Research conducted by the Crunch Group focuses on the development of stochastic multiscale methods for physical and biological applications, specifically numerical algorithms, visualization methods and parallel software for continuum and atomistic simulations in biophysics, fluid and solid mechanics, biomedical modeling and related applications. The main approach to numerical discretization is based on spectral/hp element methods, on multi-element polynomial chaos, and on stochastic molecular dynamics (DPD). The group is directed by Prof. George Em Karniadakis. We invite you to visit both our DPD Club and Crunch FPDE Club websites. 

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Research in this area targets nonlinear differential equations and dynamical systems that arise in the physical, social and life sciences. Among the equations considered are finite-dimensional dynamical systems, reaction-diffusion systems, hyperbolic conservation laws, max-plus operators and differential delay equations. Questions that are addressed for these systems include the existence and stability of nonlinear waves and patterns, kinetic theory, phase transitions, domain coarsening, and statistical theories of turbulence, to name but a few.  Even though the techniques can vary widely from case to case, a unifying philosophy is the combination of applications and theory that is in the great Brown tradition in this area of mathematics, which is being fostered by close collaboration among the members of the group:

• Constantine Dafermos:  Hyperbolic conservation laws
• Hongjie Dong: Linear and nonlinear elliptic and parabolic PDEs, fluid equations
• Yan Guo:  Partial differential equations, kinetic theory and fluids
• John Mallet-Paret:  Dynamical systems; differential–delay, lattice, and reaction–diffusion equations
• Govind Menon: Kinetics of phase transitions and models of domain coarsening, integrable systems, random matrix theory, statistical theories of turbulence.
• Bjorn Sandstede: Applied dynamical systems, nonlinear waves and patterns

The Division of Applied Mathematics is an established leader in research on mathematical modeling for physical and physiological, multi-scale fluid mechanics. In general, fluid mechanics is an enabling science that describes dynamics over a wide spectrum of scales, ranging from the global scales of climate dynamics to the transport of suspended proteins through nano-pores. The focus of research in the Division has evolved as new challenges have emerged. This goes far beyond what has been seen as traditional fluid dynamics in the past and involves a broad scientific knowledge of biological and physio-chemical processes. The main activity of the fluids group is the theoretical description and numerical simulation of complex fluids, self-organization in active suspensions and biological processes relating to blood flow in the arterial tree, brain aneurysms, diseases of blood cells and bacterial locomotion. We have ongoing interests in multi-scale phenomena in turbulence, flow-structure interactions and multiphase flows. The fluids group maintains strong connections and collaborations with other faculty in Engineering and Physics and coordinates research seminars and graduate teaching. Interactions with groups in biological and biomedical research are expanding.  

• George Em Karniadakis: Multiscale modeling of biological systems, atomistic and mesoscopic simulation methods, flow-structure interactions, micro-transport and dynamic self-assembly
• M.R. Maxey: Active suspensions, bacterial swimming, multiphase flow, turbulence, particle-based simulation methods

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The Brown University pattern theory group is working with the belief that the world is complex, and to understand it, or a part of it, requires realistic representations of knowledge about it. We create such representations using a mathematical formalism, pattern theory, that is compositional in that the representations are built from simple primitives, combined into (often) complicated structures according to rules that can be deterministic or random. This is similar to the formation of molecules from atoms connected by various forms of bonds.  Pattern theory is transformational in that groups or semigroups of transformations operate on the primitives. These transformations express the invariances of the world we are looking at.  Pattern theory is variational in that it describes the variability of the phenomena observed in different applications in terms of probability measures that are used with a Bayesian interpretation. This leads to inferences that will be realized by computational algorithms. Our aim is to realize them through codes that can be executed on currently available hardware.  Please visit our seminars page.  Professors Stuart Geman and Matt Harrison are involved in this line of research.

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The research of the group includes a broad range of topics in probability theory and stochastic processes including stochastic partial differential equations, nonlinear filtering, measure-valued processes, deterministic and stochastic control theory, probabilistic approach to partial differential equations, stability and the qualitative theory of stochastic dynamical systems, theory of large deviations, Monte Carlo simulation, Gibbs measures and phase transitions, stochastic networks. There is also a major program in numerical methods for a variety of stochastic dynamical systems, including Markov chain approximations and spectral methods. 

• Hongjie Dong: stochastic processes, stochastic control theory, probabilistic approaches of PDEs
• Paul Dupuis: applied probability, control theory, large deviation, numerical methods, Monte Carlo.
• Kavita Ramanan: Probability theory and stochastic processes, reflected diffusions, Gibbs measures and phase transitions, large deviations, measure-valued processes, stochastic networks
• Boris Rozovsky: analysis of stochastic partial differential equations (SPDEs), numerical methods for SPDEs, stochastic fluid dynamics, nonlinear filtering for hidden Markov models 
• Hui Wang: stochastic optimization, large deviations, stochastic networks, fast simulation

The Division’s research in this area mostly focuses on developing efficient and stable numerical methods for approximating solutions to partial and stochastic differential equations that arise in a wide range of engineering and science applications. For decades now the scientific computing and numerical analysis group has been at the forefront in the development of higher-order methods such as spectral methods, spectral element methods, discontinuous Galerkin methods and WENO finite difference and finite volume methods. In more recent years research has also focused on uncertainty quantification, reduced order modeling, multi-scale methods, a-posteriori estimation, adaptivity and compatible discretizations.  Parallel computing and/or GPU processors are being used for large-scale computations.  Interdisciplinary collaborations have included projects with faculty from biology, geology, and engineering, to name a few.  

• Mark Ainsworth: Finite element methods: adaptivity, a-posteriori error control, Implementation using Bernstein- Bézier techniques, Dissipative and dispersive properties.
• Jerome Darbon:  Efficient algorithms for variational/Bayesian estimations and connections with Hamilton-Jacobi PDEs • Combinatorial optimization, especially network flows and graph-based algorithms • Stochastic sampling algorithms, especially perfect samplers • Algorithm/architecture co-design including low level implementation • Applications to denoising, geomorphology, remote sensing, biological, medical, historical, radar and inverse problems in imaging sciences
• Johnny Guzmán:  Discontinuous Galerkin methods, Mixed methods/compatible discretizations, Local error analysis 
• George Em Karniadakis:  Stochastic PDEs and stochastic multi-scale modeling, Fractional PDEs, Spectral element methods, Parallel computing
• Chi-Wang Shu:  High order methods for hyperbolic and convection dominated PDEs, Computational fluid dynamics

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The Statistical Molecular Biology Group at Brown University is led by Chip Lawrence, Professor of Applied Mathematics.  The group's research energies are focused on statistical inference in molecular biology, genomics, and paleo-climatology, most specifically on several different high-dimensional (High-D) discrete inference problems in sequence data.  These problems include genome arrangements, analysis of repeated sequences in genomes, and RNA editing and structure.  Data from repeated genome sequences are well suited for processing by implementing the probabilistic models developed by the group's members. Another facet of our research concerns the growing interest in drawing statistical inferences on the history of climate and its implications for modern climate change using data from ocean sediments and ice cores.  A wide network of collaboration has brought about extensive know-how and creativity to the group's research.  Among our many collaborators are several Brown faculty members: Professors Susan Gerbi and Robert Reenan from Molecular and Cellular Biology; Professor Case Dunn from Ecology and Evolutionary Biology; Professor Ben Raphael from Computer Science; Professor Tim Herbert in Earth, Environmental and Planetary Science; and Professor Lorraine Lisiecki from the Geology Department at UC Santa Barbara.

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