H.

### Anderson Acceleration for Fixed-Point Iteration

Department of Energy

*2010 - 2013*

This grant will support a continuation of research on acceleration methods for fixed-point iteration that the PI began under an earlier NSF grant. The acceleration methods of interest are based on a method introduced by D. G. Anderson in 1965. The objectives of the research are to strengthen the theoretical understanding of Anderson acceleration, to develop extensions and variants that are more robust and more numerically sound, to demonstrate the effectiveness of the methods in a wide variety of applications, and to implement the methods within software packages developed at Lawrence Livermore National Laboratory and possibly other national laboratories. The work will build on results obtained under the earlier NSF grant, including both theoretical results that give insight into the behavior of the method and promising experimental results in applications ranging from statistical estimation to simulation of transonic flow and variably saturated porous-media flow. Learn more about the grant and Professor Walker.

## Larsen, C.

### Variational Methods for Material Defect Evolution

National Science Foundation

*2010 - 2013*

The investigator studies the evolution of defects in materials (e.g., fracture, damage, dislocation plasticity) using a progression of models from global minimization-based quasi-statics to full dynamics. Global minimization models are now well understood, but very little is understood in the more physically realistic settings of local minimization-based and dynamic problems. Specifically, existence is open for both cohesive and sharp-interface (Griffith) dynamic fracture, and effective macro-scale models for local minimization-based dislocation plasticity are unknown. The investigator seeks to make significant progress on these and related problems. The propagation of defects in materials is of obvious fundamental importance, yet in the most physically realistic settings mathematical foundations and analysis are lacking, leaving engineering models largely ad hoc. The investigator works on developing mathematical support in these areas, leading to improved models, better understanding of solutions, and improved (and justified) algorithms for computing simulations. Learn more about the grant and Professor Larsen.

## Weekes, S., Co-PI: Tilley, B.

### REU in Industrial Mathematics and Statistics

National Science Foundation

*2010 - 2011*

The WPI REU program in Industrial Mathematics and Statistics provides a unique educational experience for students of the mathematical sciences by introducing them to research in an industrial environment. It provides a glimpse of the many career possibilities which are open to students with a strong mathematical background.

The students work in teams on problems that come directly from local business and industry. Students work closely with a company representative to define the problem and develop solutions of immediate value to the company. They work closely with a faculty advisor to formulate the problem mathematically and maintain a clear focus on the mathematics at the core of the project. Industrial problems rarely fit into one area of mathematics and this fact is an important part of the research experience. Many projects require some combination of classical analysis, modelling, statistics, optimization, and computer programming.

The results from the projects completed in the REU program are of direct importance to the industrial partners and impact research and development at these companies. Learn more about the grant, Professor Weekes or Professor Tilley.

## Yakovlev, V.

### Computer Assisted Design of Microwavable Multi-Component Food Products

General Mills

*2010*

The one year project will develop models and computational experiments related to the microwave processes applied to the food industry. "The project will be focused on the development of 3D coupled electromagnetic-thermal models of the product/package systems undergoing microwave thermal processing and an optimization procedure for finding best parameters of these systems. The developed computational tools will be used for analysis of behavior and optimization of the shapes of new microwavable food products of complex compositions in specific microwave ovens." Learn more about Professor Yakovlev.

## Martin, W.

### Structure and Applications of Cometric Association Schemes

National Security Agency

*2010*

In this project, the PI proposes to study cometric (or ``Q-polynomial'') association schemes and related topics in algebraic combinatorics. While very little was known about these combinatorial structures a decade ago, recent results suggest a rich interplay of theory and application, with connections to a variety of mathematical structures, and an array of open questions ranging from fairly accessible to very hard problems.On the one hand, we propose to investigate and classify cometric association schemes with few classes. These include linked systems of designs, sets of real mutually unbiased bases, and systems of lines through the origin in Euclidean space admitting few angles. This study ties into the study of extremal codes and block designs, extremal lattices, new structures in finite geometry and certain questions in quantum information theory.

As well, we propose to apply new algebraic tools to examine the structure of cometric association schemes with an eye toward resolving one or more of the big outstanding conjectures in the area. We propose to study the ideal of polynomials which vanish on each column of the first idempotent in a $Q$-polynomial ordering. This algebraic/geometric approach unifies various tools that have been developed to date and suggests new ways of attacking outstanding problems such as the unimodality conjecture of Bannai and Ito and the absolute bound on $Q$-antipodal association schemes. Learn more about Professor Martin.

## Tang, D.

### In Vivo IVUS Image-Based Modeling for Human Coronary Plaque Assessment

National Institutes of Health

*2010*

The objective of the project is to combine anisotropic computational modeling with in vivo intravascular ultrasound (IVUS), angiography, ex vivo Magnetic Resonance Imaging (MRI), mechanical testing, and pathohistological analysis to analyze vulnerable atherosclerotic coronary plaques and identify critical blood flow and plaque stress/strain indicators for quantitative coronary plaque vulnerability assessment. The long term goals are to: a) develop computational mechanical image analysis tools for more accurate plaque assessment and possible quantitative improvement to the current American Heart Association (AHA) plaque classification scheme; b) identify critical flow and stress/strain plaque vulnerability risk indicators which could be monitored for early prediction, diagnosis, treatment, and prevention of related cardiovascular diseases. Learn more aboutthe grant and Professor Tang.

## Yakovlev, V.

### Comprehensive Electromagnetic-Thermal-Mechanical Model for Microwave Sintering of Particulate Materials

EADS Foundation

*2009*

Vadim Yakovlev was awarded a grant by the EADS Foundation (EADS is best known for being the parent company of Airbus), for the project "Comprehensive Electromagnetic-Thermal-Mechanical Model for Microwave Sintering of Particulate Materials".

WPI's portion (65,500 Euros) is a part of the total of 159,900 Euros awarded by the Foundation to the team of three research groups (including the Materials Science and Engineering Laboratory of the Grenoble Institute of Technology, France and the Microwave for Materials Processing Group of EMPA - the Swiss Federal Laboratories for Materials Science and Technology). The program is dedicated to the development of an innovative technique for modeling of microwave sintering of realistic particulate (including nano-structured) materials. The project aims to build up a comprehensive macroscopic numerical model embracing three most essential components of high temperature microwave sintering of dielectric and metallic powders. Learn more about Professor Yakovlev.

## Mitrea, I.

### Support of Sonia Kovalevsky High School Mathematics Day

Association For Women In Mathematics

*2009*

Sonia Kovalevsky High School Mathematics Day is aimed at 9th and 10th grade women students from the Worcester area. The event will consist of a mix of plenary lectures that introduce participants to the lives and mathematical accomplishments of remarkable women mathematicians and parallel sessions on different topics in mathematics of interest to high school students. It is expected that the program will bring together 60 women high school students, 15 high school teachers, and women undergraduate and graduate students in mathematics at WPI. Learn more about the grant.

## Walker, H.

### Anderson Acceleration for Fixed-Point Iteration

National Science Foundation

*2009 - 2011*

This research will focus on acceleration methods for fixed-point iteration that are based on a method introduced by D. G. Anderson in 1965. This method has been used widely and with considerable success within the computational physics, chemistry, and materials communities as a means of accelerating the self-consistent field iteration used in electronic structure computations. However, it has been untried or underexploited in many other important applications in which it seems likely to be equally successful. Moreover, it has received relatively little attention from mathematicians and numerical analysts, despite there being many significant unanswered mathematical questions. The goals of this research are to analyze the convergence of the method, to explore its effectiveness across a broad range of important applications, and ultimately to develop extensions with improved global convergence and stability properties. Learn more about the grant and Professor Walker.

## Fehribach, J., Co-PI: Vernescu, B.

### Collaborative Research: Special Meetings: The MPI Workshop

National Science Foundation

*2008 - 2011*

This proposal is a collaborative effort amongst researchers at Rensselaer Polytechnic Institute, the University of Delaware, and Worcester Polytechnic Institute to expand and strengthen the Mathematical Problems in Industry (MPI) Workshop, an annual meeting run each June in cooperation with the Graduate Student Mathematical Modeling (GSMM) Camp. The MPI Workshop is a five-day meeting which attracts leading applied mathematicians, scientists and engineers from industry, universities and national laboratories. The focus of the Workshop is a set of problems brought by contributing participants from industry. These problems span a wide range of application areas, often in fluid and solid mechanics but also in mathematical biology, data analysis, and mathematical finance, among others. Work on the problems is done in vertically integrated teams consisting of the representatives from industry, senior and junior faculty, postdocs and graduate students. The main objective of the Workshop is to provide links between mathematicians at universities and scientists and engineers from industry for the mutual benefit of both sides. Learn more about the grant, Professor Fehribach or Professor Vernescu.

## Fehribach, J.

### Vector Spaces and Kirchhoff Graphs

National Science Foundation

*2007 - 2009 *

This project concerns chemical, electrochemical, and biochemical reaction networks, reaction routes (reaction pathways) through these networks, and the depiction of these networks using Kirchhoff graphs. This work will build on a vector space approach to reaction routes to establish that one or more Kirchhoff graphs exist for any given reaction network and to develop a method for constructing these graphs. Neither of these results is obvious for arbitrary reaction networks. Learn more about the grant and Professor Fehribach.

## Gobert, J., Co-PIs: Kim, R., Heffernan, N., Ruiz, C.

### ASSISTments Meets Inquiry

National Science Foundation

*2007 - 2009 *

The Worcester Polytechnic Institute (WPI) five-year proposal addresses middle school students need to learn science more deeply through the improvement of inquiry science assessment. An outgrowth of the WPI-developed ASSISTments Systems for Math, and REC-funded Modeling Across the Curriculum, the proposal's main goal is to develop a rigorous, technology-based system for middle school standards-aligned assessment of inquiry skills (i.e., interpreting data, formulating hypotheses and predictions, conducting experiments, collecting data, mathematizing, and communicating and defending hypotheses) in six physical science content areas (i.e., Properties of Matter; Elements, Compounds, and Mixtures; Motion of Objects; Forms of Energy; and Heat Energy). Assessments will be aligned with the Massachusetts Curricular Framework and the National Science Education Standards. Learn more about the grant.

## Larsen, C.

### Damage and Fracture Evolution

National Science Foundation

*2008 - 2011*

The investigator develops and studies models for the evolution of elastic damage (i.e., regions of weakened elastic properties) and fracture. Difficulties come from, among other things, the irreversibility of these phenomena, and the fact that existing models are essentially variational (static). The general strategy is to first develop quasi-static models that are consistent with these variational principles, and prove existence and approximation results. Next, the goal is to extend these results to the fully dynamic setting. The first step has been essentially completed for damage and brittle fracture, and the investigator works on developing corresponding methods for cohesive fracture. Dynamic models and their analysis are major open problems for all these phenomena, and are the main focus of this project. Learn more about the grant and Professor Larsen.

## Sunar, B., Co-PI: Martin, W.

### Exploring Physical Functions for Lightweight and Robust Cryptography

National Science Foundation

*2008 - 2010*

Low cost devices, e.g. RFIDs, smartcards, sensor nodes, etc., are becoming crucial for building the next generation pervasive and ubiquitous heterogeneous networks. Given the massive volume, the per-unit manufacturing cost will play a key role in the adoption of these technologies. These devices are constrained in their available computational power and footprint. Yet, their cryptographic units, which tend to be among the most demanding components in the device architecture, need to be hardened against physical tampering. The project is exploiting hardware anomalies for cryptographic ends. Even as chip manufacturing progresses and becomes more precise, one can always expect slight variations along the production line which can be used to distinguish one physical device from any other manufactured on the same production line. The team is developing models, processes and hardware primitives which contribute to the goal of exploiting these individual fingerprints so that they can be used to efficiently identify devices and enable secure, tamper-resilient communication. This initiative is providing low-cost, tamper-resilient cryptography from physical functions, and thereby plays an enabling role in the adoption of a wide array of products and applications to the benefit of the national economy and national security. The results of this project include new physically unclonable function (PUF) constructions with a particular emphasis on constructions which naturally permit reduction to computationally difficult problems, PUF-enabled cryptographic building blocks (such as secure and efficient storage, tamper-resilient state machines, etc.) and PUF-enabled cryptographic primitives (e.g. authentication schemes, block ciphers, pseudo-random generators). Learn more about the grant and Professor Martin.

## Mosco, U., Co-PI: Vernescu, B.

### Transmission Problems and Large Surfaces

National Science Foundation

*2008 - 2011*

The investigator and his colleagues study mathematical problems with large surfaces and small volumes that arise in physics (e.g., highly fragmented electrical conductors, high voltage electric discharges, electrolytic deposition, diffusion-limited aggregation), chemistry (catalytic converters, surface chemistry), biology (cell membranes, vascular systems), engineering (hydraulic fracturing in oil wells, thin ramified fibers, elastic thin bodies, towers and bridges in open space). In all these problems a lower-dimensional physical body -- the "surface" -- intrudes and interacts with a full dimensional surrounding body -- the "volume." The volume is small, the surface is large, possibly fractal and with infinite area. In this project, the mathematics of fractal structures in space is focused on two fundamental issues: 1) second order transmission conditions for second order operators; 2) singular homogenization with fractal terms. New tools of analysis are developed, like Hoelder metrics and measure-valued Lagrangeans. Finite element numerical approximations provide the quantitative and flexible setting required by prospective applications such as those mentioned above. Learn more about the grant, Professor Mosco or Professor Vernescu.

## Tang, D., Co-PIs: Petruccelli, J., Walker, H.,

### Multi-Physics Modeling and Meshless Methods for Atherosclerotic Plaque Progression

National Science Foundation

*2006 - 2009*

Cardiovascular disease (CVD) is the leading cause of death in the developed world and is expected to become the leading cause of death worldwide by 2020. In the US alone, 36% of 45 year olds and 80% of those 75 and older have CVD (American Heart Association Statistics 2005). Atherosclerotic plaques may rupture without warning and cause acute cardiovascular syndromes such as heart attack and stroke. Many victims of the disease who are apparently healthy die suddenly without prior symptoms. Non-invasive screening and diagnostic methods are urgently needed to identify the victims early and avoid those tragic events. The objective of this project is to combine computational modeling, magnetic resonance imaging (MRI) and pathological analysis to simulate plaque progression and quantify critical blood flow and plaque stress/strain conditions under which plaque rupture is likely to occur. MRI and pathological analysis will be used to quantify human carotid plaque morphology and progression and to assess plaque vulnerability. For the first time, multi-year MRI patient-tracking data will be obtained to quantify human atherosclerotic plaque progression. MRI-based three-dimensional (3D) computational models with multi-component plaque structure and fluid-structure interactions (FSI) will be developed and solved by numerical methods based on the meshless local Petrov-Galerkin (MLPG) method to obtain critical flow and plaque stress/strain conditions, to identify suitable plaque rupture risk indicators for more accurate plaque assessment, and to simulate plaque progression for early prediction and diagnosis of related cardiovascular diseases. Learn more about the grant, Professor Tang, Professor Petruccelli, or Professor Walker.

## Tang, D., Co-PIs: Sotak, C., Hoffman, A., Woodard, PK.

### MRI-Based Computational Modeling for Carotid Plaque Rupture and Stroke

Department of Health and Human Services

*2004 - 2009 *

The objectives of this project are to integrate computational modeling, Magnetic Resonance Imaging (MRI) technology, ultrasound/Doppler technology (US), mechanical testing, and pathological analysis to perform quantitative mechanical analysis to atherosclerotic carotid plaques, to quantify critical blood flow and plaque stress/strain conditions under which plaque rupture is likely to occur, and to seek the potential that quantitative mechanical analysis can be integrated into state-of-the-art imaging technologies for better screening and diagnostic applications. Learn more about the grant and Professor Tang.

## Tang, D.

### Image-Based Computational Mechanical Analysis and Indexing for Cardiovascular Diseases From Invention to Commercialization

The Kalenian Award

*2008 - 2009 *

Learn more about the award and Professor Tang.

## Volkov, D.

### Reconstruction of faults from surface displacements

National Science Foundation

*2007 - 2010 *

The goals of the proposed research are (1) to measure surface displacements and use the measured surface displacements as data for the inverse problem of locating faults and portraying their geometry, and (2) to develop criteria for deciding whether a fault system is in its nucleation phase, which would suggest that an earthquake is imminent. First, the mathematical framework for a stability analysis of displacement fields near equilibrium will be developed, and the Green's tensor for the half-space fault eigenvalue problem will be derived. Next, using rigorous mathematical analysis, questions of convergence will be addressed as the depth of the medium increases, and convergence rates will be verified by high-order numerical schemes for hypersingular boundary integral equations. Finally, closed form recovery formulas hinging on the dominant term of the depth asymptotics will be derived and combined with a fast minimization method. The PI will infer robust algorithms for solving the fault inverse problem. Reconstruction methods will be tested first on synthetic data and then used on data coming from field measurements of surface dislocations. Learn more about the grant and Professor Volkov.

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