Upgrade to Pro — share decks privately, control downloads, hide ads and more …

Adaptive Stochastic Collocation on Sparse Grids

Adaptive Stochastic Collocation on Sparse Grids

Numerical simulations become more reliable if random effects are taken into account. To this end, the describing parameters can be expressed by random variables or random fields, which leads to partial differential equations (PDEs) with random parameters.
Common numerical methods to solve such problems are spectral methods of Galerkin type and stochastic collocation on sparse grids. We focus on stochastic collocation, because it decouples the random PDE into a set of deterministic equations that can be solved in parallel.
In order to keep the computational costs at a moderate level, it is inevitable to place the collocation points adaptively. Therefore, error estimates are required. For spectral methods, adjoint a posteriori error analysis has been proposed in.
We want to combine stochastic collocation with an adjoint approach in order to estimate the error of some stochastic quantity, such as the mean or variance of a solution functional. Thereby, our goal is to develop appropriate error estimates that require less computational effort then the solution itself.

Bettina Schieche

March 29, 2012
Tweet

More Decks by Bettina Schieche

Other Decks in Research

Transcript

  1. Adaptive Stochastic Collocation
    on Sparse Grids
    Bettina Schieche
    Graduate School of Computational Engineering
    Numerical Analysis and Scientific Computing
    Technische Universität Darmstadt
    Numerical
    Analysis
    GAMM 2012
    Darmstadt, March 29, 2012
    www.graduate-school-ce.de March 29, 2012

    View full-size slide

  2. Outline
    Motivation for PDEs with Random Parameters
    Adaptive Stochastic Collocation Method
    Adjoint Error Estimation
    Bettina Schieche | GAMM 2012 | 2/17

    View full-size slide

  3. Overview
    Motivation for PDEs with Random Parameters
    Adaptive Stochastic Collocation Method
    Adjoint Error Estimation
    Bettina Schieche | GAMM 2012 | 3/17

    View full-size slide

  4. General Setting: Arbitrary PDE
    Describing parameters:
    Boundary and initial conditions
    Material properties
    Forcing terms / source terms
    Topology (geometry of the system)
    Bettina Schieche | GAMM 2012 | 4/17

    View full-size slide

  5. Sources of Uncertainties
    Natural fluctuations (e.g. speed of wind)
    Human-made fluctuations (e.g. fabrication processes)
    Lack of knowledge (e.g. spread of ash cloud)
    Lack of accuracy (e.g. errors of measurements)
    Bettina Schieche | GAMM 2012 | 5/17

    View full-size slide

  6. Sources of Uncertainties
    Natural fluctuations (e.g. speed of wind)
    Human-made fluctuations (e.g. fabrication processes)
    Lack of knowledge (e.g. spread of ash cloud)
    Lack of accuracy (e.g. errors of measurements)
    → PDE with additional dimensions: space, time + parameter space
    Bettina Schieche | GAMM 2012 | 5/17

    View full-size slide

  7. Overview
    Motivation for PDEs with Random Parameters
    Adaptive Stochastic Collocation Method
    Adjoint Error Estimation
    Bettina Schieche | GAMM 2012 | 6/17

    View full-size slide

  8. How to Discretize the Random Parameter Space?
    1. Choose P parameter realizations
    → collocation points on a sparse grid
    Bettina Schieche | GAMM 2012 | 7/17

    View full-size slide

  9. How to Discretize the Random Parameter Space?
    1. Choose P parameter realizations
    → collocation points on a sparse grid
    2. Solve P deterministic problems
    A(uj, ξ(j)) = f, j = 1, . . . , P
    Bettina Schieche | GAMM 2012 | 7/17

    View full-size slide

  10. How to Discretize the Random Parameter Space?
    1. Choose P parameter realizations
    → collocation points on a sparse grid
    2. Solve P deterministic problems
    A(uj, ξ(j)) = f, j = 1, . . . , P
    3. Interpolate all solutions
    Bettina Schieche | GAMM 2012 | 7/17

    View full-size slide

  11. How to Discretize the Random Parameter Space?
    1. Choose P parameter realizations
    → collocation points on a sparse grid
    2. Solve P deterministic problems
    A(uj, ξ(j)) = f, j = 1, . . . , P
    3. Interpolate all solutions
    4. Calculate statistics
    Bettina Schieche | GAMM 2012 | 7/17

    View full-size slide

  12. How to Discretize the Random Parameter Space?
    1. Choose P parameter realizations
    → collocation points on a sparse grid
    2. Solve P deterministic problems
    A(uj, ξ(j)) = f, j = 1, . . . , P
    3. Interpolate all solutions
    4. Calculate statistics
    5. Add new collocation points adaptively
    Bettina Schieche | GAMM 2012 | 7/17

    View full-size slide

  13. How to Discretize the Random Parameter Space?
    1. Choose P parameter realizations
    → collocation points on a sparse grid
    2. Solve P deterministic problems
    A(uj, ξ(j)) = f, j = 1, . . . , P
    3. Interpolate all solutions
    4. Calculate statistics
    5. Add new collocation points adaptively
    Bettina Schieche | GAMM 2012 | 7/17

    View full-size slide

  14. How to Discretize the Random Parameter Space?
    1. Choose P parameter realizations
    → collocation points on a sparse grid
    2. Solve P deterministic problems
    A(uj, ξ(j)) = f, j = 1, . . . , P
    3. Interpolate all solutions
    4. Calculate statistics
    5. Add new collocation points adaptively
    Bettina Schieche | GAMM 2012 | 7/17

    View full-size slide

  15. How to Discretize the Random Parameter Space?
    1. Choose P parameter realizations
    → collocation points on a sparse grid
    2. Solve P deterministic problems
    A(uj, ξ(j)) = f, j = 1, . . . , P
    3. Interpolate all solutions
    4. Calculate statistics
    5. Add new collocation points adaptively
    Stopping criterion: change < TOL
    Bettina Schieche | GAMM 2012 | 7/17

    View full-size slide

  16. Stochastic Collocation in a Nutshell
    DETERMINISTIC
    PDE-SOLVER
    CHOOSE
    COLLOCATION
    POINTS
    (SPARSE GRIDS)
    INTERPOLATE and
    CALCULATE
    STATISTICS
    (QUADRATURE)
    ADAPTIVITY /
    ERROR ESTIMATION
    Bettina Schieche | GAMM 2012 | 8/17

    View full-size slide

  17. Stationary Problem: Setting
    A(u, ξ) = −∇ · (a(x, ξ)∇u) = f
    9 uniformly distributed random variables
    Bettina Schieche | GAMM 2012 | 9/17

    View full-size slide

  18. Stationary Problem: Setting
    A(u, ξ) = −∇ · (a(x, ξ)∇u) = f
    9 uniformly distributed random variables
    Quantity of interest: Q(u) = E[u]
    Bettina Schieche | GAMM 2012 | 9/17

    View full-size slide

  19. Stationary Problem: Setting
    A(u, ξ) = −∇ · (a(x, ξ)∇u) = f
    9 uniformly distributed random variables
    Quantity of interest: Q(u) = E[u]
    Stochastic collocation → uhξ
    Bettina Schieche | GAMM 2012 | 9/17

    View full-size slide

  20. Stationary Problem: Setting
    A(u, ξ) = −∇ · (a(x, ξ)∇u) = f
    9 uniformly distributed random variables
    Quantity of interest: Q(u) = E[u]
    Stochastic collocation → uhξ
    Aim: Q(u) − Q(uhξ
    )
    !
    < TOL = 10−3
    Bettina Schieche | GAMM 2012 | 9/17

    View full-size slide

  21. Stationary Problem: Results
    0 50 100 150 200
    10−4
    10−3
    10−2
    10−1
    Number of Collocation Points
    Error
    indicator
    exact
    ⇒ Much more collocation points than necessary
    → TOL
    Bettina Schieche | GAMM 2012 | 10/17

    View full-size slide

  22. Overview
    Motivation for PDEs with Random Parameters
    Adaptive Stochastic Collocation Method
    Adjoint Error Estimation
    Bettina Schieche | GAMM 2012 | 11/17

    View full-size slide

  23. Stochastic Adjoint Error Estimation
    Error: Q(u) − Q(uhξ
    ) = ?
    Bettina Schieche | GAMM 2012 | 12/17

    View full-size slide

  24. Stochastic Adjoint Error Estimation
    Error: Q(u) − Q(uhξ
    ) = ?
    Solve additional stochastic equation = adjoint problem
    A(u, ξ) = f ↔ A∗(φ, ξ) = g
    Bettina Schieche | GAMM 2012 | 12/17

    View full-size slide

  25. Stochastic Adjoint Error Estimation
    Error: Q(u) − Q(uhξ
    ) = ?
    Solve additional stochastic equation = adjoint problem
    A(u, ξ) = f ↔ A∗(φ, ξ) = g
    Residual:
    Res(uhξ
    ) = f − A(uhξ
    )
    Bettina Schieche | GAMM 2012 | 12/17

    View full-size slide

  26. Stochastic Adjoint Error Estimation
    Error: Q(u) − Q(uhξ
    ) = ?
    Solve additional stochastic equation = adjoint problem
    A(u, ξ) = f ↔ A∗(φ, ξ) = g
    Residual:
    Res(uhξ
    ) = f − A(uhξ
    )
    Error estimate:
    Q(u) − Q(uhξ
    ) = E[φRes(uhξ
    )]
    Bettina Schieche | GAMM 2012 | 12/17

    View full-size slide

  27. Stationary Problem:
    Stochastic Adjoint Error Analysis
    Error splitting: Q(u) − Q(uhξ
    )
    joint error
    = Q(u) − Q(uξ)
    stochastic error
    + Q(uξ) − Q(uhξ
    )
    deterministic error
    Bettina Schieche | GAMM 2012 | 13/17

    View full-size slide

  28. Stationary Problem:
    Stochastic Adjoint Error Analysis
    Error splitting: Q(u) − Q(uhξ
    )
    joint error
    = Q(u) − Q(uξ)
    stochastic error
    + Q(uξ) − Q(uhξ
    )
    deterministic error
    0 200 400 600 800
    0
    0.2
    0.4
    0.6
    0.8
    1
    number of adjoint collocation points
    estimate / exact error
    deterministic error
    joint error
    stochastic error
    ⇒ Much effort to capture
    stochastic error
    ⇒ Few effort to capture
    determinitic error
    Bettina Schieche | GAMM 2012 | 13/17

    View full-size slide

  29. Stationary Problem: Stochastic Collocation
    with Full Adjoint Error Estimates
    0 50 100 150 200
    10−4
    10−3
    10−2
    10−1
    Number of Collocation Points
    Error
    indicator
    exact
    full adjoint
    ⇒ Error estimates very accurate
    ⇒ Drawback: > 300 adjoint collocation points
    → TOL
    Bettina Schieche | GAMM 2012 | 14/17

    View full-size slide

  30. Idea: Order Reduction of Adjoint Problem
    1. One iteration of stochastic collocation: A(ξ(j))Uj = F
    2. Adjoint solutions in these collocation points: A∗(ξ(j))Φj = G
    3. Snapshot matrix S = (Φ1, · · · , ΦP)
    4. Singular value decomposition of S → reduced basis ϕ
    5. Galerkin projection onto ϕ:
    A∗
    R
    (ξ)ΦR(ξ) = GR, dim(A∗
    R
    ) dim(A∗) (1)
    6. Evaluation of (1) in many adjoint collocation points
    Bettina Schieche | GAMM 2012 | 15/17

    View full-size slide

  31. Stationary Problem: Stochastic Collocation
    with Reduced Adjoint Error Estimator
    0 50 100 150 200
    10−4
    10−3
    10−2
    10−1
    Number of Collocation Points
    Error
    indicator
    exact
    full adjoint
    reduced adjoint
    ⇒ Reduced adjoints very close to full adjoints
    → TOL
    Bettina Schieche | GAMM 2012 | 16/17

    View full-size slide

  32. Conclusion & Outlook
    Stochastic collocation results in
    a set of deterministic equations.
    Full adjoint stochastic collocation
    is usually not practicable.
    Reduced order models can
    reduce computational costs.
    indicator full adjoint reduced adjoint
    0
    100
    200
    300
    400
    500
    Sum of Full Collocation Points
    Bettina Schieche | GAMM 2012 | 17/17

    View full-size slide

  33. Conclusion & Outlook
    Stochastic collocation results in
    a set of deterministic equations.
    Full adjoint stochastic collocation
    is usually not practicable.
    Reduced order models can
    reduce computational costs.
    indicator full adjoint reduced adjoint
    0
    100
    200
    300
    400
    500
    Sum of Full Collocation Points
    ⇒ Extension to unsteady problems
    ⇒ Order reduction of the primal problem
    Bettina Schieche | GAMM 2012 | 17/17

    View full-size slide

  34. Conclusion & Outlook
    Stochastic collocation results in
    a set of deterministic equations.
    Full adjoint stochastic collocation
    is usually not practicable.
    Reduced order models can
    reduce computational costs.
    indicator full adjoint reduced adjoint
    0
    100
    200
    300
    400
    500
    Sum of Full Collocation Points
    ⇒ Extension to unsteady problems
    ⇒ Order reduction of the primal problem
    The work is supported by the “Excellence Initiative“ of the German Federal and State
    Governments and the Graduate School of Computational Engineering at TU Darmstadt.
    Bettina Schieche | GAMM 2012 | 17/17

    View full-size slide