# Difference between revisions of "Applied/ACMS/absF16"

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In uncertainty propagation, coefficients, boundary conditions, and other key inputs of a computational model are given as random variables and one is interested in estimating statistical moments of the corresponding model outputs. Estimating the moments with crude Monte Carlo can become prohibitively expensive in cases where a single model solve is already computationally demanding. Our multifidelity approach leverages low-cost, inaccurate surrogate models for speedup (variance reduction) and occasionally makes recourse to the expensive high-fidelity model to establish unbiased estimators, even in the absence of error control for the surrogate models. We derive an optimal sampling scheme that determines how often each model has to be evaluated to minimize the mean-squared error of the multifidelity estimator for a given computational budget. We show that an analytic, unique solution of the optimization problem exists under mild assumptions on the models and establish an asymptotic convergence theory for the multifidelity estimator. Numerical results demonstrate that our multifidelity approach achieves speedups by orders of magnitude on a linear structural mechanics problem and a nonlinear reacting flow problem. | In uncertainty propagation, coefficients, boundary conditions, and other key inputs of a computational model are given as random variables and one is interested in estimating statistical moments of the corresponding model outputs. Estimating the moments with crude Monte Carlo can become prohibitively expensive in cases where a single model solve is already computationally demanding. Our multifidelity approach leverages low-cost, inaccurate surrogate models for speedup (variance reduction) and occasionally makes recourse to the expensive high-fidelity model to establish unbiased estimators, even in the absence of error control for the surrogate models. We derive an optimal sampling scheme that determines how often each model has to be evaluated to minimize the mean-squared error of the multifidelity estimator for a given computational budget. We show that an analytic, unique solution of the optimization problem exists under mild assumptions on the models and establish an asymptotic convergence theory for the multifidelity estimator. Numerical results demonstrate that our multifidelity approach achieves speedups by orders of magnitude on a linear structural mechanics problem and a nonlinear reacting flow problem. | ||

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+ | === Yu Gu (Stanford) === | ||

+ | |||

+ | ''Fluctuations in stochastic homogenization -- when is the formal two-scale expansion valid?'' | ||

+ | |||

+ | To analyze multiscale PDEs and investigate their asymptotic behaviors, a formal expansion can be used to construct correctors and help proving the convergence. It is nevertheless a priori unclear whether those terms appearing in the expansion indeed represent the first and higher order errors. In other words, the expansion sometimes stays only on the formal level. | ||

+ | |||

+ | In this talk, we will discuss the classical problem of stochastic homogenization of elliptic operators in divergence form, and identity the first and higher order random fluctuations. It turns out that the formal expansion may or may not indicate the right answer depending on the scales on which we make the measurement. Part of the talk is based joint work with Jean-Christophe Mourrat. |

## Revision as of 10:09, 21 October 2016

## Contents

- 1 ACMS Abstracts: Fall 2016
- 1.1 Nicolas Seguin (Universite Nantes)
- 1.2 Rich Kerswell (Bristol University)
- 1.3 Yalchin Efendiev (TAMU)
- 1.4 Michael Ferris (Madison)
- 1.5 Nick Moore (FSU)
- 1.6 Gwynn Elfring (UBC)
- 1.7 Jacob Bedrossian (Maryland)
- 1.8 Clemens Heitzinger (Vienna Technical University)
- 1.9 Benjamin Peherstorfer (Madison)
- 1.10 Yu Gu (Stanford)

# ACMS Abstracts: Fall 2016

### Nicolas Seguin (Universite Nantes)

*Non-hydrostatic extension of classical shallow-water models*

When modeling incompressible flows with a free surface, many situations are compatible with the so-called shallow-water assumption: the length of the domain is much larger than the average depth. Starting from the Navier-Stokes equations or the Euler equations for water flows with free surface, average processing or asymptotic analysis may lead to the Saint-Venant equations, which is a classical hyperbolic system of conservation laws. The goal of this talk is to go one step further, accounting for vertical effects. This leads to dispersive equations, such as the well-known Green–Naghdi model. Despite the change of nature of the equations, we will see that many properties are shared by the Saint-Venant equations and the Green–Naghdi equations.

### Rich Kerswell (Bristol University)

*Using optimization to reveal scaling laws in turbulent flows*

In many fluid flow situations there is an a priori unknown global quantity, such as heat flux in convection or mass flux in pressure-driven flow, which is of overriding physical interest. Understanding how this quantity scales with the non-dimensional parameters describing the flow situation as one or more of these parameters become large (so that the flow is turbulent) is then a fundamental problem. In this talk, I will briefly review one particular approach to tackling this issue based upon optimization and then discuss how a variety of new developments augur well for future progress.

### Yalchin Efendiev (TAMU)

*A generalized multiscale model reduction technique for heterogeneous problems*

In this talk, I will discuss multiscale model reduction techniques for problems in heterogeneous media. I will describe a framework for constructing local (space-time) reduced order models for problems with multiple scales and high contrast. I will focus on a recently proposed method, Generalized Multiscale Finite Element Method, that systematically constructs local multiscale finite element basis functions on a coarse grid, which is much larger than the underlying resolved fine grid. The multiscale basis functions take into account the fine-scale information of the resolved solution space via careful choices of local snapshot spaces and local spectral decompositions. I will discuss the issues related to the construction of multiscale basis functions, main ingredients of the method, and a number of applications. These methods are intended for multiscale problems without scale separation and high contrast.

### Michael Ferris (Madison)

*Fish, cows, fields of fuel and optimization*

We describe several applications of optimization modeling to address environmental constraints. At the core of these models are complex interacting physical, biological, social and economic systems. We show how spatial visualizations of the underlying decision spaces can expose critical features of the problem to the domain experts in ways that facilitate greater understanding of optimization tradeoffs. We will detail specific optimization models underlying three applications, including a nutrient managagement system (Anmods), a fish barrier removal project (Fishwerks), and a bio-energy game (Fields of fuel).

### Nick Moore (FSU)

*How bodies erode and dissolve in fluid flows*

A variety of landscapes are formed by the action of flowing fluids, either air or water. In these settings, the development of morphology is a reciprocal process: as a structure is carved by a fluid, its changing shape can alter the local flow. In the laboratory we can examine this coevolution of shape and flow by immersing erodible or soluble bodies in fast-flowing water. I will discuss a simplified Prandtl-based model that accounts for the vanishing rates and emergent shapes observed in the experiments. In particular, determining the terminal shape can be posed as a singular Riemann-Hilbert problem, and this analysis links the different processes of erosion, dissolution, and melting.

### Gwynn Elfring (UBC)

*An active particle in a complex fluid*

Active particles are self-driven objects, biological or otherwise, which convert stored or ambient energy into systematic motion. The motion of small active particles in Newtonian fluids has received considerable attention, with interest ranging from phoretic propulsion to biological locomotion, whereas studies on active bodies immersed in complex fluids are comparatively scarce. A simple model for an active particle considers a sphere with an axisymmetric distribution of slip velocities on its surface, known as the squirmer model. This model has been helpful in developing insights into the dynamics of both biological swimmers, like Volvox and Opalina, and synthetic self-propelling colloids. In this talk we present a theory for an active squirmer-type particle in a complex fluid, and then discuss the effects of viscoelasticity and shear-thinning rheology in the context of biological locomotion and the propulsion of colloidal Janus particles.

### Jacob Bedrossian (Maryland)

*Nonlinear echoes and Landau damping with insufficient regularity*

In this talk, we will discuss recent advances towards understanding the regularity hypotheses in the theorem of Mouhot and Villani on Landau damping near equilibrium for the Vlasov-Poisson equations. We show that, in general, their theorem cannot be extended to any Sobolev space for the 1D periodic case. This is demonstrated by constructing arbitrarily small solutions with a sequence of nonlinear oscillations, known as plasma echoes, which damp at a rate arbitrarily slow compared to the linearized Vlasov equations. Some connections with hydrodynamic stability problems will be discussed if time permits.

### Clemens Heitzinger (Vienna Technical University)

*Stochastic partial differential equations with applications in nanotechnology*

Noise, fluctuations, and process variations are becoming more and more important in nanoscale devices. Their description necessitates the use of stochastic partial differential equations and gives rise to new mathematical problems. We have solved problems arising from charge transport in random environments and in multiscale settings. An important model is the stochastic drift-diffusion-Poisson system with all random coefficients. We present an existence and uniqueness result as well as optimal multi-level Monte-Carlo and multi-level randomized quasi Monte-Carlo methods. These methods, the first for a system of stochastic equations, reduce the computational cost by several orders of magnitude compared to the optimal Monte-Carlo method. The smaller the given error bound, the more effective the methods become. Our recent model to solve the multiscale problem inherent in the simulation of nanopore sensors and to calculate the stochastic properties of the transport of target molecules through such devices is also presented. Finally, simulation results for the solution of the Maxwell equations using an integral formulation and FMM expansions in a multiscale setting are also presented.

### Benjamin Peherstorfer (Madison)

*Optimal sampling in multifidelity Monte Carlo estimation for efficient uncertainty propagation*

In uncertainty propagation, coefficients, boundary conditions, and other key inputs of a computational model are given as random variables and one is interested in estimating statistical moments of the corresponding model outputs. Estimating the moments with crude Monte Carlo can become prohibitively expensive in cases where a single model solve is already computationally demanding. Our multifidelity approach leverages low-cost, inaccurate surrogate models for speedup (variance reduction) and occasionally makes recourse to the expensive high-fidelity model to establish unbiased estimators, even in the absence of error control for the surrogate models. We derive an optimal sampling scheme that determines how often each model has to be evaluated to minimize the mean-squared error of the multifidelity estimator for a given computational budget. We show that an analytic, unique solution of the optimization problem exists under mild assumptions on the models and establish an asymptotic convergence theory for the multifidelity estimator. Numerical results demonstrate that our multifidelity approach achieves speedups by orders of magnitude on a linear structural mechanics problem and a nonlinear reacting flow problem.

### Yu Gu (Stanford)

*Fluctuations in stochastic homogenization -- when is the formal two-scale expansion valid?*

To analyze multiscale PDEs and investigate their asymptotic behaviors, a formal expansion can be used to construct correctors and help proving the convergence. It is nevertheless a priori unclear whether those terms appearing in the expansion indeed represent the first and higher order errors. In other words, the expansion sometimes stays only on the formal level.

In this talk, we will discuss the classical problem of stochastic homogenization of elliptic operators in divergence form, and identity the first and higher order random fluctuations. It turns out that the formal expansion may or may not indicate the right answer depending on the scales on which we make the measurement. Part of the talk is based joint work with Jean-Christophe Mourrat.