## Matthew Simpson : The mathematics of Hirschsprungs Disease

- Applied Math and Analysis ( 156 Views )Hirschsprung's Disease is a relatively common human congenital defect where the nervous system supporting our gut (the enteric nervous system) fails to develop properly. During embryonic development, the enteric nervous system forms as a result of neural crest cell invasion. Neural crest cells migrate from the hindbrain to the anal end of the gastrointestinal tract. This is one of the longest known cell migration paths, both spatially and temporally, occurring during vertebrate embryogenesis. Neural crest cell invasion is complicated by the simultaneous expansion of underlying tissues and the influence of several growth factors. This presentation outlines a combined experimental and mathematical approach used to investigate and deduce the mechanisms responsible for successful neural crest cell colonization. This approach enables previously hypothesized mechanisms for neural crest cell colonization of the gut tissues to be refuted and refined. The current experimental and mathematical results are focused on population-scale approaches. Further experimental details of cell-scale properties thought to play an important role will be presented. Preliminary discrete modelling results aiming to realize the cell-scale phenomena will also be discussed and outlined as future work.

## Gitta Kutyniok : Frames and Sparsity

- Applied Math and Analysis ( 131 Views )Frames are nowadays a standard methodology in applied mathematics, computer science, and engineering when redundant, yet stable expansions are required. Sparsity is a new paradigm in signal processing, which allows for significantly reduced measurements yet still highly accurate reconstruction. In this talk, we will focus on the main two links between these exciting, rapidly growing areas. Firstly, the redundancy of a frame promotes sparse expansions of signals, thereby strongly supporting sparse recovery methods such as Compressed Sensing. After providing an overview of sparsity methodologies, we will discuss new results on sparse recovery for structured signals, in particular, which are a composition of `distinct' components. Secondly, in very high dimensions, frame decompositions might be intractable in applications with limited computing budget. This problem can be addressed by requiring sparsity of the frame itself, and we will show how to derive optimally sparse frames. Finally, we will discuss how some of the presented results generalize to the novel notion of fusion frames, which was introduced a few years ago for modeling distributed processing applications.

## Gadi Fibich : Aggregate Diffusion Dynamics in Agent-Based Models with a Spatial Structure

- Applied Math and Analysis ( 126 Views )The diffusion or adoption of new products (such as fax machines, skype, facebook, Ipad, etc.) is one of the key problems in Marketing research. In recent years, this problem was often studied numerically, using agent-based models (ABMs). In this talk I will focus on analysis of the aggregate diffusion dynamics in ABMs with a spatial structure. In one-dimensional ABMs, the aggregate diffusion dynamics can be explicitly calculated, without using the mean-field approximation. In multidimensional ABMs, we introduce a clusters-dynamics approach, and use it to derive an analytic approximation of the aggregate diffusion dynamics. The clusters-dynamics approximation shows that the aggregate diffusion dynamics does not depend on the average distance between individuals, but rather on the expansion rate of clusters of adopters. Therefore, the grid dimension has a large effect on the aggregate adoption dynamics, but a small-world structure and heterogeneity among individuals have only a minor effect. Our results suggest that the one-dimensional model and the fully-connected Bass model provide a lower bound and an upper bound, respectively, for the aggregate diffusion dynamics in agent-based models with "any" spatial structure. This is joint work with Ro'i Gibori and Eitan Muller

## Mark Hoefer : Eulerian Dispersive Shock Waves and Instabilities

- Applied Math and Analysis ( 125 Views )Recent experimental and theoretical research in Bose-Einstein condensation and nonlinear optics have demonstrated novel supersonic, fluid-like phenomena. Shock waves in these and other systems are modeled by a dispersive regularization of Euler's equations, implemented by use of the Whitham averaging technique. Normal and oblique dispersive shock waves (DSWs) connecting two constant states are constructed. Numerical computations of supersonic, dispersive flow over a corner in the special case of systems modeled by the Nonlinear Schrodinger equation (NLS) exhibit stable pattern formation (oblique DSWs) or instability (turbulent-like behavior) depending on the flow parameters. A combination of analytical and computational approaches are used to demonstrate that this change in behavior can be identified with the transition from convective to absolute instability of dark solitons. The linearized NLS behavior about the dark soliton DSW trailing edge is studied in detail to identify the separatrix between convective and absolute instabilities.

## Lei Li : Some algorithms and analysis for first order interacting particle systems

- Applied Math and Analysis ( 123 Views )We focus on first order interacting particle systems, which can be viewed as overdamped Langevin equations. In the first part, we will look at the so-called random batch methods (RBM) for simulating the interacting particle systems. The algorithms are motivated by the mini-batch idea in machine learning. For some special cases, we show the convergence of RBMs for the first marginal under Wasserstein distance. In the second part, we look at the Coulomb interaction in 3D space. We show that as the number of particles go to infinity, almost surely, the empirical measure converges in law to weak solutions of the limiting nonlinear Fokker-Planck equation. This talk is based on joint works with Shi Jin (Shanghai Jiao Tong), Jian-Guo Liu (Duke University) and Pu Yu (Peking University).

## Xin Yang Lu : EVOLUTION EQUATIONS FROM EPITAXIAL GROWTH

- Applied Math and Analysis ( 122 Views )Epitaxial growth is a process in which a thin film is grown above a much thicker substrate. In the simplest case, no deposition is considered, and all the interactions are assumed to be purely elastic. However, since the film may potentially have different rigidity constant from the substate, such growth leads to a nonuniform film thickness. The equations governing epitaxial growth are high order (generally fourth order), nonlocal, and highly nonlinear. In this talk I will present some recent results about the regularity of solutions to several equations arising from epitaxial growth. Joint work with I. Fonseca and G.Leoni.

## Michael Siegel : Modeling, analysis, and computations of the influence of surfactant on the breakup of bubbles and drops in a viscous fluid

- Applied Math and Analysis ( 121 Views )We present an overview of experiments, numerical simulations, and mathematical analysis of the breakup of a low viscosity drop in a viscous fluid, and consider the role of surface contaminants, or surfactants, on the dynamics near breakup. As part of our study, we address a significant difficulty in the numerical computation of fluid interfaces with soluble surfactant that occurs in the important limit of very large values of bulk Peclet number Pe. At the high values of Pe in typical fluid-surfactant systems, there is a narrow transition layer near the drop surface or interface in which the surfactant concentration varies rapidly, and its gradient at the interface must be determined accurately to find the dropÂ?s dynamics. Accurately resolving the layer is a challenge for traditional numerical methods. We present recent work that uses the narrowness of the layer to develop fast and accurate `hybridÂ? numerical methods that incorporate a separate analytical reduction of the dynamics within the transition layer into a full numerical solution of the interfacial free boundary problem.

## Ben Murphy : Random Matrices, Spectral Measures, and Transport in Composite Media

- Applied Math and Analysis ( 120 Views )We consider composite media with a broad range of scales, whose effective properties are important in materials science, biophysics, and climate modeling. Examples include random resistor networks, polycrystalline media, porous bone, the brine microstructure of sea ice, ocean eddies, melt ponds on the surface of Arctic sea ice, and the polar ice packs themselves. The analytic continuation method provides Stieltjes integral representations for the bulk transport coefficients of such systems, involving spectral measures of self-adjoint random operators which depend only on the composite geometry. On finite bond lattices or discretizations of continuum systems, these random operators are represented by random matrices and the spectral measures are given explicitly in terms of their eigenvalues and eigenvectors. In this lecture we will discuss various implications and applications of these integral representations. We will also discuss computations of the spectral measures of the operators, as well as statistical measures of their eigenvalues. For example, the effective behavior of composite materials often exhibits large changes associated with transitions in the connectedness or percolation properties of a particular phase. We demonstrate that an onset of connectedness gives rise to striking transitional behavior in the short and long range correlations in the eigenvalues of the associated random matrix. This, in turn, gives rise to transitional behavior in the spectral measures, leading to observed critical behavior in the effective transport properties of the media.

## Geoffrey Schiebinger : Analyzing Developmental Stochastic Processes with Optimal Transport

- Applied Math and Analysis ( 119 Views )Single-cell RNA sequencing (scRNA-Seq) has emerged as a powerful tool to sample the complexity of large populations of cells and observe biological processes at unprecedented molecular resolution. This offers the exciting prospect of understanding the molecular programs that guide cellular differentiation during development. Here, we introduce Waddington-OT: a mathematical framework for understanding the temporal dynamics of development based on snapshots of expression profiles. The central challenge in analyzing these data arises from the fact that scRNA-Seq is destructive, which means that one cannot directly measure the trajectory of any given cell over time. We model the population of developing cells mathematically with a time-varying probability distribution (i.e. stochastic process) on a high-dimensional gene expression space, and we propose to recover the temporal coupling of the process with optimal transport. We demonstrate the power of Waddington-OT by applying the approach to study 315,000 scRNA-seq profiles collected at 40 time points over 16 days during reprogramming of fibroblasts to induced pluripotent stem cells. We construct a high-resolution map of reprogramming that rediscovers known features; uncovers new alternative cell fates including neural- and placental-like cells; predicts the origin and fate of any cell class; and implicates regulatory models in particular trajectories. Of these findings, we highlight Obox6, which we experimentally show enhances reprogramming efficiency. Our approach provides a general framework for investigating cellular differentiation.

## Wencai Liu : Spectral transitions for Schr\odinger operators with decaying potentials and Laplacians on asymptotically flat (hyperbolic) manifolds

- Applied Math and Analysis ( 116 Views )We apply piecewise constructions and gluing technics to construct asymptotically flat (hyperbolic) manifolds such that associated Laplacians have dense embedded eigenvalues or singular continuous spectra. The method also allows us to provide various examples of operators with embedded singular spectra, including perturbed periodic operators, periodic Jacobi operators, and Stark operators. We establish the asymptotic behavior (WKB for example) of eigensolutions under small perturbations, which implies certain rules for the absence of singular spectra. As a result, several sharp spectral transitions (even criteria) for a single (finitely many or countably many) embedded eigenvalues, singular continuous spectra and essential supports of spectral measures are obtained. The talk is based on several papers, some joint with Jitomirskaya and Ong.

## Stefan Steinerberger : Vibration and the local structure of elliptic partial differential equations

- Applied Math and Analysis ( 116 Views )If you put sand on a metal plate and start inducing vibrations with a violin bow, the sand jumps around and arranges itself in the most beautiful patterns - this used to be a circus trick in the late 18th century: Napoleon was a big fan and put a prize on giving the best mathematical explanation. Today we know that the sand moves to lines where a certain Laplacian eigenfunction vanishes but these remain mysterious. I will show pictures of sand and demonstrate a new approach: the key ingredient is to make the elliptic equation parabolic and then work with two different interpretations of the heat equation at the same time. If time allows, I will sketch another application of this philosophy to localization phenomena for Schroedinger operators.

## Charlie Doering : Optimal bounds and extremal trajectories for time averages in nonlinear dynamical systems

- Applied Math and Analysis ( 114 Views )For any quantity of interest in a system governed by nonlinear differential equations it is natural to seek the largest (or smallest) long-time average among solution trajectories. Upper bounds can be proved a priori using auxiliary functions, the optimal choice of which is a convex optimization. We show that the problems of finding maximal trajectories and minimal auxiliary functions are strongly dual. Thus, auxiliary functions provide arbitrarily sharp upper bounds on maximal time averages. They also provide volumes in phase space where maximal trajectories must lie. For polynomial equations, auxiliary functions can be constructed by semidefinite programming which we illustrate using the Lorenz and Kuramoto-Sivashinsky equations. This is joint work with Ian Tobasco and David Goluskin, part of which appears in Physics Letters A 382, 382Â?386 (2018).

## Yao Yao : Long time behavior of solutions to the 2D Keller-Segel equation with degenerate diffusion

- Applied Math and Analysis ( 112 Views )In this talk I will discuss the Keller-Segel equation, which is a nonlocal PDE modeling the collective motion of cells attracted by a self-emitted chemical substance. When this equation is set up in 2D with a degenerate diffusion term, it is known that solutions exist globally in time, but their long-time behavior remain unclear. To answer this question, we investigate a general aggregation equation with degenerate diffusion, and prove that all stationary solutions must be radially symmetric up to a translation. As a consequence, this enables us to obtain a convergence result for solutions to 2D Keller-Segel equation with degenerate diffusion as the time goes to infinity. This is a joint work with J. Carrillo, S. Hittmeir and B. Volzone.

## Yossi Farjoun : Solving Conservation Law and Balance Equations by Particle Management

- Applied Math and Analysis ( 112 Views )Conservation equations are at the heart of many interesting and important problems. Examples come from physics, chemistry, biology, traffic and many more. Analytically, hyperbolic equations have a beautiful structure due to the existence of characteristics. These provide the possibility of transforming a conservation PDE into a system of ODE and thus greatly reducing the computational effort required to solve such problems. However, even in one dimension, one encounters problems after a short time.

The most obvious difficulty that needs to be dealt with has to do with the creation of shocks, or in other words, the crossing of characteristics. With a particle based method one would like to avoid a situation when one particle overtakes a neighboring one. However, since shocks are inherent to many hyperbolic equations and relevant to the problems that one would like to solve, it would be good not to ``smooth away'' the shock but rather find a good representation of it and a good solution for the offending particles.

In this talk I will present a new particle based method for solving (one dimensional, scalar) conservation law equations. The guiding principle of the method is the conservative property of the underlying equation. The basic method is conservative, entropy decreasing, variation diminishing and exact away from shocks. A recent extension allows solving equations with a source term, and also provides ``exact'' solutions to the PDE. The method compares favorably to other benchmark solvers, for example CLAWPACK, and requires less computation power to reach the same resolution. A few examples will be shown to illustrate the method, with its various extensions. Due to the current limitation to 1D scalar, the main application we are looking at is traffic flow on a large network. Though we still hope to manage to extend the method to either systems or higher dimensions (each of these extensions has its own set of difficulties), I would be happy to discuss further possible applications or suggestions for extensions.

## Zhizhen (Jane) Zhao : Multi-segment Reconstruction with Invariant Features

- Applied Math and Analysis ( 111 Views )Multi-segment reconstruction (MSR) problem consists of recovering a signal from noisy segments with unknown positions of the observation windows. One example arises in DNA sequence assembly, which is typically solved by matching short reads to form longer sequences. Instead of trying to locate the segment within the sequence through pair-wise matching, we propose a new approach that uses shift-invariant features to estimate both the underlying signal and the distribution of the positions of the segments. Using the invariant features, we formulate the problem as a constrained nonlinear least-squares. The non-convexity of the problem leads to its sensitivity to the initialization. However, with clean data, we show empirically that for longer segment lengths, random initialization achieves exact recovery. Furthermore, we compare the performance of our approach to the results of expectation maximization and demonstrate that the new approach is robust to noise and computationally more efficient.

## Alexandr Labovschii : High accuracy numerical methods for fluid flow problems and turbulence modeling

- Applied Math and Analysis ( 109 Views )We present several high accuracy numerical methods for fluid flow problems and turbulence modeling.

First we consider a stabilized finite element method for the Navier-Stokes equations which has second order temporal accuracy. The method requires only the solution of one linear system (arising from an Oseen problem) per time step.

We proceed by introducing a family of defect correction methods for the time dependent Navier-Stokes equations, aiming at higher Reynolds' number. The method presented is unconditionally stable, computationally cheap and gives an accurate approximation to the quantities sought.

Next, we present a defect correction method with increased time accuracy. The method is applied to the evolutionary transport problem, it is proven to be unconditionally stable, and the desired time accuracy is attained with no extra computational cost.

We then turn to the turbulence modeling in coupled Navier-Stokes systems - namely, MagnetoHydroDynamics. We consider the mathematical properties of a model for the simulation of the large eddies in turbulent viscous, incompressible, electrically conducting flows. We prove existence, uniqueness and convergence of solutions for the simplest closed MHD model. Furthermore, we show that the model preserves the properties of the 3D MHD equations.

Lastly, we consider the family of approximate deconvolution models (ADM) for turbulent MHD flows. We prove existence, uniqueness and convergence of solutions, and derive a bound on the modeling error. We verify the physical properties of the models and provide the results of the computational tests.

## Sung Ha Kang : Efficient methods for curvature based variational imaging models

- Applied Math and Analysis ( 109 Views )Starting with an introduction to multiphase image segmentation, this talk will focus on inpainting and illusory contour using variational models with curvature terms. Recent developments of fast algorithms, based on operator splitting, augmented Lagrangian, and alternating minimization, enabled us to efficiently solve functional with higher order terms. Main ideas of the models and algorithms, some analysis and numerical results will be presented.

## Tom Beale : Uniform error estimates for fluid flow with moving boundaries using finite difference methods

- Applied Math and Analysis ( 107 Views )Recently there has been extensive development of numerical methods for fluid flow interacting with moving boundaries or interfaces, using regular finite difference grids which do not conform to the boundaries. Simulations at low Reynolds number have demonstrated that, with certain choices in the design of the method, the velocity can be accurate to about O(h^2) while discretizing near the interface with truncation error as large as O(h). We will describe error estimates which verify that such accuracy can be achieved in a simple prototype problem, even near the interface, using corrections to difference operators as in the immersed interface method. We neglect errors in the interface location and derive uniform estimates for the fluid velocity and pressure. We will first discuss maximum norm estimates for finite difference versions of the Poisson equation and diffusion equation with a gain of regularity. We will then describe the application to the Navier-Stokes equations.

## Antoine Mellet : Existence and regularity of extremal solutions for a mean-curvature equation

- Applied Math and Analysis ( 107 Views )We study a mean curvature problem arising in particular in the study of MEMS (Micro-Electro-Mechanical Systems) and pendent capillary drops. The underlying mathematical problem is similar to the classical semilinear equation -\Delta u=\lambda f(u) but involves the mean-curvature operator instead of the Laplacian.

## Alexander Kiselev : Regularity and blow up in ideal fluid

- Applied Math and Analysis ( 106 Views )The incompressible Euler equation of fluid mechanics has been derived in 1755. It is one of the central equations of applied analysis, yet due to its nonlinearity and non-locality many fundamental properties of solutions remain poorly understood. In particular, the global regularity vs finite time blow up question for incompressible three dimensional Euler equation remains open. In two dimensions, it has been known since 1930s that solutions to Euler equation with smooth initial data are globally regular. The best available upper bound on the growth of derivatives of the solution has been double exponential in time. I will describe a construction showing that such fast generation of small scales can actually happen, so that the double exponential bound is qualitatively sharp. This work has been motivated by numerical experiments due to Hou and Luo who propose a new scenario for singularity formation in solutions of 3D Euler equation. The scenario is axi-symmetric. The geometry of the scenario is related to the geometry of 2D Euler double exponential growth example and involves hyperbolic points of the flow located at the boundary of the domain. If time permits, I will discuss some recent attempts to gain insight into the three-dimensional fluid behavior in this scenario.

## Maja Taskovic : Tail behavior of solutions to the Boltzmann equation

- Applied Math and Analysis ( 105 Views )The Boltzmann equation models the evolution of the probability density of gas particles that interact through predominantly binary collisions. The equation consists of a transport operator and a collision operator. The latter is a bilinear integral with a non-integrable angular kernel. For a long time the equation was simplified by assuming that the kernel is integrable (so called Grad's cutoff), with a belief that such an assumption does not affect the equation significantly. Recently, however, it has been observed that a non-integrable singularity carries regularizing properties, which motivates further analysis of the equation in this setting. We study the behavior in time of tails of solutions to the Boltzmann equation in the non-cutoff regime, by examining the generation and propagation in time of $L^1$ and $L^\infty$ exponentially weighted estimates and the relation between them. We show how the singularity rate of the angular kernel affects the order of tails that can be propagated. The result uses Mittag-Leffler functions, which are a generalization of exponential functions. This is based on joint works with Alonso, Gamba, Pavlovic and with Gamba, Pavlovic.

## Gabriel Stoltz : Langevin dynamics at equilibrium and out of equilibrium: from hypocoercivity to efficient sampling

- Applied Math and Analysis ( 105 Views )I will present various results on the Langevin dynamics, both from theoretical and numerical perspectives. This dynamics is quite popular for sampling purposes in computational statistical physics. It can be seen as a Hamiltonian dynamics perturbed by an Ornstein-Uhlenbeck process on the momenta. I will start on the theoretical side with an account of the hypocoercive approach by Dolbeault, Mouhot and Schmeiser, which is a key technique to prove that the asymptotic variance of time averages is well defined, and also to obtain quantitative bounds on it. I will then discuss various extensions/modifications of the standard Langevin dynamics, such as replacing the standard quadratic kinetic energy by a more general one, constructing control variates relying on a simplified Poisson equation, proving the convergence of nonequilibrium versions such as the one encountered in the Temperature Accelerated Molecular Dynamics method, etc.

## Zhouping Xin : On Gases Expanding into Vacuum with or without Self-Gravitations

- Applied Math and Analysis ( 104 Views )In this talk I will discuss several issues concerning the motions of gases expanding into vacuum with or without self-gravitations which are governed by a free-boundary value problem for the 3-dimnesional compressible Euler system with/or without Poisson equation. A general uniqueness theorem for classical solutions to such a free boundary-value problem is presented for physical vacuums. A typical physical vacuum solution includes the famous Lane-Emdan solution in astrophysics. The uniqueness is proved by a relative entropy argument. Then a local well-posedness theory for spherically symmetric motions is established in a less regular space by a deliberate choice of weighted functional to overcome difficulties arising both at the free surface and the symmetry center. Finally, the uniqueness of the spherically symmetric motions is discussed for general equation of state without self-gravitations. This is a joint work with Professor Tao Luo and Professor Huihui Zeng.

## Nathan Glatt-Holtz : Invisicid Limits for the Stochastic Navier Stokes Equations and Related Systems

- Applied Math and Analysis ( 104 Views )One of the original motivations for the development of stochastic partial differential equations traces it's origins to the study of turbulence. In particular, invariant measures provide a canonical mathematical object connecting the basic equations of fluid dynamics to the statistical properties of turbulent flows. In this talk we discuss some recent results concerning inviscid limits in this class of measures for the stochastic Navier-Stokes equations and other related systems arising in geophysical and numerical settings. This is joint work with Peter Constantin, Vladimir Sverak and Vlad Vicol.

## Bob Pego : Merging-splitting group dynamics via Bernstein function theory (or: How to count fish using mathematics)

- Applied Math and Analysis ( 101 Views )We study coagulation-fragmentation equations inspired by a simple model derived in fisheries science to explain data on the size distribution of schools of pelagic fish. The equations lack detailed balance and admit no H-theorem, but we are anyway able to develop a rather complete description of equilibrium profiles and large-time behavior, based on complex function theory for Bernstein and Pick (Herglotz) functions. The generating function for discrete equilibrium profiles also generates the Fuss-Catalan numbers that count all ternary trees with $n$ nodes. The structure of equilibrium profiles and other related sequences is explained through a new and elegant characterization of the generating functions of completely monotone sequences, as those Pick functions analytic and nonnegative on a half line. This is joint work with Jian-Guo Liu and Pierre Degond.

## Zongming Ma : Community detection in stochastic block models and beyond

- Applied Math and Analysis ( 99 Views )In this talk, I will start with a simple algorithm for community detection in stochastic block models and discuss its statistical optimality. After that, we will discuss two related issues. One is model selection for stochastic block models. The other is the extension to community detection in degree-corrected block models. We shall pay close attention to the achievability of statistical optimality by computationally feasible procedures throughout the talk.

## Ken Kamrin : A hierarchy of continuum models for granular flow

- Applied Math and Analysis ( 98 Views )Granular materials are common in everyday life but are historically difficult to model. This has direct ramifications owing to the prominent role granular media play in multiple industries and terrain dynamics. One can attempt to track every grain with discrete particle methods, but realistic systems are often too large for this approach and a continuum model is desired. However, granular media display unusual behaviors that complicate the continuum treatment: they can behave like solid, flow like liquid, or separate into a "gas", and the rheology of the flowing state displays remarkable subtleties that have been historically difficult to model. To address these challenges, in this talk we develop a family of continuum models and solvers, permitting quantitative modeling capabilities for a variety of applications, ranging from general problems to specific techniques for problems of intrusion, impact, driving, and locomotion in grains.

To calculate flows in general cases, a rather significant nonlocal effect is evident, which is well-described with our recent nonlocal model accounting for grain cooperativity within the flow rule. This model enables us to capture a number of seemingly disparate manifestations of particle size-effects in granular flows including: (i) the wide shear-band widths observed in many inhomogeneous flows, (ii) the apparent strengthening exhibited in thin layers of grains, and (iii) the fluidization observed due to far-away motion of a boundary. On the other hand, to model only intrusion forces on submerged objects, we will show, and explain why, many of the experimentally observed results can be captured from a much simpler tension-free frictional plasticity model. This approach gives way to some surprisingly simple general tools, including the granular Resistive Force Theory, and a broad set of scaling laws inherent to the problem of granular locomotion. These scalings are validated experimentally and in discrete particle simulations suggesting a new down-scaled paradigm for granular locomotive design, on earth and beyond, to be used much like scaling laws in fluid mechanics.

## Bruce Pitman : Where Are You Going To Go When The Volcano Blows?

- Applied Math and Analysis ( 96 Views )We discuss one approach to determining the hazard threat to a locale due to a large volcanic avalanche. The methodology employed includes large-scale numerical simulations, field data reporting the volume and runout of flow events, and a detailed statistical analysis of uncertainties in the modeling and data. The probability of a catastrophic event impacting a locale is calculated, together with a estimate of the uncertainty in that calculation. By a careful use of simulations, a hazard map for an entire region can be determined. The calculation can be turned around quickly, and the methodology can be applied to other hazard scenarios.

## Johann Guilleminot : Stochastic Modeling and Simulations of Random Fields in Computational Nonlinear Mechanics

- Applied Math and Analysis ( 94 Views )Accounting for system-parameter and model uncertainties in computational models is a highly topical issue at the interface of computational mechanics, materials science and probability theory. In addition to the construction of efficient (e.g. Galerkin-type) stochastic solvers, the construction, calibration and validation of probabilistic representations are now widely recognized as key ingredients for performing accurate and robust simulations. This talk is specifically focused on the modeling and simulation of spatially-dependent properties in both linear and nonlinear frameworks. Information-theoretic models for matrix-valued random fields are first introduced. These representations are typically used, in solid mechanics, to define tensor-valued coefficients in elliptic stochastic partial differential operators. The main concepts and tools are illustrated, throughout this part, by considering the modeling of elasticity tensors fluctuating over nonpolyhedral geometries, as well as the modeling and identification of random interfaces in polymer nanocomposites. The latter application relies, in particular, on a statistical inverse problem coupling large-scale Molecular Dynamics simulations and a homogenization procedure. We then address the probabilistic modeling of strain energy functions in nonlinear elasticity. Here, constraints related to the polyconvexity of the potential are notably taken into account in order to ensure the existence of a stochastic solution. The proposed framework is finally exemplified by considering the modeling of various soft biological tissues, such as human brain and liver tissues.