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Efficient Uncertainty Quantification via Sparse Representation

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Efficient Uncertainty Quantification via Sparse Representation Powered By Docstoc
					Efficient Uncertainty Quantification via
       Sparse Representation

            Hermann G. Matthies
    Alexander Litvinenko, Dishi Liu, Elmar Zander

       Institute of Scientific Computing
                           a
     Technische Universit¨t Braunschweig
              Brunswick, Germany
             wire@tu-bs.de
        http://www.wire.tu-bs.de
                                                                                                   2
                           Overview

1. Uncertainty and stochastic models

2. General problem description

3. Solution in tensor product space

4. Model reduction and sparse representation

5. Low rank representation and algorithms

6. Conclusion




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TU Braunschweig                                Institute of Scientific Computing   Sci              g
                                                                                                      3
                   Sources of Uncertainty

 Mechanical/ physical systems may contain uncertain elements, as
 some details are not precisely known.
 • Action on the system from the rest of the world (surrounding
   environment).
 • The system itself may contain only incompletely known para-
   meters, processes or fields (not possible or too costly to measure)
 • There may be small, unresolved scales in the model, they act as
   a kind of background noise.
 All these items introduce some uncertainty in the model.


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TU Braunschweig                                   Institute of Scientific Computing   Sci              g
                                                                                                      4
                  Ontology and Modelling

 A bit of ontology:
 • Uncertainty may be aleatoric, which means random and not
   reducible (action from outside is often in this category), or

 • epistemic, which means due to incomplete knowledge (uncer-
   tainty in the system or model is often of this kind).
 Stochastic models give quantitative information about uncertainty,
 here used in modelling both types of uncertainty.

 Possible areas of use: Reliability, heterogeneous materials, upsca-
 ling, incomplete knowledge of details, uncertain [inter-]action with
 environment, random loading, etc.
                                                                                           ifiCompu
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TU Braunschweig                                   Institute of Scientific Computing   Sci              g
                                                                                                      5
                           Probability

What is probability?
We may understand probability as
• A mathematical concept — theory of a finite measure.
• Applies to aleatoric phenomena, i.e. frequencies of occurrence —
  Bernoulli’s law of large numbers.
• Applies also to epistemic concepts — extension of Aristotelian pro-
  positional logic to uncertain propositions (Cox’s theorem). Realm
  of Bayesian ideas and methods.
Exlusive application to first area is today often labelled classical or
frequentist, Bernoulli and Laplace had both in mind.


                                                                                           ifiCompu
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TU Braunschweig                                   Institute of Scientific Computing   Sci              g
                                                                                                             6
                      General Problem Description

 Consider mechanical / physical system (stationary for simplicity)
                          A(u) = f       u ∈ U, f ∈ F,
   where A models the system, F — space of actions / loadings,
         U — dual space of possible states of the system.

                     Solution is usually by first discretisation
                   A(u) = f        u ∈ UN ⊂ U, f ∈ FN ⊂ F,
                  and then (iterative) numerical solution process
                        uk+1 = Φ(uk ),        lim uk = u.
                                             k→∞

            Process is efficient, if Φ can be evaluated efficiently,
                 and if # of iterations k can be kept low.
                                                                                                  ifiCompu
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TU Braunschweig                                          Institute of Scientific Computing   Sci              g
                                                                                                            7
                      Model with Uncertainties
With uncertainties modelled by apropriate probab. space (Ω, P, A):
                    A[ω](u(ω)) = f (ω)         a.s. in ω ∈ Ω,
               State u(ω) is U - valued random variable (RV),
              viewed as element of tensor product S ⊗ U =: W,
              with R-valued RVs S. Similarly after discretisation
                   A[ω](u(ω)) = f (ω)          a.s. in ω ∈ Ω,
assume {v j }N a basis in UN , then the approx. solution in S ⊗ UN
             j=1
                                      N
                             u(ω) =         uj (ω)v j
                                      j=1
  is represented by N real-valued RVs uj (ω), to be approximated.


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TU Braunschweig                                         Institute of Scientific Computing   Sci              g
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               Direct Integration / Sampling Solution

                Builds on fact that ultimately E (Ψ (u)) is wanted.
                                                    Z
           E (Ψ (u)) =        Ψ (ω, u(ω)) P(dω) ≈         wz Ψ (ωz , u(ωz ))
                          Ω                         z=0
Pick (e.g. Monte Carlo) {ω}Z points, for each ωz do solution process
                           z=0
                           uk+1(ωz ) = Φ[ωz ](uk (ωz )),
                    giving u(ωz ) = j uj (ωz )v j = j uz v j .
                                                           j
                (Usually u(ωz ) discarded after use in integration.)

        Random state represented by collection [u(ω0), . . . , u(ωZ )],
                   or the tensor uZ := {uz }z=0,...,Z .
                                  N        j j=1,...,N



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  TU Braunschweig                                            Institute of Scientific Computing   Sci              g
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                           Solution by Emulation

          Emulation is replacement of expensive simulation by
                 inexpensive approximation −→ emulation
        ( alias response surfaces, proxy / surrogate models, etc.)

              Choose subspace SB ⊂ S with basis {Xβ }B ,
                                                       β=0
             make ansatz for each uj (ω) ≈ β uβ Xβ (ω), giving
                                              j


                  u(ω) =         uβ Xβ (ω)v j =
                                  j                     uβ Xβ (ω) ⊗ v j .
                                                         j
                           j,β                    j,β

  Solution is in tensor product WB,N := SB ⊗ UN ⊂ S ⊗ U = W.

   Random state u(ω) represented by tensor uB := {uβ }β=0,...,B .
                                            N      j j=1,...,N


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TU Braunschweig                                                 Institute of Scientific Computing   Sci              g
                                                                                                       10
                     Tensor Product Structure

          Story does not end here as one may choose S = m Sm,
                                      M
            approximated by SB = m=1 SBm with SBm ⊂ Sm.

           Solution represented as a tensor of grade M + 1
                                    M
                   in WB,N =        m=1 SBm ⊗ UN .
                — but that is a story for another talk.
 For higher grade tensor product structure, more reduction is possible,
               — here we stay with M = 1, i.e. grade 2.
                      Sparse representation entails
• reduce uB := [uβ ] to important info u ≈ uB ,
          N      j                          N

• never store all of uB , but only u,
                      N

• operate efficiently on sparse representation u.
                                                                                            ifiCompu
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   TU Braunschweig                                 Institute of Scientific Computing   Sci               g
                                                                                                              11
                        Computation of Co-Efficients

  Co-efficients computed by neural nets, Gaussian process emulation,
       Kriging, stochastic collocation, stochastic Galerkin, . . .

            Weighted residual method with weighting {Yγ (ω)}B ,
                                                            γ=1
                          ∀γ :     E    f − A(uB ) Yγ = 0.
                                               N
 Often Xα(ω) = Hα(ω), Norbert Wiener’s polynomial chaos expansion
(PCE) with Hermite polynomials in Gaussian variables. Many extensions
              (gPCE, ME-gPCE, wavelets, Fourier, ...).

      In stochastic collocation (and many other methods) you take
                           Yz (ω) = δ(ω − ωz ).

                    In stochastic Galerkin you take Yβ (ω) = Xβ (ω).

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  TU Braunschweig                                         Institute of Scientific Computing   Sci               g
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                    Model Reduction / Discretisation

            On continuous level discretisation is choice of subspace
                     WB,N := SB ⊗ UN ⊂ S ⊗ U = W
                and—important for computation—basis in it.

       On discrete level reduced models find subspace WR ⊂ WB,N
           with smaller dimensionalty dim WR = R     B × N.
                  They can work on SB or UN , or both.
             Different approaches to choose reduced model:
• Before the solution process (e.g. proper generalised decomposition).
• After the solution process (essentially data compression).
• During solution, computing solution and reduction simultaneously.


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  TU Braunschweig                                      Institute of Scientific Computing   Sci               g
                                                                                                             13
                       Low-Rank Approximation

Fokus on array of numbers uB := [uβ ], which one may view as matrix
                           N      j
                                 B   N
                                           uβ eβ ⊗ ej
                                            j
                                 β=0 j=1
                  with unit vectors eβ , ej .
The sum has M = (B + 1) ∗ N terms, the number of entries in uB .
                                                             N

                            An approximation with
                        B   N                    R
                                 uβ eβ ⊗ ej ≈
                                  j                   y ⊗w
                       β=0 j=1                   =1
                        is called a rank-R representation.
                   It contains only R ∗ (B + 1 + N ) numbers.

                                                                                                  ifiCompu
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 TU Braunschweig                                         Institute of Scientific Computing   Sci               g
                                                                                                    14
                  Use in Sampling Solution I

        Example: UQ-computations of RANS-flow around airfoils.




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TU Braunschweig                                 Institute of Scientific Computing   Sci               g
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                     Use in Sampling Solution II

                                                     e
Here reduction is achieved by truncating Karhunen-Lo`ve expansion
            (alias: singular value decomposition (SVD),
             proper orthogonal decomposition (POD)):

                     Relative errors, memory requirements:
            rank R     pressure   turb. kin. energy   memory [MB]
            10         1.9e-2     4.0e-3                       21
            20         1.4e-2     5.9e-3                       42
            50         5.3e-3     1.5e-4                      104

  Each tensor ∈ R260000×600. Dense matrix format costs 1.25 GB.

Made from 600 MC Simulations, SVD is updated every 10 samples.

                                                                                                ifiCompu
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TU Braunschweig                                        Institute of Scientific Computing   Sci               g
                                                                                                   16
                    Use in Time-Space Randomness I

Example: UQ-computations of time-dependent shallow-water flow.
               1:50 Scale model of Toce river (Italy)




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  TU Braunschweig                              Institute of Scientific Computing   Sci               g
                                                                                                      17
                    Time-Space Randomness II

                  Topography in model - random elevation




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                                                                                        ent        tin
TU Braunschweig                                   Institute of Scientific Computing   Sci               g
                                                                                                      18
                  Time-Space Randomness III

                     5 % exceedance water level




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TU Braunschweig                                   Institute of Scientific Computing   Sci               g
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                                  Use in Emulation
          Example: Diffusion equation with random conductivity.
                    Solution via stochastic Galerkin.

     Discretisation with finite elements and PCE, solution process
                                       uk+1 = Φ(uk )
                                     may be written as
                                                         M
                  uk+1 = uk − Ξ(uk ) = uk −                    Y i ⊗ G (uk ).
                                                         i=1
                                        R0
                      With u0 =         j=1 y 0,j   ⊗ g 0,j , this gives
                           R0                      M
                    u1 =         y 0,j ⊗ g 0,j −         Y i(u0) ⊗ Gi(u0)
                           j=1                     i=1

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TU Braunschweig                                                     Institute of Scientific Computing   Sci               g
                                                                                                                        20
                                  Truncated Iteration
      If iteration and rank-truncation are alternated, rank stays low.
                     Rank-truncation by updated SVD.

                             Rk                      M
                    ˆ
                    uk+1 =         y k,j ⊗ g k,j −         Y i(uk ) ⊗ Gi(uk ),
                             j=1                     i=1

                                                u
                                      uk+1 = T (ˆk+1).

It can be shown that truncated iteration converges until stagnation for
• super-linearly convergent iterative process (Hackbusch, Tyrtyshnikov),
• linearly convergent process with enlarged stagnation range (Zander).

                                                                                                             ifiCompu
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  TU Braunschweig                                                   Institute of Scientific Computing   Sci               g
                                                                                                        21
                        Truncation Accuracy

                  Accuracy of k-term tensor approximation.




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TU Braunschweig                                     Institute of Scientific Computing   Sci               g
                                                                                                                                     22
                                                Iteration Accuracy

                                        Convergence of truncated iteration.
                                    0
                                                                             ε=0.01
                                                                             ε=0.001
                                                                             ε=0.0001
                                   −1                                        ε=1e−05
                                                                             ε=1e−06
                                                                             ε=1e−07
                                                                             ε=1e−08
                                                                             ε=1e−09
                                   −2
                                                                             ε=1e−10
                                                                             ε=1e−11
                                                                             ε=1e−12
                  ||u −u||/||u||




                                                                             ε=1e−13
                                   −3
                        ε




                                   −4




                                   −5




                                   −6
                                   −10      0     10   20    30    40   50      60        70
                                                            iter




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TU Braunschweig                                                                  Institute of Scientific Computing   Sci               g
                                                                                                                            23
                                      Number of Iterations

                                Iteration count of truncated iteration.

                                60




                                50




                                40
                  #iterations




                                30




                                20




                                10




                                0
                                −14   −12    −10     −8       −6   −4           −2
                                                   log10(ε)




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                                                                                                              ent        tin
TU Braunschweig                                                         Institute of Scientific Computing   Sci               g
                                                                                                        24
                             Conclusion

• Stochastic calculations produce huge amount of data.

• For efficency try and use sparse representation throughout: ansatz in
  tensor products, as well as storage of solution and residuum—and
  iterartor in tensor products, sparse grids for integration.

• Works also for non-linear problems and solvers.

• Works also for time-dependent problems.




                                                                                             ifiCompu
                                                                                          ent        tin
  TU Braunschweig                                   Institute of Scientific Computing   Sci               g

				
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posted:3/16/2013
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