S. Loire: Spatial Filter and Backward Time Approach of Probabilistic Method to Advection Diffusion...

8/6/2019 S. Loire: Spatial Filter and Backward Time Approach of Probabilistic Method to Advection Diffusion Equation

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Dynamical Systems and Nonlinear Control Theory

Spatial Filter and Backward Time

Approach of Probabilistic Method toAdvection Diffusion Equation

Sophie LoireIgor Mezi

Department of Mechanical Engineering,

University of California, Santa Barbara


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Monday, May 22, 2011 Spatial Filter and Backward Time Approach of Probabilistic Method to Advection Diffusion EquationSophie Loire

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IntroductionStudy of parabolic type equation: for example advection diffusion equationNeed efficient numerical method

For systems with:

delta like initial concentration,

advection dominated problems,

highly chaotic advection,high velocity gradient,

space varying diffusivity tensor.

Finite element methods can lead to:

excessive numerical dispersion,

artificial oscillations,

instability,

negative values of concentration.

Probabilistic Method

Example: Perturbed cellular, divergence-free velocity field and small diffusion


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Spatial Filter and Backward Time Approach


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Backward Time Approach,

Backward Probabilistic method


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Monte Carlo Averaging

Solution = Expected value of the functional:

Monte Carlo Averaging Approach:

Converge as 1/N

N is problem depend.


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Averaging with a spatial filterMonte Carlo Averaging

Filter used as point spreadfunctions by convolution,

Instead of averaging with

respect to differentrealizations,

Gain in computational time


Spatial Filtering


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Convergence


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Convergence

To illustrate this theorem,we first study a 1D example:

v(x) = -sin(x)

M i R h G


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1D example

M i R h G


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Convergence

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L: number of grid points 2M+1: number of points in filter

N: number of runs at each gridpoints

(2M+1)N: total number of pointsfor the calculation of theexpected value

LxN: computational effort

Backward Time Approach:

Numerical Efficiency

The minimum of grid points forsmaller computational effort for

the spatial filter method scalesas:

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Perturbed Cellular,

Divergence-Free Velocity Field

And Small Diffusion

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2D example

Advection-Diffusion Equation

with

v(x,t) = v1(x) + a

p*cos(2*pi*t)*v

2(x)

and

c(x,t=0) = sin(2*pi*y)

v1(x)=

v2(x)=

v1(x) v2(x)

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D=0, ap=0,

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e esea c G oup



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D=0, ap=0.1,

Mesohyperbolicity

Gives an identification of

the mixing regions Enhanced mixing in those

regions with diffusion?

MS60:5:10-5:30 Koopman Operator, TimeAverages and the Big Oil Spill, Igor Mezic

A New Mixing Diagnostic and Gulf Oil Spill Movement,Igor Mezi, S. Loire, Vladimir A. Fonoberov, and P. Hogan,Science 22 October 2010: 330 (6003),

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p



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D=0, ap=0.1,

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D=0, ap=0.1,

What is the effect of diffusion on the mixing in this flow?

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D=10-2, ap=0.1

For Large D, rapid mixing!

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S C


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D=10-5 , ap=0.1

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D i l S t d N li C t l Th


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D=10-6 , ap=0.1


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Mixing Measure,

Mix-Norm and Mix-variance

Mix-Norm

Mix-Variance

H-1/2 Norm

To quantify the mixing in this case, we use

The mix-variance measure.

The Mix-Norm is: a multiscale measure for mixing

to quantify the degree of mixedness of a

densityfield.

is based on averaging the function over all

scales and integrating the L2 norms of the

averaged functions over all scales.

MixVariance decreases towards 0 for perfect

mixing.

(Mathew et al. 2005

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Mixing Measure,

Mix-variance

D=0 D=0, D=1e-2.5 to D=1e-1.5D=0, D=0, D=1e-2.5 to D=1e-1.5

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Mixing Measure,

Mix-variance

D=0, D=1e-2.5 to D=1e-1.5 D=0, D=1e-3 to 1e-1.5

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Mixing Measure,

Mix-variance

D=0, D=1e-9 to 1e-1.5D=0, D=1e-3 to 1e-1.5

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y y y


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Mixing Measure,

Mix-variance

Finite Time Mixing versus Peclet numberD=0, D=1e-9 to 1e-1.5

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y y y


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Conclusion

Advection diffusion equations can be studied using a probabilistic

approach to analyze transport of densities.

We introduced a new method using backward time integration andspatial averaging.

We applied this method to the study of the transport of densities by a

perturbed cellular, divergence-free velocity field and small diffusion.

S. Loire: Spatial Filter and Backward Time Approach of Probabilistic Method to Advection Diffusion...

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