Bath Institute For Complex Systems - ENS de Cachan - Antenne de ...

More documents

Recommendations

Info

Figure 2: Left: Plot of E [ |u K ∗ ,h ∗ − u K ∗ ,h| H 1 (D)]versus 1/h for model problem (2.8) with d = 1,λ = 0.1 and σ 2 = 3. Right: Corresponding L 2 -error E [ ‖u K ∗ ,h ∗ − u K ∗ ,h‖ L 2 (D)]. The gradient ofthe dash-dotted (resp. dotted) line is −1/2 (resp. -1).Figure 3: Left: Plot of |E [ ‖u K ∗ ,h‖ L 2 (D) − ‖u K ∗ ,h ∗‖ L 2 (D)]|, for model problem (2.8) with d = 1,λ = 0.1, σ 2 = 3, K ∗ = 5000 and h ∗ = 1/2048. Right: Corresponding plot of the varianceV [ ‖u K ∗ ,h‖ L 2 (D) − ‖u K ∗ ,2h‖ L 2 (D)]. The gradient of the dotted (resp. dashed) line is −1 (resp. −2).5.3 Multilevel Monte Carlo ConvergenceWe first want to confirm Assumptions M1 and M2 in Theorem 4.1, for Q = ‖u‖ L 2 (D). In otherwords, we want to confirm the rate of decay for the quantities |E [ ‖u K ∗ ,h ∗‖ L 2 (D) − ‖u K ∗ ,h‖ L 2 (D)]|and V [ ‖u K ∗ ,h‖ L 2 (D) − ‖u K ∗ ,2h‖ L 2 (D)]. Again, we choose K ∗ = 5000 and h ∗ = 1/2048 and look atthe model problem (2.8) for d = 1. The results are shown in Figure 3. A dotted line indicateslinear convergence, and a dashed line indicates quadratic convergence.For the variance V [ ‖u K ∗ ,h‖ L 2 (D) − ‖u K ∗ ,2h‖ L 2 (D)](right plot), we observe the quadratic convergencepredicted by Proposition 4.3. The expected value in the left plot seems to converge slightlyfaster than the predicted linear convergence. In our experience, this faster convergence dependson the choice of model problem and quantity of interest and will need to be investigated further,since it directly affects the cost of the multilevel Monte Carlo method through the size of the ratioβ/α in the bound in Theorem 4.1.The actual performance of the standard MC and MLMC estimators in estimating Q = ‖u‖ L 2 (D)is shown in Figure 4. In the right plot the accuracy is scaled by the expected value of the quantity19
Figure 4: Left: Number of samples N l per level. Right: Plot of the cost scaled by ε −2 of theMLMC and standard MC estimators for d = 1, with λ = 0.1 and σ 2 = 3. The coarsest mesh sizein all tests is h 0 = 1/16.of interest. We see a clear advantage of the multilevel Monte Carlo method. For more numericalresults, in particular results in 2D, we refer the reader to [7].6 Conclusions and Further WorkOne of the major bottlenecks in probabilistic uncertainty quantification is the efficient numericalsolution of PDEs with random coefficients. A typical model problem that arises in subsurface flowis an elliptic PDE with log-normal coefficients. The resulting PDE is not uniformly elliptic orbounded with respect to the random variable ω and usually lacks also full regularity. In this paper,we carry out for the first time a careful finite element error analysis under these weaker assumptionsand provide optimal error estimates — with respect to the given regularity of the coefficients —for all finite moments of the finite element error (including quadrature errors). This then allowsus to rigorously analyse the convergence and the cost of multilevel Monte Carlo methods, recentlyproposed for elliptic PDEs with random coefficients, in particular in the practically important caseof log-normal coefficients with exponential covariance which has hitherto been unproved.Via duality arguments, the results in this paper are readily applicable also to other functionalsof the solution of more practical relevance. Other important aspects which we want to investigatein the future are the influence of the correlation length λ on the MLMC convergence, as well as thesuperconvergence in the expected value of certain model problems and quantities of interest (ashighlighted in the discussion of Figure 2). Another issue which will require further research is thechoice of quadrature rule. So far we have only used simple midpoint or trapezoidal rules. In futurework we would like to also explore multiscale methods and model reduction on coarser grids.ADetailed Regularity ProofIn this appendix, we give more detailed proofs of Proposition 3.1 and Lemmas 3.2 and 3.3. Theproofs follow those of Hackbusch [18, Theorems 9.1.8, 9.1.11 and 9.1.16], making explicit thedependence of all the constants that appear on the PDE coefficient a. The proof follows theclassical Nirenberg translation method and consists in three main steps.20
Page 1 and 2: BICSBath Institute for Complex Syst
Page 3 and 4: only Hölder continuous with expone
Page 5 and 6: for almost all ω ∈ Ω. The diffe
Page 7 and 8: 2.4 Truncated Karhunen-Loève Expan
Page 9 and 10: 3.1 Regularity of the SolutionPropo
Page 11 and 12: Theorem 3.4. Let Assumptions A1-A3
Page 13 and 14: Corollary 3.10. Let Assumptions A1-
Page 15 and 16: 3.4 Truncated FieldsCombining the r
Page 17 and 18: Since all the expectations E[Y l ]
Page 19: Figure 1: Let d = 1 and σ 2 = 1 in
Page 23 and 24: therefore we getA min ‖(R i h )
Page 25 and 26: ∥∂which means that2 ˜w∂x i
Page 27 and 28: Proof. Since supp(u 0 ) ⊂ D 0 , w
Page 29 and 30: We now extend A i to R d +. Since w

Bath Institute For Complex Systems - ENS de Cachan - Antenne de ...

Create successful ePaper yourself

Delete template?

Save as template?