Skip to main content

Multi-index Stochastic Collocation Convergence Rates for Random PDEs with Parametric Regularity

Abstract

We analyze the recent Multi-index Stochastic Collocation (MISC) method for computing statistics of the solution of a partial differential equation (PDE) with random data, where the random coefficient is parametrized by means of a countable sequence of terms in a suitable expansion. MISC is a combination technique based on mixed differences of spatial approximations and quadratures over the space of random data, and naturally, the error analysis uses the joint regularity of the solution with respect to both the variables in the physical domain and parametric variables. In MISC, the number of problem solutions performed at each discretization level is not determined by balancing the spatial and stochastic components of the error, but rather by suitably extending the knapsack-problem approach employed in the construction of the quasi-optimal sparse-grids and Multi-index Monte Carlo methods, i.e., we use a greedy optimization procedure to select the most effective mixed differences to include in the MISC estimator. We apply our theoretical estimates to a linear elliptic PDE in which the log-diffusion coefficient is modeled as a random field, with a covariance similar to a Matérn model, whose realizations have spatial regularity determined by a scalar parameter. We conduct a complexity analysis based on a summability argument showing algebraic rates of convergence with respect to the overall computational work. The rate of convergence depends on the smoothness parameter, the physical dimensionality and the efficiency of the linear solver. Numerical experiments show the effectiveness of MISC in this infinite dimensional setting compared with the Multi-index Monte Carlo method and compare the convergence rate against the rates predicted in our theoretical analysis.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Notes

  1. Recall that, given \(q \ge 1\), \(L^q(\Gamma ;V) = \left\{ v : \Gamma \rightarrow V \text { strongly measurable, such that } \int _\Gamma \left\| u \right\| _{V} ^q~{\text {d}}\mu < \infty \right\} \).

  2. We recall that \(H^{{{\varvec{l}}}}({\mathscr {B}})\) is the completion of formal sums \(v=\sum _{k=1}^{K} v_{1,k}v_{2,k}\cdots v_{D,k}\) with \(v_{i,k} \in H^{l_i}({\mathscr {B}}_i)\) with respect to the norm induced by the inner product

    $$\begin{aligned} (v,w)_{H^{{{\varvec{l}}}}({\mathscr {B}})} = \sum _{k,i} (v_{1,k},w_{1,i})_{H^{l_1}({\mathscr {B}}_1)}(v_{2,k},w_{2,i})_{H^{l_2}({\mathscr {B}}_2)}\cdots (v_{D,k},w_{D,i})_{H^{l_D}({\mathscr {B}}_D)}. \end{aligned}$$

References

  1. I. Babuška, F. Nobile, and R. Tempone, A stochastic collocation method for elliptic partial differential equations with random input data, SIAM Review, 52 (2010), 317–355.

    MathSciNet  Article  MATH  Google Scholar 

  2. A. Barth, C. Schwab, and N. Zollinger, Multi-level Monte Carlo finite element method for elliptic PDEs with stochastic coefficients, Numerische Mathematik, 119 (2011), 123–161.

    MathSciNet  Article  MATH  Google Scholar 

  3. J. Beck, F. Nobile, L. Tamellini, and R. Tempone, On the optimal polynomial approximation of stochastic PDEs by Galerkin and collocation methods, Mathematical Models and Methods in Applied Sciences, 22 (2012), 1250023.

    MathSciNet  Article  MATH  Google Scholar 

  4. M. Bieri, A sparse composite collocation finite element method for elliptic SPDEs., SIAM Journal on Numerical Analysis, 49 (2011), 2277–2301.

    MathSciNet  Article  MATH  Google Scholar 

  5. V. I. Bogachev, Measure Theory, Vol. 1, Springer Berlin Heidelberg, 2007.

    Book  MATH  Google Scholar 

  6. H. J. Bungartz and M. Griebel, Sparse grids, Acta Numerica, 13 (2004), 147–269.

    MathSciNet  Article  MATH  Google Scholar 

  7. H. J. Bungartz, M. Griebel, D. Röschke, and C. Zenger, Pointwise convergence of the combination technique for the Laplace equation, East-West Journal of Numerical Mathematics, 2 (1994), 21–45.

    MathSciNet  MATH  Google Scholar 

  8. J. Charrier, Strong and weak error estimates for elliptic partial differential equations with random coefficients, SIAM Journal on Numerical Analysis, 50 (2012), 216–246.

    MathSciNet  Article  MATH  Google Scholar 

  9. J. Charrier, R. Scheichl, and A. Teckentrup, Finite element error analysis of elliptic pdes with random coefficients and its application to multilevel Monte Carlo methods, SIAM Journal on Numerical Analysis, 51 (2013), 322–352.

    MathSciNet  Article  MATH  Google Scholar 

  10. A. Chkifa, On the Lebesgue constant of Leja sequences for the complex unit disk and of their real projection, Journal of Approximation Theory, 166 (2013), 176–200.

  11. A. Cohen, R. Devore, and C. Schwab, Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDE’S, Analysis and Applications, 9 (2011), 11–47.

    MathSciNet  Article  MATH  Google Scholar 

  12. N. Collier, A.-L. Haji-Ali, F. Nobile, E. von Schwerin, and R. Tempone, A continuation multilevel Monte Carlo algorithm, BIT Numerical Mathematics, 55 (2015), 399–432.

    MathSciNet  Article  MATH  Google Scholar 

  13. G. M. Constantine and T. H. Savits, A multivariate Faà di Bruno formula with applications, Transactions of the American Mathematical Society, 348 (1996), 503–520.

    MathSciNet  Article  MATH  Google Scholar 

  14. D. Dũng and M. Griebel, Hyperbolic cross approximation in infinite dimensions, Journal of Complexity, 33 (2016), 55–88.

  15. B. Ganapathysubramanian and N. Zabaras, Sparse grid collocation schemes for stochastic natural convection problems, jcp, 225 (2007), 652–685.

    MathSciNet  MATH  Google Scholar 

  16. M. B. Giles, Multilevel Monte Carlo path simulation, Operations Research, 56 (2008), 607–617.

    MathSciNet  Article  MATH  Google Scholar 

  17. W. J. Gordon and C. A. Hall, Construction of curvilinear co-ordinate systems and applications to mesh generation, International Journal for Numerical Methods in Engineering, 7 (1973), 461–477.

    MathSciNet  Article  MATH  Google Scholar 

  18. I. G. Graham, R. Scheichl, and E. Ullmann, Mixed finite element analysis of lognormal diffusion and multilevel Monte Carlo methods, Stochastic Partial Differential Equations: Analysis and Computations, (2015), 1–35.

  19. M. Griebel and H. Harbrecht, On the convergence of the combination technique, in Sparse Grids and Applications - Munich 2012, J. Garcke and D. Pflüger, eds., vol. 97 of Lecture Notes in Computational Science and Engineering, Springer International Publishing, 2014, 55–74.

  20. M. Griebel and S. Knapek, Optimized general sparse grid approximation spaces for operator equations, Mathematics of Computation, 78 (2009), 2223–2257.

    MathSciNet  Article  MATH  Google Scholar 

  21. M. Griebel, M. Schneider, and C. Zenger, A combination technique for the solution of sparse grid problems, in Iterative Methods in Linear Algebra, P. de Groen and R. Beauwens, eds., IMACS, Elsevier, North Holland, 1992, pp. 263–281.

    Google Scholar 

  22. A.-L. Haji-Ali, F. Nobile, L. Tamellini, and R. Tempone, Multi-index stochastic collocation for random PDEs, Computer Methods in Applied Mechanics and Engineering, 306 (2016), 95–122.

  23. A.-L. Haji-Ali, F. Nobile, and R. Tempone, Multi-index Monte Carlo: when sparsity meets sampling, Numerische Mathematik, 132 (2015), 767–806.

    MathSciNet  Article  MATH  Google Scholar 

  24. A.-L. Haji-Ali, F. Nobile, E. von Schwerin, and R. Tempone, Optimization of mesh hierarchies in multilevel Monte Carlo samplers, Stochastic Partial Differential Equations: Analysis and Computations, 4 (2015), 76–112.

    MathSciNet  Article  Google Scholar 

  25. H. Harbrecht, M. Peters, and M. Siebenmorgen, On multilevel quadrature for elliptic stochastic partial differential equations, in Sparse Grids and Applications, vol. 88 of Lecture Notes in Computational Science and Engineering, Springer, 2013, 161–179.

  26. M. Hegland, J. Garcke, and V. Challis, The combination technique and some generalisations, Linear Algebra and its Applications, 420 (2007), 249–275.

    MathSciNet  Article  MATH  Google Scholar 

  27. S. Heinrich, Multilevel Monte Carlo methods, in Large-Scale Scientific Computing, vol. 2179 of Lecture Notes in Computer Science, Springer Berlin Heidelberg, 2001, 58–67.

  28. T. J. R. Hughes, J. A. Cottrell, and Y. Bazilevs, Isogeometric analysis: CAD, finite elements, nurbs, exact geometry and mesh refinement, Computer Methods in Applied Mechanics and Engineering, 194 (2005), 4135–4195.

    MathSciNet  Article  MATH  Google Scholar 

  29. F. Y. Kuo, C. Schwab, and I. Sloan, Multi-level Quasi-Monte Carlo Finite Element Methods for a Class of Elliptic PDEs with Random Coefficients, Foundations of Computational Mathematics, 15 (2015), 411–449.

    MathSciNet  Article  MATH  Google Scholar 

  30. S. Martello and P. Toth, Knapsack problems: algorithms and computer implementations, Wiley-Interscience series in discrete mathematics and optimization, J. Wiley & Sons, 1990.

  31. A. Narayan and J. D. Jakeman, Adaptive Leja Sparse Grid Constructions for Stochastic Collocation and High-Dimensional Approximation, SIAM Journal on Scientific Computing, 36 (2014), A2952–A2983.

    MathSciNet  Article  MATH  Google Scholar 

  32. F. Nobile, L. Tamellini, and R. Tempone, Comparison of Clenshaw-Curtis and Leja Quasi-Optimal Sparse Grids for the Approximation of Random PDEs, in Spectral and High Order Methods for Partial Differential Equations - ICOSAHOM ’14, R. M. Kirby, M. Berzins, and J. S. Hesthaven, eds., vol. 106 of Lecture Notes in Computational Science and Engineering, Springer International Publishing, 2015, 475–482.

  33. F. Nobile, L. Tamellini, and R. Tempone, Convergence of quasi-optimal sparse-grid approximation of Hilbert-space-valued functions: application to random elliptic PDEs, Numerische Mathematik, 134(2) (2016), 343–388.

  34. F. Nobile, R. Tempone, and C. Webster, An anisotropic sparse grid stochastic collocation method for partial differential equations with random input data, SIAM Journal on Numerical Analysis, 46 (2008), 2411–2442.

    MathSciNet  Article  MATH  Google Scholar 

  35. F. Nobile, R. Tempone, and C. Webster, A sparse grid stochastic collocation method for partial differential equations with random input data, SIAM Journal on Numerical Analysis, 46 (2008), 2309–2345.

    MathSciNet  Article  MATH  Google Scholar 

  36. C. Schillings and C. Schwab, Sparse, adaptive Smolyak quadratures for Bayesian inverse problems, Inverse Problems, 29 (2013), 065011.

    MathSciNet  Article  MATH  Google Scholar 

  37. A. Teckentrup, P. Jantsch, C. G. Webster, and M. Gunzburger, A Multilevel Stochastic Collocation Method for Partial Differential Equations with Random Input Data, SIAM/ASA Journal on Uncertainty Quantification, 3 (2015), 1046–1074.

    MathSciNet  Article  MATH  Google Scholar 

  38. H. W. van Wyk, Multilevel sparse grid methods for elliptic partial differential equations with random coefficients, arXiv preprint arXiv:1404.0963, 2014.

  39. G. W. Wasilkowski and H. Wozniakowski, Explicit cost bounds of algorithms for multivariate tensor product problems, Journal of Complexity, 11 (1995), 1–56.

    MathSciNet  Article  MATH  Google Scholar 

  40. D. Xiu and J. Hesthaven, High-order collocation methods for differential equations with random inputs, SIAM Journal on Scientific Computing, 27 (2005), 1118–1139.

    MathSciNet  Article  MATH  Google Scholar 

  41. C. Zenger, Sparse grids, in Parallel Algorithms for Partial Differential Equations, W. Hackbusch, ed., vol. 31 of Notes on Numerical Fluid Mechanics, Vieweg, 1991, pp. 241–251.

Download references

Acknowledgments

F. Nobile and L. Tamellini received support from the Center for ADvanced MOdeling Science (CADMOS) and partial support by the Swiss National Science Foundation under the Project No. 140574 “Efficient numerical methods for flow and transport phenomena in heterogeneous random porous media”. L. Tamellini also received support from the Gruppo Nazionale Calcolo Scientifico - Istituto Nazionale di Alta Matematica “Francesco Severi” (GNCS-INDAM). R. Tempone is a member of the KAUST Strategic Research Initiative, Center for Uncertainty Quantification in Computational Sciences and Engineering. R. Tempone received support from the KAUST CRG3 Award Ref: 2281.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Raúl Tempone.

Additional information

Communicated by Albert Cohen.

Appendices

Appendix 1: Summability of Series Expansion

We start by recalling a useful multivariate Faà di Bruno formula taken from [13, Theorem 2.1].

Lemma 11

Let \({\mathscr {B}}\subset {\mathbb {R}}^d\) be an open domain, \(g:{\mathscr {B}}\rightarrow {\mathbb {R}}\) and \(f:{\mathbb {R}}\rightarrow {\mathbb {R}}\) be functions of class \(C^s(\mathscr {B})\) and denote \(h=f\circ g : {\mathscr {B}}\rightarrow {\mathbb {R}}\). For any multi-index \({{\varvec{i}}}\in {\mathbb {N}}^d\), \(|{{\varvec{i}}}| \le s\), and any \({{\varvec{x}}}\in {\mathscr {B}}\),

$$\begin{aligned} D^{{\varvec{i}}}h({{\varvec{x}}}) = {{\varvec{i}}}!\sum _{\lambda =1}^{|{{\varvec{i}}}|} f^{(\lambda )}(g({{\varvec{x}}})) \sum _{r=1}^{\lambda } \sum _{p_r({{\varvec{i}}},\lambda )} \prod _{j=1}^r \frac{(D^{{\varvec{\ell }}_j}g({{\varvec{x}}}))^{k_j}}{k_j! ({\varvec{\ell }}_j!)^{k_j}}, \end{aligned}$$
(47)

holds, where

$$\begin{aligned}&p_r({{\varvec{i}}},\lambda ) = \{ (k_j,{\varvec{\ell }}_j)\in {\mathbb {N}}\times {\mathbb {N}}^d_0, \; j=1,\ldots ,r: \;\; \varvec{0}\prec {\varvec{\ell }}_1\prec {\varvec{\ell }}_2 \prec \cdots \prec {\varvec{\ell }}_r, \\&\quad \sum _{j=1}^r k_j = \lambda , \;\;\sum _{j=1}^r k_j{\varvec{\ell }}_j = {{\varvec{i}}}\} \end{aligned}$$

and \(\prec \) denotes the lexicographic ordering of multi-indices. The set \(p_r({{\varvec{i}}},\lambda )\) denotes the set of possible decompositions of \({{\varvec{i}}}\) as a sum of \(\lambda \) multi-indices with \(r\le \lambda \) distinct multi-indices, \({\varvec{\ell }}_j\), taken with multiplicity \(k_j\) such that \(\sum _{j=1}^r k_j=\lambda \).

Also from [13, Corollary 2.9], we have that, for any \({{\varvec{i}}}\in {\mathbb {N}}^d\),

$$\begin{aligned} {{\varvec{i}}}! \sum _{r=1}^{\lambda } \sum _{p_r({{\varvec{i}}},\lambda )} \prod _{j=1}^r \frac{1}{k_j! ({\varvec{\ell }}_j!)^{k_j}} = S_{|{{\varvec{i}}}|,\lambda }, \end{aligned}$$

where \(S_{n,k}\) is the Stirling number of the second kind, which counts the number of ways to partition a set of n objects into k non-empty subsets. Similarly, the Bell number, \(B_n=\sum _{k=0}^{n} S_{n,k}\), counts the number of partitions of a set of n objects, whereas the ordered Bell numbers are defined by \(\tilde{B}_n = \sum _{k=0}^{n} k! S_{n,k}\) and satisfy the recursive relation \(\tilde{B}_n = \sum _{k=0}^{n-1} \left( {\begin{array}{c}n\\ k\end{array}}\right) \tilde{B}_k\). Clearly, \(B_n\le \tilde{B}_n\). Moreover, the bound

$$\begin{aligned} B_n\le \tilde{B}_n \le n!/(\log 2)^n \end{aligned}$$
(48)

was given in [3, Lemma A.3]. We now use these results to show the following result

Lemma 12

Let \({\mathscr {B}} \subset {\mathbb {R}}^d\) be an open-bounded domain and \(\kappa \in C^s(\overline{{\mathscr {B}}})\) (real or complex valued) for \(s \ge 0\). Then, \(a = e^\kappa \in C^s(\overline{{\mathscr {B}}})\) and we have the estimate

$$\begin{aligned} \Vert a\Vert _{C^s(\overline{{\mathscr {B}}})} \le \frac{s!}{(\log 2)^s}\Vert a\Vert _{C^0(\overline{{\mathscr {B}}})}(1+\Vert \kappa \Vert _{C^s(\overline{{\mathscr {B}}})})^s. \end{aligned}$$

Proof

Using formula (47), we have for any \({{\varvec{i}}}\in {\mathbb {N}}^d\), \(|{{\varvec{i}}}|\le s\) and any \({{\varvec{x}}}\in {\mathscr {B}}\)

$$\begin{aligned} |D^{{\varvec{i}}}e^{\kappa ({{\varvec{x}}})}|&= {{\varvec{i}}}!\sum _{\lambda =1}^{|{{\varvec{i}}}|} e^{\kappa ({{\varvec{x}}})} \sum _{r=1}^{\lambda } \sum _{p_r({{\varvec{i}}},\lambda )} \prod _{j=1}^r \frac{|D^{{\varvec{\ell }}_j}\kappa ({{\varvec{x}}})|^{k_j}}{k_j! ({\varvec{\ell }}_j!)^{k_j}} \le \Vert a\Vert _{C^0(\overline{{\mathscr {B}}})} \sum _{\lambda =1}^{|{{\varvec{i}}}|} \Vert \kappa \Vert _{C^s(\overline{{\mathscr {B}}})}^\lambda S_{|{{\varvec{i}}}|,\lambda }\\&\le \Vert a\Vert _{C^0(\overline{{\mathscr {B}}})} (1+\Vert \kappa \Vert _{C^s(\overline{{\mathscr {B}}})})^{|{{\varvec{i}}}|} B_n. \end{aligned}$$

The result then follows from the bound on the Bell numbers in (48). \(\square \)

\(L^p(\Gamma )\) Summability, Pointwise in Space

We now consider a diffusion coefficient as in Assumption A2:

$$\begin{aligned} a({{\varvec{x}}},{{\varvec{y}}}) = \exp \left\{ \sum _{j\ge 1} \psi _j({{\varvec{x}}})y_j\right\} = \prod _{j=1}^\infty e^{y_j\psi _j({{\varvec{x}}})}, \qquad {{\varvec{x}}}\in {\mathscr {B}}, \end{aligned}$$

with \(y_j\), \(j \ge 1\), independent random variables, all uniformly distributed in \([-1,1]\) and recall the definition of the sequence \({{\varvec{b}}}_{s}=\{b_{s,j}\}_{j\ge 1}\), for all \(s\in {\mathbb {N}}\) in (6).

Lemma 13

If \({{\varvec{b}}}_0\in \ell ^2\) then \({\mathbb {E}}\left[ a({{\varvec{x}}})^p\right] <\infty \) for all \(0<p<\infty \) and \(\forall {{\varvec{x}}}\in {\mathscr {B}}\).

Proof

For any \({{\varvec{x}}}\in {\mathscr {B}}\), we estimate the p-th moment of \(a({{\varvec{x}}},{{\varvec{y}}})\), exploiting the independence of the random variables, \(y_j\):

$$\begin{aligned} {\mathbb {E}}\left[ a({{\varvec{x}}})^p\right]&= {\mathbb {E}}\left[ \prod _{j=1}^\infty e^{py_j\psi _j({{\varvec{x}}})}\right] = \prod _{j=1}^\infty {\mathbb {E}}\left[ e^{py_j\psi _j({{\varvec{x}}})}\right] = \prod _{j=1}^\infty \frac{\sinh (p\psi _j({{\varvec{x}}}))}{p\psi _j({{\varvec{x}}})} \\&= \exp \left\{ \sum _{j=1}^\infty \log \left( \frac{\sinh (p\psi _j({{\varvec{x}}}))}{p\psi _j({{\varvec{x}}})}\right) \right\} \end{aligned}$$

where in the last two equalities we have implicitly assumed that \(\sinh (z)/z=1\) for \(z=0\). Setting \(\theta _0(p;{{\varvec{x}}}) = \prod _{j=1}^\infty \frac{\sinh (p\psi _j({{\varvec{x}}}))}{p\psi _j({{\varvec{x}}})}\) and observing that \(\log (\sinh (z)/z) \sim z^2/6\), we have

$$\begin{aligned} {\mathbb {E}}\left[ a({{\varvec{x}}})^p\right] = \theta _0(p;{{\varvec{x}}})<\infty \quad \forall {{\varvec{x}}}\in {\mathscr {B}}, \;\; 0<p<\infty \qquad \Longleftrightarrow \qquad \sum _{j=1}^\infty \psi _j({{\varvec{x}}})^2 <\infty . \end{aligned}$$

Since \(\sum _{j=1}^\infty b_{0,j}^2 <\infty \) implies \(\sum _{j=1}^\infty \psi _j({{\varvec{x}}})^2 <\infty \) for any \({{\varvec{x}}}\in {\mathscr {B}}\), this concludes the proof. \(\square \)

A similar result holds for higher-order derivatives of a.

Lemma 14

Let \(s\in {\mathbb {N}}_+\). If \({{\varvec{b}}}_s\in \ell ^2\), then for any \({{\varvec{i}}}\in {\mathbb {N}}^d\), \(|{{\varvec{i}}}|=s\), \({\mathbb {E}}\left[ (D^{{\varvec{i}}}a({{\varvec{x}}}))^{2p}\right] <\infty \) for all \(0<p<\infty \) and \(\forall {{\varvec{x}}}\in {\mathscr {B}}\).

Proof

Since the calculations are tedious, we prove the result here for \(s=1\) only. Using the chain rule, we have

$$\begin{aligned} (\partial _{x_i}a({{\varvec{x}}},{{\varvec{y}}}))^{2p}&= \left( \sum _{j \ge 1} a({{\varvec{x}}},{{\varvec{y}}})\partial _{x_i}\psi _j({{\varvec{x}}})y_j\right) ^{2p} \\&= a({{\varvec{x}}},{{\varvec{y}}})^{2p}{\mathop {\mathop {\sum }\limits _{{{\varvec{q}}}\in {\mathbb {N}}^{{\mathbb {N}}}}}\limits _{|{{\varvec{q}}}|=2p}} (2p)! \prod _{j=1}^\infty \frac{1}{q_j!}(\partial _{x_i}\psi _j({{\varvec{x}}})y_j)^{q_j}\\&={\mathop {\mathop {\sum }\limits _{{{\varvec{q}}}\in {\mathbb {N}}^{{\mathbb {N}}}}}\limits _{|{{\varvec{q}}}|=2p}} (2p)! \prod _{j=1}^\infty \frac{1}{q_j!}(\partial _{x_i}\psi _j({{\varvec{x}}})y_j)^{q_j}e^{2py_j\psi _j({{\varvec{x}}})}. \end{aligned}$$

Hence,

$$\begin{aligned} {\mathbb {E}}\left[ (\partial _{x_i}a({{\varvec{x}}},{{\varvec{y}}}))^{2p}\right] = {\mathop {\mathop {\sum }\limits _{{{\varvec{q}}}\in {\mathbb {N}}^{{\mathbb {N}}}}}\limits _{|{{\varvec{q}}}|=2p}} (2p)! \prod _{j=1}^\infty (\partial _{x_i}\psi _j({{\varvec{x}}}))^{q_j}{\mathbb {E}}\left[ \frac{1}{q_j!}y_j^{q_j}e^{2py_j\psi _j({{\varvec{x}}})}\right] . \end{aligned}$$

We now distinguish between even or odd \(q_j\). For even \(q_j\), we have

$$\begin{aligned} {\mathbb {E}}\left[ \frac{1}{q_j!}y_j^{q_j}e^{2py_j\psi _j({{\varvec{x}}})}\right] \le {\mathbb {E}}\left[ \frac{1}{q_j!}e^{2py_j\psi _j({{\varvec{x}}})}\right] = \frac{1}{q_j!} \frac{\sinh (2p\psi _j({{\varvec{x}}}))}{2p\psi _j({{\varvec{x}}})}, \end{aligned}$$

while for \(q_j\) odd we have

$$\begin{aligned} {\mathbb {E}}\left[ \frac{1}{q_j!}y_j^{q_j}e^{2py_j\psi _j({{\varvec{x}}})}\right]&=\! \frac{1}{q_j!}\int _{-1}^1 \frac{1}{2}y^{q_j}e^{2py\psi _j({{\varvec{x}}})}dy \!=\! \frac{1}{q_j!}\int _{0}^1 y^{q_j}\sinh (2py\psi _j({{\varvec{x}}}))dy \\&=\frac{1}{q_j!}\sum _{n=0}^\infty \frac{(2p\psi _j({{\varvec{x}}}))^{2n+1}}{(2n+1)!}\int _0^1y^{2n+1+q_j}dy \\&= \frac{1}{q_j!}\sum _{n=0}^\infty \frac{(2p\psi _j({{\varvec{x}}}))^{2n+1}}{(2n+1)!(2n+2+q_j)} \\&\le \frac{1}{(q_j+1)!}\sinh (2p|\psi _j({{\varvec{x}}})|) \le \frac{2pb_{1,j}}{(q_j+1)!} \frac{\sinh (2p\psi _j({{\varvec{x}}}))}{2p\psi _j({{\varvec{x}}})}. \end{aligned}$$

Hence, defining the function

$$\begin{aligned} f(q_j) = {\left\{ \begin{array}{ll} \frac{1}{q_j!} &{} {\text {for }}q_j {\text { even}},\\ \frac{2pb_{1,j}}{(q_j+1)!} &{} {\text {for }}q_j {\text { odd}}, \end{array}\right. } \end{aligned}$$

we have

$$\begin{aligned} {\mathbb {E}}\left[ (\partial _{x_i}a({{\varvec{x}}},{{\varvec{y}}}))^{2p}\right]&\le {\mathop {\mathop {\sum }\limits _{{{\varvec{q}}}\in {\mathbb {N}}^{{\mathbb {N}}}}}\limits _{|{{\varvec{q}}}|=2p}} (2p)! \prod _{j=1}^\infty b_{1,j}^{q_j} f(q_j) \frac{\sinh (2p\psi _j({{\varvec{x}}}))}{2p\psi _j({{\varvec{x}}})} \\&= \theta _0(2p;{{\varvec{x}}}){\mathop {\mathop {\sum }\limits _{{{\varvec{q}}}\in {\mathbb {N}}^{{\mathbb {N}}}}}\limits _{|{{\varvec{q}}}|=2p}} (2p)! \prod _{j=1}^\infty b_{1,j}^{q_j} f(q_j)\\&\le \theta _0(2p;{{\varvec{x}}}){\mathop {\mathop {\sum }\limits _{{{\varvec{q}}}\in {\mathbb {N}}^{{\mathbb {N}}}}}\limits _{|{{\varvec{q}}}|=2p, {{\varvec{q}}}\text { even}}} (2p)! (1+2p)^{|{{\varvec{q}}}|_0} \prod _{j=1}^\infty \frac{b_{1,j}^{q_j}}{q_j!} \\&\le (1+2p)^{p} \theta _0(2p;{{\varvec{x}}}){\mathop {\mathop {\sum }\limits _{{{\varvec{q}}}\in {\mathbb {N}}^{{\mathbb {N}}}}}\limits _{|{{\varvec{q}}}|=p}} (2p)! \prod _{j=1}^\infty \frac{b_{1,j}^{2q_j}}{(2q_j)!} \\&\le (1+2p)^{p} (2p)^p \theta _0(2p;{{\varvec{x}}}){\mathop {\mathop {\sum }\limits _{{{\varvec{q}}}\in {\mathbb {N}}^{{\mathbb {N}}}}}\limits _{|{{\varvec{q}}}|=p}} p! \prod _{j=1}^\infty \frac{(b_{1,j}^2)^{q_j}}{q_j!} \\&= (1+2p)^{p} (2p)^p \theta _0(2p;{{\varvec{x}}}) \left( \sum _{j\ge 1} b_{1,j}^2\right) \end{aligned}$$

from which we see that \({\mathbb {E}}\left[ (\partial _{x_i}a({{\varvec{x}}},{{\varvec{y}}}))^{2p}\right] \) is bounded for any \(0\le p<\infty \) and any \({{\varvec{x}}}\in {\mathscr {B}}\) if \({{\varvec{b}}}_1\in \ell ^2\). \(\square \)

\(L^p(\Gamma )\) Summability, Uniform in Space

Assuming now that \({{\varvec{b}}}_s\in \ell ^2\) so that the random field, a, is s-times differentiable in an \(L^p(\Gamma )\) sense according to Lemma 14, we show that this implies some uniform \(L^p(\Gamma )\) summability as detailed in the next lemma.

Lemma 15

Let \(s\in {\mathbb {N}}_+\). If \({{\varvec{b}}}_s\in \ell ^2\) then \({\mathbb {E}}\left[ \Vert a\Vert ^p_{W^{\upsilon ,\infty }({\mathscr {B}})}\right] <\infty \) for all \(1\le p<\infty \) and \(\upsilon <s\).

Proof

We exploit the Sobolev embedding, \(W^{\upsilon +\frac{d}{2q},2q}({\mathscr {B}}) \subseteq W^{\upsilon ,\infty }({\mathscr {B}})\), for all \(\upsilon \ge 0\) and \(q\ge 1\). For \(q\ge \max \{d/2(s-\upsilon ),p/2\}\), we have

$$\begin{aligned}&{\mathbb {E}}\left[ \Vert a\Vert ^{p}_{W^{\upsilon ,\infty }({\mathscr {B}})}\right] \le {\mathbb {E}}\left[ \Vert a\Vert ^{2q}_{W^{s-\frac{d}{2q},\infty }({\mathscr {B}})}\right] \lesssim {\mathbb {E}}\left[ \Vert a\Vert ^{2q}_{W^{s,2q}({\mathscr {B}})}\right] \\&\quad = {\mathbb {E}}\left[ \sum _{|{{\varvec{i}}}|\le s} \int _{{\mathscr {B}}} (D^{{\varvec{i}}}a({{\varvec{x}}}))^{2q}d{{\varvec{x}}}\right] = \sum _{|{{\varvec{i}}}|\le s} \int _{{\mathscr {B}}} {\mathbb {E}}\left[ (D^{{\varvec{i}}}a({{\varvec{x}}}))^{2q}\right] d{{\varvec{x}}}< \infty , \end{aligned}$$

where the last term is bounded from Lemma 14. \(\square \)

Now, we directly observe by taking \(\upsilon =0\) in the previous result that \(a_{\max } = \Vert a\Vert _{L^\infty ({\mathscr {B}})}\) has bounded moments,

$$\begin{aligned}{\mathbb {E}}\left[ a_{\max }^p\right] <\infty , \end{aligned}$$

for all \(1\le p<\infty \) and \(0<s\). Finally, by observing that, due to (2), in we have that \(a_{\min } = \frac{1}{\Vert a^{-1}\Vert _{L^\infty ({\mathscr {B}})}}\) has the same distribution as \(a_{\max }\). As a consequence, \(a_{\min }\) has bounded moments as well. This implies in turn that (3) holds and thus problem (1) is well posed in the Bochner space, \(L^p\left( \Gamma ;H^1_0({\mathscr {B}})\right) \). That is,

Corollary 16

(Well-posedness with log uniform coefficient) We have for \(0<\nu \) that the problem in Example 1 is well posed in the Bochner space \(L^p\left( \Gamma ;H^1_0({\mathscr {B}})\right) \). The corresponding solution, u, satisfies

$$\begin{aligned} \Vert u\Vert _{L^p(\Gamma ;H^1_0({\mathscr {B}}))} \le C {\mathbb {E}}\left[ \frac{1}{a_{\min }^p}\right] ^{1/p} \Vert \varsigma \Vert _{H^{-1}({\mathscr {B}})}. \end{aligned}$$

We observe that higher regularity of the solution, u, can be obtained by using larger values of s in Lemma 15. This in turn yields control on moments of \(W^{\upsilon ,\infty }({\mathscr {B}})\) norms of the coefficient, a, and following, for instance, estimates similar to (2.10) in [18, Theorem 2.4], we can estimate moments of the \(H^{1+s}({\mathscr {B}})\) norm of the solution, u. These regularity estimates, once combined with pathwise error estimates for the combination technique, can be further used to show the corresponding \(\nu \)-dependent convergence rates of MIMC [23], for Example 1, similar to what was presented in Sect. 5 for MISC in the current work.

Appendix 2: Shift Theorem for Problem (1)

Here, we seek to establish a shift theorem for the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} - {\text {div}}(a({{\varvec{x}}})\,\nabla u({{\varvec{x}}})) = \varsigma ({{\varvec{x}}}) &{} \text {in}\quad {\mathscr {B}}=[0,1]^D \\ u({{\varvec{x}}}) \,=\, 0 &{} \text {on}\quad \partial {\mathscr {B}}, \end{array}\right. } \end{aligned}$$
(49)

under suitable assumptions on a and \(\varsigma \).

With respect to problem (1), for convenience, we drop the dependence on the parameter vector, \({{\varvec{y}}}\). We consider an odd periodic extension of \(\varsigma \), on \([-1,1]^D\), and an even periodic extension of the coefficient a on \([-1,1]^D\), named, respectively, \(\tilde{\varsigma }\) and \(\tilde{a}\). More precisely, for \({{\varvec{j}}}=\{0,1\}^D\), we denote by \({{\varvec{x}}}_{{\varvec{j}}}= ((-1)^{j_1}x_1,\ldots ,(-1)^{j_D}x_D)\) and

$$\begin{aligned}&\tilde{\varsigma }({{\varvec{x}}}_{{\varvec{j}}}+2{{\varvec{k}}}) = (-1)^{|{{\varvec{j}}}|}\varsigma ({{\varvec{x}}}), \quad \tilde{a}({{\varvec{x}}}_{{\varvec{j}}}+2{{\varvec{k}}})=a({{\varvec{x}}}), \quad \forall {{\varvec{x}}}\in [0,1]^2, \;\; {{\varvec{j}}}\in \{0,1\}^D, \quad {{\varvec{k}}}\in {\mathbb {N}}^D. \end{aligned}$$

The following Shift theorem holds for problem (49).

Lemma 17

If the coefficient a is such that its periodic extension satisfies \(\tilde{a}\in W^{s,\infty }({\mathbb {R}}^D)\), \(s\ge 0\) and \(\varsigma \in C^\infty _0({\mathscr {B}})\) then \(u\in H^{s+1}({\mathscr {B}})\).

Proof

We define the extended problem

$$\begin{aligned} {\left\{ \begin{array}{ll} - {\text {div}}(\tilde{a}({{\varvec{x}}})\,\nabla \tilde{u}({{\varvec{x}}})) = \tilde{\varsigma }({{\varvec{x}}}) &{} \text {in}\quad \tilde{{\mathscr {B}}}=[-1,1]^D \\ \int _{\tilde{{\mathscr {B}}}} u({{\varvec{x}}}) \,=\, 0 \\ {\text {periodic boundary conditions on }}\partial \tilde{{\mathscr {B}}}. \end{array}\right. } \end{aligned}$$

Since by assumption \(\tilde{a}\in L^\infty ({\mathbb {R}}^D)\) and \(\tilde{\varsigma }\in L^2(\tilde{{\mathscr {B}}})\), this problem has a unique solution, \(\tilde{u}\in H^1_{per}(\tilde{{\mathscr {B}}})\setminus {\mathbb {R}}\), where we denote with \(H^s_{per}(\tilde{{\mathscr {B}}})\) the space of periodic functions with (periodic) square integrable derivatives up to order s. It is easy to check that the solution \(\tilde{u}\) is odd, that is \(\tilde{u}({{\varvec{x}}}_{{\varvec{j}}})=(-1)^{{{\varvec{j}}}}\tilde{u}({{\varvec{x}}})\), \(\forall {{\varvec{x}}}\in [0,1]^D\), hence \(\tilde{u}=0\) (in the sense of traces) on \(\partial {\mathscr {B}}\) and it coincides with the (unique) solution of (49) on \({\mathscr {B}}\). Moreover, standard elliptic regularity arguments allow us to say that \(\tilde{u}\in H^s_{per}(\tilde{{\mathscr {B}}})\), hence \(u\in H^s({\mathscr {B}})\). \(\square \)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Haji-Ali, AL., Nobile, F., Tamellini, L. et al. Multi-index Stochastic Collocation Convergence Rates for Random PDEs with Parametric Regularity. Found Comput Math 16, 1555–1605 (2016). https://doi.org/10.1007/s10208-016-9327-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10208-016-9327-7

Keywords

  • Multi-level
  • Multi-index Stochastic Collocation
  • Infinite dimensional integration
  • Elliptic partial differential equations with random coefficients
  • Finite element method
  • Uncertainty quantification
  • Random partial differential equations
  • Multivariate approximation
  • Sparse grids
  • Stochastic Collocation methods
  • Multi-level methods
  • Combination technique

Mathematics Subject Classification

  • 41A10 (approx by polynomials)
  • 65C20 (models, numerical methods)
  • 65N30 (Finite elements)
  • 65N05 (Finite differences)