### Research Grants

#### NSF Grant 2016-2019: A novel paradigm for nonlinear convection models and large systems of particles

PI: Pierre-Emmanuel Jabin (CSCAMM/Mathematics, University of Maryland)

Convection or transport mechanisms are a critical feature of several phenomena in Physics and the Bio-Sciences. In particular the proposal focuses on compressible Fluid Mechanics and systems of many "particles". Compressible Fluid Mechanics includes a largeset of models in very diverse settings: Geophysical fluids with gravity in large scale fluids (Earth atmosphere), biological fluids (such as swimming bacteria) or ``exotic'' examples such as solar events or photon radiation. Systems of particles typicallyinvolve a very large number of coupled equations, one for each particle. Particles can here represent many different objects: In Physics they can represent ions and electrons in plasmas, or molecules in a fluid, or even galaxies in some cosmological models;in the Biosciences particles typically model micro-organisms (cells or bacteria); in Economics or Social Sciences, particles are individual ``agents''. Instabilities can develop in all those systems and may manifest as oscillations in the mass density of thefluid or collisions (or near collisions) between particles. The main challenge to reduce the complexity of such systems is to understand how and over which time scales the convection or transport (of mass in the fluid or of the particles) can amplify such oscillations.

#### NSF Grant 2016-2019: Agent-based Models, Nonlinear Transport and Social Hydrodynamics

PI: Eitan Tadmor (CSCAMM/Mathematics/IPST, University of Maryland)

Nature and human societies offer many examples of collective dynamics that tend to self-organize into large-scale patterns. The ultimate goal of this proposal is to study nonlinear transport equations which arise in diverse applications of collective dynamics, e.g., opinion dynamics, flocking hydrodynamics and chemotaxis. We focus our attention on the persistence of global features of such equations, including propagation of regularity and regularizing effects, self-organization and the emergence of patterns such as large-scale clustering into consensus, flocking and leaders.

To this end, we will pursue research with collaborators, postdocs and graduate students to be supported by this grant, in studying the following.

(i) We plan to study the ensemble of agent-based dynamics in order to predict how different rules of local engagement affect the large-time emergence of one or more clusters. (ii) We propose a new paradigm, in which communication takes place along the projections of those agents who are ``moving ahead'' and explore the dynamics which takes into account the ``tendency'' to move ahead and the emergences of leaders. (iii) It is known that smooth solutions of collective social hydrodynamics governed by nonlinear transport must flock. When does the smoothness of these equations persist? we will study the regularity of nonlinear transport equations in social hydrodynamics, and their dependence on critical thresholds in the space of initial configurations. (iv) Chemotaxis is a canonical example for group dynamics driven by (chemo-) attraction and (volume-filling) repulsion. When the population of bacteria is `over-crowded', there is a repulsion peak and leads to new incompressible model for chemotaxis. We will study the vanishing viscosity limit of such `peaked'' repulsion. (v) We will investigate nonlinear transport associated with social hydrodynamics driven by degenerate local Laplacians. To this end we plan to “import'' velocity-averaging techniques used to study the regularization of nonlinear conservation laws with degenerate diffusion.

The project provides a great educational experience through research for the graduate students and postdoctoral fellows involved.

**NSF Grant 2016-2021: CAREER: Inviscid Limits and Stability at High Reynolds Numbers**

PI: Jacob Bedrossian (CSCAMM/Mathematics, University of Maryland)

In many applications of fluid mechanics, such as those arising in atmosphere and ocean sciences, aerospace engineering, and high-energy or fusion plasma physics, understanding the dynamics of fluids and plasmas where dissipative forces (e.g., friction) are weak is crucial. For example, the stability of the layer of air over a wing, and the dissipation of energy nearby, can have major implications for the aircraft, such as drastically changing the fuel efficiency. The focus of the project is to better understand dissipation and stability of equilibrium configurations and related problems in this regime using mathematical analysis. The project helps lay the foundation for a wider mathematical theory on mixing, dissipation, and stability in fluid mechanics. As it requires both pure and applied mathematical innovations, it also presents an excellent opportunity to train new, versatile researchers who are fluent in both scientific applications and sophisticated mathematical analysis. Graduate students are included in the work of the project.

The project aims to further elucidate basic questions of nonlinear stability, direct cascades, and the dissipation of enstrophy or energy at small scales in fluids and to expand the rigorous mathematical theory for understanding these phenomena. The investigator and his collaborators focus on fundamental questions that will have broad impact due to intrinsic interest, as will the new tools developed to solve them. Three general areas are studied: (A) analysis tools for understanding enhanced dissipation and transient unmixing; (B) linear and nonlinear stochastically forced problems in order to understand statistically stationary direct cascades in mathematically accessible settings; (C) estimating the subcritical transition thresholds and understanding instabilities for laminar flows, such as pipe flow, in infinite and finite regularity. Finally, natural extensions to hypoelliptic problems, such as collisionless limits in kinetic theory, may also be considered. The work is primarily mathematical analysis; however, computer experiments may be performed in order to provide preliminary insights and to provide accessible training opportunities for undergraduates in applied mathematics. Graduate students are included in the research activities.

**NSF Grant 2010-2015: Nonlinear Transport, Degenerate Diffusion, Critical Regularity and Self-organized Dynamics**

PI: Eitan Tadmor (CSCAMM/Mathematics/IPST, University of Maryland)

The ultimate goal of this project is to construct, analyze and simulate time-dependent problems which are governed by nonlinear Partial Differential Equations (PDEs) and develop related novel computational schemes. The underlying equations involve nonlinear transport models, self-organized dynamics, and possibly different small scale decompositions into particle dynamics, kinetic distributions, or intensity of pixels; they arise in diverse applications, including fluid dynamics, collective behavioral sciences and image decomposition and de-noising. We will focus on the unifying mathematical content of the equations, using a synergy of modern analytical tools and novel computational algorithms, to study the persistence of global features in these equations.

We will investigate, in particular, the following five prototype nonlinear time-dependent problems. (i) Critical regularity in Eulerian dynamics: we will use spectral dynamics to investigate a new framework for vanishing viscosity solutions of the pressure-less Euler equations and global regularity of Euler-Poisson equations subject to sub-critical initial data; (ii) Entropy stability and well-balanced schemes; (iii) Self-organized dynamics: we will study the long-time behavior of models driven by velocity-alignment and address two interrelated issues. When does flocking occur with local interactions, depending on the connectivity of the underlying graph, and how is it realized in hydrodynamic models of flocking? We will also explore new models of self-organized dynamics in which inter-particle communication is scaled by their relative distance. (iv) Regularizing effects in quasi-linear transport-diffusion equations; and (v) Integro-differential equations for multi-scale decomposition of images: we will study the localization properties of new multi-scale integro-differential equations for image de-noising and de-blurring.

The project provides a great educational experience through research for the graduate students and postdoctoral fellows involved.

**NSF Grant 2009-2010: Algorithms, Scientific Computing, and Numerical Studies in Classical and Quantum General Relativity**

PI: Manuel Tiglio, (CSCAMM/Physics, University of Maryland)

A comprehensive set of new algorithmic and scientific computing tools to numerically investigate the modeling of binary black hole collisions with higher efficiency will be developed. Symbolic manipulation tools to automatize and obtain higher order post-Newtonian expansions of the Einstein equations using techniques from Quantum Field Theory will be developed. The use of heterogeneous computing and, more specifically, general purpose Graphics Processing Units (GPUs) in numerical relativity will be explored.

The tools developed in this project aim to explore the physics of binary black hole collisions, expected to be one of the main sources of gravitational waves to be detected by ground- and space-based interferometers such as LIGO and LISA. The parameter space is so large that new computing paradigms are needed to explore it. The results of this project should be of interest in the broad areas of general relativity, gravitational waves, symbolic computing algebra, field theory, and high performance computing.

**NSF Grant 2008-2011: Numerical simulations of Einstein's equations**

PI: Manuel Tiglio, (CSCAMM/Physics, University of Maryland)

This award supports research in numerical simulations of binary compact objects (black holes and neutron stars) accurate and efficient enough to be used for parameter estimation of sources of gravitational wave signals to be measured by detectors such as LIGO.These simulations will also be used to calibrate and drive the construction of faithful semi-analytical gravitational wave templates for parameter estimation in (e.g.) LIGO data analysis. The main numerical techniques to be used for both black hole and neutron star simulations will be high order and pseudo-spectral multi-domain methods. Special effort will be spent on methods for extending binary simulations of orbiting compact objects using these techniques all the way to the merger and ring-down regimes.

This project will address current obstacles in the ability to study with very high accuracy the gravitational waves emitted by coalescing binary compact objects, by combining the application and development of novel analytical and numerical techniques. The final goal is to develop both numerical and semi-analytical models that can be used to infer the sources of gravitational waves from their detected signals. The results of this project will have a broad impact on areas of computational relativity, modeling and detection of gravitational waves, and will likely drive the development of some new numerical techniques, which should be useful in rather general fields of computational physics involving time-dependent problems.

**NVIDIA professor partnership award, 2009-2010**

PI: Manuel Tiglio, (CSCAMM/Physics, University of Maryland)

**NSF Grant 2009-2014: CAREER: Thermodynamic and Kinetic Approaches for Epitaxial Material Systems**

PI: Dionisios Margetis (CSCAMM/Mathematics/IPST, University of Maryland)

The investigator focuses on research and education goals at the crossroads of mathematics with materials science and physics.
This research links macroscopic principles for crystal interfacial phenomena to microscale processes. The physical mechanisms are described by discrete "particle" schemes for atomic line defects. However, mesoscopic and macroscopic consequences are best understood via continuous models. Linkages between models are sought by kinetic theory, partial differential equations, and stochastic analysis. The education part includes the training of a graduate student, development of innovative courses in applied mathematics, activities for mathematics awareness and outreach, authorship of a book on integral equations, and organization of working group seminars.
This Career award is supported by the MPS Division of Mathematical Sciences and by the MPS Division of Materials Research.

**ONR Grant 2009-2013: Multi-Model Ensemble Approaches to Data Assimilation Using the 4D-Local Ensemble Transform Kalman Filter**

PI: Kayo Ide (CSCAMM/AOSC/IPST, University of Maryland)

Uncertainties in the numerical prediction using a computational model of a physical system arise from two primary sources: i) errors within the model itself; and ii) imperfect knowledge of (a) the initial conditions to start the model and (b) boundary conditions and the forcing that is required to run the model. One way to examine these uncertainties is the multi-model approach, i.e., to compare results from multiple models. However, the multi-model approach cannot completely address either (i) or (ii) due to lack of knowledge of the real state. Another way is to compare the results with the "observations" that sample the real state. However, the observations introduce another source of uncertainties, i.e., iii) imperfect knowledge and/or improper assumptions within the observations including sampling error.

The ultimate objective of this project is to develop a framework for two purposes: one is the maximum reduction of the reducible uncertainties and the other is the diagnosis of the irreducible uncertainties in the numerical prediction. We use data assimilation
combined with multi-model. The Local Ensemble Transform Kalman Filter
(LETKF) is our choice of the data assimilation method. Because it uses an ensemble to estimate the state uncertainties, it offers a perfect vehicle for the multi-model approach. In addition, a number of advantageous algorithms have been developed for the quantification and the reduction of uncertainties of all three types (i) - (iii), including both model-bias correction and observation-bias correction.
Bias corrections, in a sense, transform part of the irreducible uncertainties (by other methods) into the reducible uncertainties. By integrating it into the multi-model approach, the LETKF will gain a powerful additional advantage: the combination of the ensemble weights and the calibration of the model will lead to improved performance over a single model LETKF. The resulting uncertainties are irreducible by the multi-model LETKF. To develop our framework, we will use Observing System Simulation Experiments.

**ONR Grant 2009-2011: Computational Methods for the Shallow-Water Equations and Related Problem****
**PI: Eitan Tadmor (CSCAMM/Mathematics/IPST, University of Maryland)

The overall goal of this proposal is to construct, analyze and implement novel high-resolution, "faithful" approximations of shallow-water (SW) and related rotational flows.

We will continue the development of high-resolution central schemes which offer a simple and versatile approach for computing approximate solutions of non-linear systems of hyperbolic conservation laws and related PDEs. In particular, we will address the issues of

*entropy stability* and

*constrained transport* which arise, for example, with the energy-conserving SW equations with variable topography. We will continue the recent development of central

*Discontinuous Galerkin* methods and will implement new

*spectral viscosity* approximations to these problems. These developments will be linked to the study of

*regularizing effect* of rotational forces.

**NSF Grant 2008-2011: Nonlinear Signal Processing and
Wireless Communications using Frames and Operator Theory**

PI: Radu Balan
(CSCAMM/Mathematics, University of Maryland)

This proposal develops a new mathematical framework for
nonlinear signal processing and wireless communication
channels. The nonlinear signal processing problem
addressed here is signal reconstruction from the
magnitude of a redundant linear representation. When the
linear redundant representation is associated to a group
representation (such as the Weyl-Heisenberg grgoup), the
relevant Hilbert-Schmidt operators inherit this
invariance property. Thus a fast (nonlinear)
reconstruction algorithm seems possible. A wireless
communication channel is modeled as a linear operator
that describes how transmit signals propagate to a
receiver. For ultrawide band (UWB) signals, the Doppler
effect no longer can be modeled as a frequency shift.
Instead it is captured as a time dilation operator. A
continuous superposition of time-scale shifts is used to
model a UWB communication channel, and consequences to
pseudo-differential operator theory are analyzed.

The solutions to these problems have a strong impact in
the strategic area of information technology. Important
applications such as signal processing, X-ray
crystallography, and quantum computing are impacted by
solutions to the first problem. High-impact applications
related to the second problem include UWB through-wall
imaging systems, higher throughput 802.15 Wireless
Personal Area Networks, and wireless sensor networks.

NIH R01, 2008-2012: Interplay Between Cancer and
Immune Cells on Targeted Therapy

PI: Peter Lee, MD (Hematology,
Stanford University)

Co-PI: Doron Levy, PhD (CSCAMM/Mathematics, University of Maryland)

**
**

The role of the host immune response in relation to
current therapies in controlling cancer remains
unclear. We hypothesize that novel molecular targeted
therapies, such as imatinib for chronic myelogenous
leukemia (CML), may render leukemic cells immunogenic as
patients enter remission. This relates to the relative
selectivity of imatinib over standard chemotherapy for
leukmic cells, thus sparing normal immune cells. In
addition, cancer has potent immunsuppressive activity.
As leukemic cell population decreases with imatinib,
immune function may be restored while cancer antigens
are still present at significant levels, thus leading to
maintain disease control (remission), and may be further
expanded to eliminate residual leukemic cells for a
durable cure. In preliminary studies, we used a novel
combined experimental and modeling approach to address
the immune response to leukemia. We showed that the
majority of CML patients develop robust anti-leukemia T
cell responses upon remission on imatinib.
Intriguingly, CD4+ T cells producing TNF-
represent
the dominant response, with levels reaching 40% in one
patient. Analysis of IFN-
production
by CD8+ T cells alone, as commonly done today, would not
reveal the entire anti-leukemia T cell response. We
studied the dynamics of these responses and showed that
they peak around the time patients enter cytogenetic
remission, but wane to undetectable levels shortly
thereafter. We utilized mathematical modeling to gain
insights into the dynamics of leukemia and the immune
response, and the complex interplay between these
populations. Our results suggest that anti-leukemia T
cell responses may contribute to the maintenance of
remission under imatinib therapy. The goal of this
proposal is to expand on these findings by conducting
and integrating additional experiments and mathematical
modeling. We will analyze in detail 10 patients per
year over 5 years, for a total of 50 patients using an
array of novel immunological techniques (Aim 1). We
will develop new mathematical models of the interplay
between leukemia and immune cells. These models will
extend our preliminary findings to independently
consider CD4 T cells, CD8 T cells, NK cells, B cell
response (antibodies), and different cytokine patterns
(Aim 2). We will use the new experimental data for
evaluating patient-dependent model parameters, and study
whether the magnitude, timing, and dynamics of the
immune response can be correlated with the clinical
outcome. Lastly, we will develop novel therapeutic
strategies in silico (Aim 3). Using our mathematical
models and real patient parameters, we will study cancer
vaccines, structured treatment interruptions, and
mini-transplants. If successful, this work will provide
important insights into the role of the host immune
response in CML under targeted therapy, and may lead to
novel treatment strategies combining therapy with
immunotherapy.

**
NSF Focus Research Group Grant 2008-2011:
Collaborative Research on Kinetic Description of
Multiscale Phenomena: Modeling, Theory and Computation**

PI: Eitan Tadmor
(CSCAMM/Mathematics/IPST, University of Maryland)

Co-PI: Thanos Tzavaras (Mathematics, University of
Maryland)

Kinetic equations play a central role in many
areas of mathematical physics, from micro- and nano-physics
to continuum mechanics. They are an indispensable tool
in the mathematical description of applications in
physical and social sciences, from semi-conductors,
polymers and plasma to traffic networking and swarming.
The ultimate goal of this proposal is to develop novel
analytical and numerical methods based on kinetic
descriptions of complex phenomena with multiple scales
and with a wide range of applications. The objective is
to achieve a better understanding of problems which are
in the forefront of current research and to contribute
to the solution of long standing problems by synergetic
collaboration of theory, modeling and numerics.

To this end, this Focus Research Group (FRG) will
provide a platform, led by leading researchers from
Universities of Maryland, Brown, Iowa State,
Wisconsin-Madison, Arizona State, Austin-Texas and
Toulouse, France, who will merge their expertise in the
construction, analysis and implementation of kinetic
descriptions for a selected suite of problems with
crossing scales from quantum and micro scales to the
macro scales. Topics to be discussed include:

● Kinetic descriptions of microscopic and quantum
phenomena

● Kinetic descriptions of macroscopic phenomena

and finally, as a recent novel example for the kinetic
methodology we will use kinetic descriptions to study

● hyperbolic flows for complex supply chains.

The theoretical and modeling aspects of this research
program, on both microscopic and macroscopic scales,
will be integrated with kinetic based numerical methods
for capturing “smaller scales phenomena". The rationale
behind this proposal is a timely effort to address
several important issues in modern applied mathematics.
Kinetic theories are not new. Yet, there has been many
major developments in kinetic modeling, kinetic theories
and related numerical methods, with the potential for a
considerable impact on emerging new fields in physical
and social sciences.

The proposed effort will significantly strengthen the
leading role that the US researchers can play in
pursuing cutting-edge research and training a new
generation of applied mathematicians in this important
field.

We expect this project to contribute to the development
of scientific workforce by advanced training for
doctoral and postdoctoral researchers and by providing a
platform for interdisciplinary interactions with
researchers from related disciplines. Interactions -
internal and external, should be maintained through
synergetic cooperations which will come to fruition
during the three annual workshops planned to be held in
Maryland (YEAR 1), France and Brown (YEAR 2) and
Wisconsin (YEAR 3). In this context, we plan to hold
(inter-)national meetings, as part of the series of
interdisciplinary focused workshops organized by the
Center for Scientific Computation and Mathematical
Modeling (CSCAMM) in the University of Maryland. We also
expect to gain from collaboration with the DOE Center
for Multiscale Plasma Dynamics housed in CSCAMM, the DOE
Ames Laboratory in Iowa State University and the
Institute for Computational Engineering and Sciences
(ICES) in UT Austin.

NSF Grant 2007-2010: Regularity and Critical
Thresholds in Nonlinear Transport-Diffusion Equations

PI: Eitan Tadmor
(CSCAMM/Mathematics/IPST, University of Maryland)

The ultimate goal of this proposal is to study the
persistence of global features in nonlinear
transport-diffusion equations which arise in a wide
variety of applications. Examples include nonlinear
conservation laws with degenerate diffusion which model
sedimentation, traffic flows and data driven
applications in image processing, the ubiquitous
Eulerian dynamics governing a range of phenomena from
the small scale of semi-conductors through the largest
scale of stars formation, or chemotaxis models found in
biological applications. We focus our attention on the
unifying mathematical content of the underlying
transport-diffusion equations. Of primary interest are
problems with critical regularity properties which hinge
on a borderline balance between the nonlinear convection
mechanisms, the nonlinear diffusion processes and the
possibly nonlinear forcing driving such problems.

We plan to use modern mathematical tools complemented by
novel computational simulations to examine the phenomena
of regularizing effects, critical thresholds, time
decay, entropy stability, scaling and more, where the
following questions will be addressed in the context of
(i) Transport-diffusion equations: how do diffusion and
entropy dissipation dictate the regularizing effect in
such equations? (ii) Eulerian dynamics: how does the
competition between rotation and pressure forcing
determine the overall stability? (iii) Chemotaxis and
related bio-related transport-diffusion models: how
should we interpret the solutions beyond their critical
time? (iv) Hierarchical decompositions of images: how
does the inverse scale space can be adapted to the data
for effective image processing?

The principle investigator together with his
collaborators plan to pursue the development of new
analytical and computational tools to explore
transport-diffusion models, which are expected to impact
our overall understanding of the dynamics of such
realistic models in a variety of applications.

DOE Grant 2004-2009: Center for Multiscale Plasma Dynamics

PI: William Dorland
(CSCAMM/Physics, University of
Maryland)

The Center
for Multiscale Plasma dynamics is a joint UCLA/Maryland fusion science center
focused on the interaction of microscale and macroscale dynamics in key plasma
physics problems. Foremost among these problems are the sawtooth crash, the
growth of neoclassical magnetic islands, and the formation and collapse of
transport barriers -- all of central importance to the fusion program. Each
involves large scale flows and magnetic fields tightly coupled to the small
scale, kinetic dynamics of turbulence, particle acceleration, and energy
cascade. The interaction between these vastly disparate scales controls the
evolution of the system. The enormous range of temporal and spatial scales
associated with these problems renders direct simulation intractable, even in
computations that use the largest existing parallel computers. Powerful new
multiscale algorithms
are emerging from the applied
mathematics and engineering communities. A central mission of the center is
to adapt, modify and extend these ideas to model multiscale plasma dynamics.
This mission puts the center in one of the most active and rapidly advancing
areas of computational research.

Activities funded by the Center include a post-doctoral fellowship program, a graduate
student fellowship program, contributions to targeted experimental programs,
advanced courses (at the senior graduate student or post-doctoral level,
convened weekly in video
conference), and an annual Winter School.

**NSF Grant 2004-2007: Critical Regularity Phenomena in Nonlinear Balance Laws**

PI: Eitan Tadmor
(CSCAMM/Mathematics/IPST,
University of Maryland)

The overall goal of this project is the study of critical regularity associated with different nonlinear balance laws.
We focus on borderline cases where intrinsic features of the solutions such as smoothness vs. generic finite time
breakdown, time decay, etc., hinge on a delicate balance between nonlinear convection and a variety of (possibly
nonlinear) forcing mechanisms. These are precisely the problems which are of great interest in various
applications. Often this balance is maintained by global invariants of the flow. These include spectral
invariants, which in turn led to critical threshold phenomena, or borderline invariant regularity spaces.

A main focal topic of this project are balance laws governed by Eulerian dynamics. A precise spectral dynamics
quantified the decisive role of the eigenvalues of the corresponding strain matrix. Here we seek extensions to
include more realistic models driven by global and non-isotropic forcing.
What happens when global pressure terms are being added? what can be said about the critical threshold of different
systems augmented with different potentials? More realistic 3D extensions are sought; here the major issue is: what
quantity(-ies) play the role of the 2D spectral gap?

The proposed research is expected to identify new (spectral) invariants which will have a direct impact on the construction of
more faithful simulations for such models.

**Living With a Star (LWS) NASA Program**

**NASA grant 2002-2005: Integrated Numerical Simulation of Solar Terrestrial Environment for the LWS**

PI: Allen Sussman (Computer Science, University of
Maryland)

Co-PI: Jim Drake (Physics/IPST, University of Maryland)

The aim of this project is to model the physics of the entire Sun to Earth system - space weather. With funding from NASA and NSF,
a group of space and computer scientists at Maryland and several other sites including Boston U., Dartmouth College, Rice U.
and the National Center for Atmospheric Research (NCAR), are building simulation-based models that will be used to predict
the effects of solar radiation on the Earth's local environment. This is particularly important during solar events
(storms), when the Sun's radiation can have a severe impact on man-made systems, such as satellites and electrical power grids.