Research Grants

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.

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.

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.

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.

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.

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.

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.

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-α represent the dominant response, with levels reaching 40% in one patient. Analysis of IFN-γ 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.

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.

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.

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.

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.

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.