BRINKMAN PROBLEM STOCHASTIC THE MULTI-LEVEL MONTE CARLO FINITE ELEMENT METHOD FOR A

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DBAB205397

Doc.ID: Revision: Status:

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THE MULTI-LEVEL MONTE CARLO FINITE ELEMENT METHOD FOR A

STOCHASTIC

BRINKMAN PROBLEM

**CLAUDE GITTELSON*, JUHO KÖNNÖ**, CHRISTOPH SCHWAB* AND ROLF STENBERG*

15 July 2011

1 The MLMC-FEM for a stochastic Brinkman Problem/ Juho Könnö

* ETH Zürich, Seminar für angewandte Mathematik

** Wärtsilä Finland Oy, R&D/C&S

*** Aalto University, Department of Mathematics and Systems Analysis

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Outline

• The Brinkman model

• Stochastic permeability

• Karhunen-Loève expansion

• Main ideas of the MLMC method

• Requirements from the finite element method – Implementation

• Numerical results

15 July 2011

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The Brinkman model

• Describes the flow of a viscous fluid in a porous medium – Cracks and flow channels in porous media

– Heat pipes, oil filters, composite resin infusion – Combination of the Darcy and Stokes models

15 July 2011

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Norms and variational spaces

• The nature of the problem changes when going to the Darcy limit

• The following norms are employed

15 July 2011

Stokes Darcy

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The weak form

• Find a velocity-pressure pair

15 July 2011

such that

in which

and

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Stochastic permeability

• The permeability tensor is assumed to be a random field

• A log-normal model is employed for the permeability

• Often only statistical data is available for the permeability

15 July 2011

• G is a Gaussian field and M ⁰ is positive definite

• M is integrable with respect to the probability measure

• Implies integrability of the velocity-pressure pair

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K-L expansion

• The Gaussian field can be expanded as the Karhunen-Loève series

15 July 2011

• Next, the series is approximated by truncating after N terms Normalized eigenpairs of

the covariance operator

Normal random variables

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K-L expansion

15 July 2011

• The following truncation estimate holds for the Gaussian field G

• The parameter s depends on the convergence speed of the eigenvalues of the covariance operator

• Holds for any

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Truncated log-normal field

• For the solution of the problem with the truncated permeability field it holds

• How to decide what N should be?

15 July 2011 The MLMC-FEM for a stochastic Brinkman Problem/ Juho Könnö 9

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The MLMC ideology

• Multi level idea: balance the stochastic truncation error and discretization error

• Different eigenfunctions correspond to different „frequencies‟

– Sparse mesh is sufficient for low frequencies

• Traditional Monte Carlo with M samples

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The MLMC method

• Based on the following telescoping sum property:

• Use different level of approximation on each level l

• “Compute successive increments, not the whole field”

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Error estimates

• The following bounds hold for a L-level method

• We have to balance the MC error compared to

– FEM errors

– Truncation error

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Number of samples and workload

• Assuming uniform mesh refinement we choose the truncation level and number of MC samples as

• Yields optimal order of convergence with respect to the polynomial degree

• Workload is

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Requirements towards FEM

• Telescoping sum property must hold for the discrete solution – In practice, nested FE spaces are required

• Problems with bubble degrees of freedom, e.g. MINI element

– Interpolation could be done, but too expensive

• Stabilization parameters might depend on stochastic data – Must be easily computable or estimated

• Evaluating the K-L series is a major computational effort – Low-order methods, reduced integration

15 July 2011

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Our approach to MLMC-FEM

• Stabilized equal order Stokes-based elements

• Mesh-dependent norm

• Residual stabilization

– Parameter easily computed for linear elements

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FEM error estimates

• Estimates follow those presented by Juntunen & Stenberg (Calcolo 2009) for the Brinkman problem

– Nature of the problem changes numerically at t=h – Modified to include stochastic permeability

– Constants must be tracked carefully to assure integrability

15 July 2011

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Numerical results

• Test case: simple square domain with explicitly known covariance

15 July 2011

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Numerical results

• We can control the convergence rate of the series

• Problem loaded by a smooth boundary condition enforced with Nitsche‟s method

• Number of samples and truncation chosen as

15 July 2011

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Fast series - error

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Fast series - workload

15 July 2011

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Moderate series - error

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Moderate series - workload

15 July 2011

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Slow series - error

15 July 2011

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Slow series - workload

15 July 2011

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BRINKMAN PROBLEM STOCHASTIC THE MULTI-LEVEL MONTE CARLO FINITE ELEMENT METHOD FOR A

THE MULTI-LEVEL MONTE CARLO FINITE ELEMENT METHOD FOR A

STOCHASTIC

BRINKMAN PROBLEM

CLAUDE GITTELSON*, JUHO KÖNNÖ**, CHRISTOPH SCHWAB* AND ROLF STENBERG***

* ETH Zürich, Seminar für angewandte Mathematik

** Wärtsilä Finland Oy, R&D/C&S

*** Aalto University, Department of Mathematics and Systems Analysis

Outline

• The Brinkman model

• Stochastic permeability

• Karhunen-Loève expansion

• Main ideas of the MLMC method

• Requirements from the finite element method – Implementation

• Numerical results

The Brinkman model

• Describes the flow of a viscous fluid in a porous medium – Cracks and flow channels in porous media

– Heat pipes, oil filters, composite resin infusion – Combination of the Darcy and Stokes models

Norms and variational spaces

• The nature of the problem changes when going to the Darcy limit

• The following norms are employed

Stokes Darcy

The weak form

• Find a velocity-pressure pair

such that

in which

and

Stochastic permeability

• The permeability tensor is assumed to be a random field

• A log-normal model is employed for the permeability

• Often only statistical data is available for the permeability

• G is a Gaussian field and M 0 is positive definite

• M is integrable with respect to the probability measure

• Implies integrability of the velocity-pressure pair

K-L expansion

• The Gaussian field can be expanded as the Karhunen-Loève series

• Next, the series is approximated by truncating after N terms Normalized eigenpairs of

the covariance operator

Normal random variables

K-L expansion

• The following truncation estimate holds for the Gaussian field G

• The parameter s depends on the convergence speed of the eigenvalues of the covariance operator

• Holds for any

Truncated log-normal field

• For the solution of the problem with the truncated permeability field it holds

• How to decide what N should be?

The MLMC ideology

• Multi level idea: balance the stochastic truncation error and discretization error

• Different eigenfunctions correspond to different „frequencies‟

– Sparse mesh is sufficient for low frequencies

• Traditional Monte Carlo with M samples

The MLMC method

• Based on the following telescoping sum property:

• Use different level of approximation on each level l

• “Compute successive increments, not the whole field”

Error estimates

• The following bounds hold for a L-level method

• We have to balance the MC error compared to

– FEM errors

– Truncation error

Number of samples and workload

• Assuming uniform mesh refinement we choose the truncation level and number of MC samples as

• Yields optimal order of convergence with respect to the polynomial degree

• Workload is

Requirements towards FEM

• Telescoping sum property must hold for the discrete solution – In practice, nested FE spaces are required

• Problems with bubble degrees of freedom, e.g. MINI element

– Interpolation could be done, but too expensive

• Stabilization parameters might depend on stochastic data – Must be easily computable or estimated

• Evaluating the K-L series is a major computational effort – Low-order methods, reduced integration

Our approach to MLMC-FEM

• Stabilized equal order Stokes-based elements

• Mesh-dependent norm

• Residual stabilization

– Parameter easily computed for linear elements

FEM error estimates

• Estimates follow those presented by Juntunen & Stenberg (Calcolo 2009) for the Brinkman problem

– Nature of the problem changes numerically at t=h – Modified to include stochastic permeability

– Constants must be tracked carefully to assure integrability

Numerical results

**CLAUDE GITTELSON*, JUHO KÖNNÖ**, CHRISTOPH SCHWAB* AND ROLF STENBERG*

• G is a Gaussian field and M ⁰ is positive definite