Faculty members from the Integrated Applied Mathematics Program participate in an array of research areas, listed below. If interested in learning more about a particular research area, please feel free to contact one of the listed faculty members in the specific area of interest.
UNH researchers are devising state-of-the-art mathematical tools for analyzing strongly nonlinear PDEs whose solutions exhibit the phenomenon of emergence: the arising of novel and coherent structures, patterns and properties during the process of self-organization in high-dimensional nonlinear dynamical systems. A primary achievement has been the development of new dynamical systems techniques for extracting equilibrium, traveling wave, and periodic orbit solutions of turbulent shear flows (with over 100,000 degrees of freedom!), which UNH researchers have shown organize the flow and give rise to coherent structures. The unstable periodic orbits, in particular, form a nonlinear basis for analyzing statistics of chaotic systems, using a methodology known as periodic orbit theory. This formalism has yielded promising results in the context of wall-bounded shear flow turbulence and is pointing the way toward feasible control algorithms for reducing turbulent drag over aerodynamic surfaces and in pipelines. A complementary approach, also being pioneered by UNH researchers, involves the construction of novel lower-order ODE models of turbulent flows using a rigorous mathematical technique for bounding global transport quantities known as upper bound theory. Unlike popular but empirical approaches based, e.g., on the Proper Orthogonal Decomposition (POD), this new methodology does not require extensive data sets from DNS or experiments and, thus, is fully predictive. Finally, UNH researchers are employing multiscale asymptotic analysis to derive asymptotically-exact reduced PDE models of constrained turbulent flows by exploiting the anisotropy of emergent flow structures (e.g. elongated streamwise vortices in turbulent boundary layers over solid surfaces); interaction equations for wave turbulence in plasmas; and coupled systems of PDEs that systematically retain the dominant multiscale interactions, as for ocean surface boundary layer interactions with "submesoscale'' flows.
UNH has a strong research focus on methods for numerical solution of PDEs as they occur in various applied science contexts, in particular in computational fluid dynamics, fluid descriptions of plasmas, materials physics, numerical relativity, and biophysics. These systems are often characterized by a large range of spatial and temporal scales, making these problems computationally very expensive. UNH researchers are involved in a number of active research areas aimed at making such computations more feasible. For example, in the area of magnetic reconnection or the physics of the magnetosphere, the flows are quite smooth in a large fraction of the domain, while strong thin current sheets occur in localized regions. Block-structured adaptive mesh refinement (AMR) is a technique in which the grid resolution is automatically adapted to the spatial scales of the solution, providing high resolution only where needed and hence reducing the degrees of freedom of the global problem. Similarly, implicit time integrators filter fast time scales associated with high-frequency waves that are not of direct physical interest, while still properly modeling the dynamics on "slow manifolds". Newton direct-solver and Newton-Krylov methods with physics-based preconditioning are being used to solve the algebraic systems of equations resulting from the implicit time discretization. Finally, in concert with these algorithmic advances, efficient implementation of the numerical algorithms on modern computer systems is crucial. In particular, heterogeneous architectures like the Cell processor and GPGPUs (general purpose graphics processing units) hold the promise of increasing computing power by an order of magnitude or more, but writing efficient code for modern processors is a rather involved undertaking. We have developed an automated code generation system that takes stencil computations in symbolic form as input, and automatically generates optimized code for various modern processors, shielding the domain scientist from having to learn the specifics of a given hardware architecture and programming model.
Research has shown that unstable periodic behavior often can provide a framework for understanding the dynamics of nonlinear ODEs. Once they have been identified, manipulation of these unstable periodic orbits provides novel opportunities for controlling nonlinear systems. In the context of nonlinear ODEs, it has been shown that a simple control scheme can be used to robustly stabilize thousands of unstable periodic orbits (called "cupolets") in a wide variety of systems, including, e.g., the double scroll oscillator, the Lorenz system, and unimodal maps. Further, by studying the control mechanism and the dynamics of the periodic orbits, it has been possible to develop an algorithm that uses graph theory to find the most efficient connecting route from one periodic orbit to another, using only periodic orbits as intermediaries (and hence accessible through the control mechanism). These dynamical systems theoretical developments provide a framework against which it has been possible to analyze the dynamics of experimental data in a wide range of areas. A key tool in this process has been the introduction of super-resolution techniques for frequency analysis. These tools have played a fundamental role in the construction of media compression algorithms where, e.g., the nonstationary data in music or audio signals are analyzed and then mapped onto a suitable basis of unstable periodic orbits from a chaotic system. Similarly, the tools have proven useful in analyzing turbulent noise in flow over an airplane wing, where the super-resolution techniques were used to select a low dimensional number of modes to model the data, and these modes could be updated as the turbulent noise evolved. The data thus drives a reduced model. This empirical approach can be compared to theoretical techniques such as the upper-bound/Galerkin methods described above, and can provide a real-world test that indicates what modes must be retained in a reduced model and what degrees of freedom must be maintained in order to allow the modes to evolve in keeping with the actual physical system.
Machine learning studies algorithms that automatically learn from data. Most machine learning algorithms combine mathematical modeling with ideas from statistics coupled with mathematical optimization. The learning typically happens by computing a model that minimizes a well-chosen loss function. The selection of an appropriate loss function can be informed by a better understanding of the underlying system dynamics, which may include the numerical or analytic solutions to differential equations such as those being studied by other IAM research groups. To minimize this loss function, one usually turns to mathematical optimization algorithms like gradient descent, Newton's method, or the simplex method. Machine learning research at UNH advances techniques for learning reliably even from small and noisy datasets. To achieve reliability, we rely on ideas from robust optimization and compute the best models for the plausible worst-case realization of data errors.
Fluid dynamics research is a particular area of strength at UNH, where the overarching philosophy is that the best approach to applications associated with complex fluid dynamical problems is through fundamental understanding of the underlying physical processes. This understanding is being pursued using a variety of applied mathematical tools including multiscale asymptotic analysis, nonlinear hydrodynamic stability theory, state-of-the-art dynamical systems algorithms, and high-performance numerical simulations. Complementary laboratory experiments, also being performed at UNH, and field studies serve to inform these applied mathematical investigations. Presently, UNH researchers are studying a diverse array of flow phenomena, including:
- Coherent structures in shear and convective turbulence
- Nonlinear waves and instabilities in stratified, rotating flows and in aeroelastic flutter of airplane wings
- Transport and mixing in geophysical and environmental (e.g. porous media) flows, including two-fluid/phase and non-Newtonian flows
- Biofluid dynamics including spontaneous capillary blood flow oscillations and pulmonary alveolar stability
- Thermo-acoustic streaming in energy systems
Most of the visible matter in the universe is in the form of plasma - high temperature material containing an admixture of ionized atoms and free electrons. In some ways, plasmas behave like fluids or gases, responding to pressure forces for example. However, unlike gases or fluids, the charged particles within a plasma produce electric and magnetic fields and are in turn strongly influenced by these fields. This additional layer of complexity leads to a fascinating array of behaviors and processes, many of which are crucial to the understanding of astrophysical systems. The Space Plasma Theory Group at UNH focuses on the physics of plasmas in the solar atmosphere, the interplanetary medium, and Earth's magnetosphere using a combination of high-performance computing, analytical theory, and analysis of spacecraft measurements and remote observations. Areas of particular interest include: i. Magnetic reconnection, ii. Plasma turbulence, iii. Solar flares, iv. The solar wind.
Additional information may be found at the Center for Integrated Computation and Analysis of Reconnection and Turbulence.
The Numerical Relativity group at UNH focuses on the study of colliding compact objects (black holes, neutron stars). Merging compact objects are powerful sources of gravitational waves and electromagnetic signals (gamma-ray bursts, kilonovae), and are one of the most likely sources of production of about half of the elements heavier than iron (e.g. gold, platinum). Currently, these mergers can only be studied through complex 3D simulations including general relativity, magnetohydrodynamics, and relativistic neutrino radiation transport. The UNH group works on the development of numerical methods for the study of these merger events, and on the modeling of the observable signals that they power.
Faculty Expertise: Germaschewski.
Magnetospheric research at UNH centers on the development of global numerical models to study and predict the interaction of the solar wind with the magnetosphere. The Open Geospace General Circulation Model (OpenGGCM) is one of the worldwide leading magnetosphere models, which is now a collaboration between UNH, Rice University, Colorado University, NASA Goddard Space Flight Center, and the National center for Atmospheric Research (NCAR) at Boulder, lead by UNH. At its core, the OpenGGCM is a magnetohydrodynamic (MHD) model of the outer parts of Earth's magnetosphere, coupled with models of the ionosphere / thermosphere system, the inner magnetosphere (ring current), and the radiation belts. It has been a community model available for runs on demand by the scientific community at the Community Coordinated Modeling Center (CCMC, NASA Goddard Space Flight Center) since 2002. Although the OpenGGCM has been in the making for some 20 years, it is continually further improved for efficiency and physical realism. Besides addressing fundamental problems in magnetospheric physics, we are porting the model to IBM/Cell and GPU based architectures, work on more efficient input/output strategies, adaptive mesh refinement (AMR), less diffusive/dispersive numerical algorithms, and more efficient visualization techniques. Several Ph.D. theses have been based on work with the OpenGGCM. More details about the model can be found on the OpenGGCM page and at the CCMC page. Some of the latest results can be found on Raeder's home page
Encompassing an immense range of temporal and spatial scales, the fluid dynamics of the world's air and water play a pivotal role in weather and climate, the transport and dispersion of pollutants (e.g. spilled crude oil and smokestack plumes) and other chemicals, and the distribution and proliferation of biota. Over sufficiently large scales, ocean dynamics are constrained by the Earth's rotation and by the evolving interior density stratification, giving rise to a complex mosaic of inertia--gravity waves, intense jets and coherent vortices. A central challenge in physical oceanography is to understand the two-way coupling between these large-scale, 2D features and small-scale, fully 3D processes including Ekman boundary layer instabilities, surface and internal wave propagation, Langmuir circulation and thermal convection. UNH researchers are investigating these issues by studying canonical problems, aimed at revealing the fundamental physical mechanisms involved, using novel multiscale asymptotic and numerical methods. As in the ocean, transport and mixing in atmospheric flows involves a complex interplay between emergent nonlinear wave motions and quasi-coherent eddies, fronts, and filaments. Two key parameterization challenges facing modern computational weather and climate models are the treatment of internal gravity wave packets (i.e. buoyancy driven waves that propagate within the interior of the oceans and atmosphere) and of multiscale convective storm systems, which often excite these internal waves. UNH faculty are using advanced analytical and computational techniques, validated by in situ balloon measurements and remotely sensed lidar data, to perform detailed process studies of these phenomena with an eye toward development of improved parameterizations.
Research in the area of multi-scale mechanics involves the use of advanced theoretical and computational methods to investigate the elastic, plastic, and fracture behavior of manufactured and naturally-occurring materials over a wide range of length scales. Current areas of theoretical focus include the development of continuum mathematical models of the wrinkling behavior of wet soft matter arising in biology and of the deformation of glaciers, ice sheets, and sea ice in geomechanics. Multiscale asymptotic and variational methods are used to analyze the resulting models and to elucidate the mechanisms controlling deformation. A particular area of focus of the computational multi-scale mechanics research group is on the prediction of the behavior of manufactured materials via the development of advanced numerical tools that integrate multi-scale material models and through the use of novel high-performance computing (including GPU) platforms. The aim is to obtain a quantitative understanding of material behavior under complex loading conditions, to develop new constitutive material models, and to inform materials design for a wide variety of industrial applications.
Faculty Expertise: Vashisth.
Biophysics is the study of biological systems via theories and methods rooted in physics, especially statistical and molecular physics. Biophysical studies at the atomic scale are inherently challenging due to the large number of degrees of freedom in biomolecules (proteins, nucleic acids DNA/RNA, and lipids). To overcome this challenge, biomolecular dynamics are often explored by combining statistical physics methods with high performance computing resources to generate dynamical trajectories of atomic motions, most commonly using molecular dynamics (MD) simulations. However, conventional MD simulations of large biomolecular systems are often non-ergodic because of sampling of a limited part of the accessible conformational space. The theoretical and computational biophysics research at UNH is aimed at addressing spatiotemporal limitations of conventional simulation methods and explore new frontiers in methods development to enhance conformational sampling. These new and improved methods allow more detailed studies of biophysical phenomena and facilitate discovery of new biological materials and therapeutics. Graduate students from the IAM program interested in research activities in this area will learn about new computer simulation methods, programming paradigms, and applications to exciting problems in biophysics and chemical physics.