Collocation methods for uncertainty quantification in PDE models with random data

Transcription

1 Collocation methods for uncertainty quantification in PDE models with random data Fabio Nobile CSQI-MATHICSE, EPFL, Switzerland Acknowlegments: L. Tamellini, G. Migliorati (EPFL), R. Tempone, (KAUST), E. von Schwerin (KTH), J. Beck (UCL) Workshop Advances in Uncertainty Quantification Methods, Algorithms and Applications, KAUST, January 6-10, 2014 Italian project FIRB-IDEAS ( 09) Advanced Numerical Techniques for Uncertainty Quantification in Engineering and Life Science Problems Center for Advanced Modeling and Science F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

2 Outline 1 Introduction PDEs with random parameters 2 Polynomial approximation by sparse grid collocation Quasi optimal sparse grid construction Convergence result for elliptic PDEs with random inclusions Numerical results 3 Discrete L 2 projection using random evaluations Convergence analysis Numerical results 4 Conclusions F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

3 Outline Introduction PDEs with random parameters 1 Introduction PDEs with random parameters 2 Polynomial approximation by sparse grid collocation Quasi optimal sparse grid construction Convergence result for elliptic PDEs with random inclusions Numerical results 3 Discrete L 2 projection using random evaluations Convergence analysis Numerical results 4 Conclusions F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

4 Introduction PDEs with random parameters UQ for deterministic PDE models Consider a deterministic PDE model find u : L(y)(u) = F in D R d (1) with suitable boundary / initial conditions. The operator L(y) depends on a vector of N parameters: y = (y 1,..., y N ) R N (N = when dealing with distributed fields). Often in applications the parameters y are not perfectly known or are intrinsically variable. Examples are: subsurface modeling: porous media flows; seismic waves; basin evolutions;... modeling of living tissues: mechanical response; growth models; material science: properties of composite materials Probabilistic approach: y is a random vector with probability density function ρ : Γ R +. F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

5 Introduction PDEs with random parameters UQ for deterministic PDE models Consider a deterministic PDE model find u : L(y)(u) = F in D R d (1) with suitable boundary / initial conditions. The operator L(y) depends on a vector of N parameters: y = (y 1,..., y N ) R N (N = when dealing with distributed fields). Often in applications the parameters y are not perfectly known or are intrinsically variable. Examples are: subsurface modeling: porous media flows; seismic waves; basin evolutions;... modeling of living tissues: mechanical response; growth models; material science: properties of composite materials Probabilistic approach: y is a random vector with probability density function ρ : Γ R +. F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

6 Introduction PDEs with random parameters UQ for deterministic PDE models Consider a deterministic PDE model find u : L(y)(u) = F in D R d (1) with suitable boundary / initial conditions. The operator L(y) depends on a vector of N parameters: y = (y 1,..., y N ) R N (N = when dealing with distributed fields). Often in applications the parameters y are not perfectly known or are intrinsically variable. Examples are: subsurface modeling: porous media flows; seismic waves; basin evolutions;... modeling of living tissues: mechanical response; growth models; material science: properties of composite materials Probabilistic approach: y is a random vector with probability density function ρ : Γ R +. F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

7 Introduction PDEs with random parameters UQ for deterministic PDE models Consider a deterministic PDE model find u : L(y)(u) = F in D R d (1) with suitable boundary / initial conditions. The operator L(y) depends on a vector of N parameters: y = (y 1,..., y N ) R N (N = when dealing with distributed fields). Often in applications the parameters y are not perfectly known or are intrinsically variable. Examples are: subsurface modeling: porous media flows; seismic waves; basin evolutions;... modeling of living tissues: mechanical response; growth models; material science: properties of composite materials Probabilistic approach: y is a random vector with probability density function ρ : Γ R +. F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

8 Introduction PDEs with random parameters UQ for deterministic PDE models Assumption: y Γ the problem admits a unique solution u V in a Hilbert space V. Moreover, u(y) V C(y) F V Then, the PDE (1) induces a map u = u(y) : Γ V. if C(y) L p ρ(γ), then u L p ρ(γ, V ). Goals: Compute statistics of the solution pointwise Expected value: ū(x) = E[u(x, )], x D pointwise Variance: Var[u](x) = E[(u(x, ) ū(x)) 2 ](x) two points corr.: Cov u (x 1, x 2 ) = E[(u(x 1, ) ū(x 1 ))(u(x 2, ) ū(x 2 ))] We can also look at specific Quantities of Interest Q(u) : V R. Then φ(y) = Q(u(y)) is a real-valued function of the random vector y and we would like to approximate its law. F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

9 Introduction PDEs with random parameters UQ for deterministic PDE models Assumption: y Γ the problem admits a unique solution u V in a Hilbert space V. Moreover, u(y) V C(y) F V Then, the PDE (1) induces a map u = u(y) : Γ V. if C(y) L p ρ(γ), then u L p ρ(γ, V ). Goals: Compute statistics of the solution pointwise Expected value: ū(x) = E[u(x, )], x D pointwise Variance: Var[u](x) = E[(u(x, ) ū(x)) 2 ](x) two points corr.: Cov u (x 1, x 2 ) = E[(u(x 1, ) ū(x 1 ))(u(x 2, ) ū(x 2 ))] We can also look at specific Quantities of Interest Q(u) : V R. Then φ(y) = Q(u(y)) is a real-valued function of the random vector y and we would like to approximate its law. F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

10 Introduction PDEs with random parameters UQ for deterministic PDE models Assumption: y Γ the problem admits a unique solution u V in a Hilbert space V. Moreover, u(y) V C(y) F V Then, the PDE (1) induces a map u = u(y) : Γ V. if C(y) L p ρ(γ), then u L p ρ(γ, V ). Goals: Compute statistics of the solution pointwise Expected value: ū(x) = E[u(x, )], x D pointwise Variance: Var[u](x) = E[(u(x, ) ū(x)) 2 ](x) two points corr.: Cov u (x 1, x 2 ) = E[(u(x 1, ) ū(x 1 ))(u(x 2, ) ū(x 2 ))] We can also look at specific Quantities of Interest Q(u) : V R. Then φ(y) = Q(u(y)) is a real-valued function of the random vector y and we would like to approximate its law. F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

11 Introduction PDEs with random parameters Example: Elliptic PDE with random coefficients { div(a(y, x) u(y, x)) = f (x) x D, y Γ, u(y, x) = 0 x D, y Γ with a min (y) = inf x D a(y, x) > 0 for all y Γ and f L 2 (D). Then y Γ, u(y) V H 1 0 (D), and u(y) V C P a min (y) f L 2 (D). Inclusions problem y describes the conductivitiy in each inclusion N a(y, x) = a 0 + y n1 Dn (x) n=n Random fields problem a(y, x) is a random field, e.g. lognormal: a(y, x) = e γ(y,x) with γ expanded e.g. in Karhunen-Loève series γ(y, x) = λnynb n(x), n=1 random field with L c =1/ y n N(0, 1) i.i.d. F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

12 Introduction PDEs with random parameters Example: Elliptic PDE with random coefficients { div(a(y, x) u(y, x)) = f (x) x D, y Γ, u(y, x) = 0 x D, y Γ with a min (y) = inf x D a(y, x) > 0 for all y Γ and f L 2 (D). Then y Γ, u(y) V H 1 0 (D), and u(y) V C P a min (y) f L 2 (D). Inclusions problem y describes the conductivitiy in each inclusion N a(y, x) = a 0 + y n1 Dn (x) n=n Random fields problem a(y, x) is a random field, e.g. lognormal: a(y, x) = e γ(y,x) with γ expanded e.g. in Karhunen-Loève series γ(y, x) = λnynb n(x), n=1 random field with L c =1/ y n N(0, 1) i.i.d. F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

13 Outline Polynomial approximation by sparse grid collocation 1 Introduction PDEs with random parameters 2 Polynomial approximation by sparse grid collocation Quasi optimal sparse grid construction Convergence result for elliptic PDEs with random inclusions Numerical results 3 Discrete L 2 projection using random evaluations Convergence analysis Numerical results 4 Conclusions F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

14 Polynomial approximation by sparse grid collocation Stochastic multivariate polynomial approximation The parameter-to-solution map u(y) : Γ V is often smooth (even analytic for the elliptic diffusion model). It is therefore sound to approximate it by global multivariate polynomials. Let Λ N N be an index set of cardinality Λ = M, and consider the multivariate polynomial space { N } P Λ (Γ) = span n=1 y n pn, with p = (p 1,..., p N ) Λ We seek an approximation P Λ u P Λ (Γ) V. Collocation approaches Construct a polynomial approximation of u(y) : Γ V using only point evaluations u i = u(y i ) where {y i } M i=1 is a set of suitable collocation points, with M M. F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

15 Polynomial approximation by sparse grid collocation Stochastic multivariate polynomial approximation The parameter-to-solution map u(y) : Γ V is often smooth (even analytic for the elliptic diffusion model). It is therefore sound to approximate it by global multivariate polynomials. Let Λ N N be an index set of cardinality Λ = M, and consider the multivariate polynomial space { N } P Λ (Γ) = span n=1 y n pn, with p = (p 1,..., p N ) Λ We seek an approximation P Λ u P Λ (Γ) V. Collocation approaches Construct a polynomial approximation of u(y) : Γ V using only point evaluations u i = u(y i ) where {y i } M i=1 is a set of suitable collocation points, with M M. F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

16 Polynomial approximation by sparse grid collocation Stochastic multivariate polynomial approximation The parameter-to-solution map u(y) : Γ V is often smooth (even analytic for the elliptic diffusion model). It is therefore sound to approximate it by global multivariate polynomials. Let Λ N N be an index set of cardinality Λ = M, and consider the multivariate polynomial space { N } P Λ (Γ) = span n=1 y n pn, with p = (p 1,..., p N ) Λ We seek an approximation P Λ u P Λ (Γ) V. Collocation approaches Construct a polynomial approximation of u(y) : Γ V using only point evaluations u i = u(y i ) where {y i } M i=1 is a set of suitable collocation points, with M M. F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

17 Polynomial approximation by sparse grid collocation Collocation on a (generalized) Sparse Grid Let i = [i 1,..., i N ] N N + and m(i) : N + N + an increasing function 1 1D polynomial interpolant operators: Un m(in) on m(i n ) abscissas. We use either Clenshaw-Curtis (extrema on Chebyshev polynomials) Gauss points w.r.t. the weight ρ n, assuming that the probability density factorizes as ρ(y) = N 2 Detail operator: m(in) n = U m(in) n n=1 ρ n(y n ) Un m(in 1), U m(0) n = 0. 3 Hierarchical surplus: m(i) = N n=1 m(in) n. 4 Sparse grid approximation: on an index set I N N S I u = i I m(i) [u] Assumption: The set I is downward closed: i I i e n I, n = 1,..., N. F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

18 Polynomial approximation by sparse grid collocation Collocation on a (generalized) Sparse Grid Let i = [i 1,..., i N ] N N + and m(i) : N + N + an increasing function 1 1D polynomial interpolant operators: Un m(in) on m(i n ) abscissas. We use either Clenshaw-Curtis (extrema on Chebyshev polynomials) Gauss points w.r.t. the weight ρ n, assuming that the probability density factorizes as ρ(y) = N 2 Detail operator: m(in) n = U m(in) n n=1 ρ n(y n ) Un m(in 1), U m(0) n = 0. 3 Hierarchical surplus: m(i) = N n=1 m(in) n. 4 Sparse grid approximation: on an index set I N N S I u = i I m(i) [u] Assumption: The set I is downward closed: i I i e n I, n = 1,..., N. F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

19 Polynomial approximation by sparse grid collocation Collocation on a (generalized) Sparse Grid Let i = [i 1,..., i N ] N N + and m(i) : N + N + an increasing function 1 1D polynomial interpolant operators: Un m(in) on m(i n ) abscissas. We use either Clenshaw-Curtis (extrema on Chebyshev polynomials) Gauss points w.r.t. the weight ρ n, assuming that the probability density factorizes as ρ(y) = N 2 Detail operator: m(in) n = U m(in) n n=1 ρ n(y n ) Un m(in 1), U m(0) n = 0. 3 Hierarchical surplus: m(i) = N n=1 m(in) n. 4 Sparse grid approximation: on an index set I N N S I u = i I m(i) [u] Assumption: The set I is downward closed: i I i e n I, n = 1,..., N. F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

20 Polynomial approximation by sparse grid collocation Collocation on a (generalized) Sparse Grid Let i = [i 1,..., i N ] N N + and m(i) : N + N + an increasing function 1 1D polynomial interpolant operators: Un m(in) on m(i n ) abscissas. We use either Clenshaw-Curtis (extrema on Chebyshev polynomials) Gauss points w.r.t. the weight ρ n, assuming that the probability density factorizes as ρ(y) = N 2 Detail operator: m(in) n = U m(in) n n=1 ρ n(y n ) Un m(in 1), U m(0) n = 0. 3 Hierarchical surplus: m(i) = N n=1 m(in) n. 4 Sparse grid approximation: on an index set I N N S I u = i I m(i) [u] Assumption: The set I is downward closed: i I i e n I, n = 1,..., N. F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

21 Polynomial approximation by sparse grid collocation Equivalent formulation S I u = c(i)u m(i 1) 1 U m(i N) N u. i I with c(i) = ( 1) j j N, and c(i) = 0 if i + 1 I j {0,1} N (i+j) I linear combination of tensor grids (each with relatively few points!) Theorem ([Back-Nobile-Tamellini-Tempone, 2010]) Let Λ(I, m) = {p N N : p m(i) 1, i I}. Then S I : C 0 (Γ) P Λ(I,m) (Γ) S I v = v, v P Λ(I,m) (Γ) F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

22 Polynomial approximation by sparse grid collocation Equivalent formulation S I u = c(i)u m(i 1) 1 U m(i N) N u. i I with c(i) = ( 1) j j N, and c(i) = 0 if i + 1 I j {0,1} N (i+j) I linear combination of tensor grids (each with relatively few points!) Theorem ([Back-Nobile-Tamellini-Tempone, 2010]) Let Λ(I, m) = {p N N : p m(i) 1, i I}. Then S I : C 0 (Γ) P Λ(I,m) (Γ) S I v = v, v P Λ(I,m) (Γ) F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

23 Polynomial approximation by sparse grid collocation Quasi optimal sparse grid construction Quasi-optimal sparse grid construction S I u = i I m(i) [u] = u S I u = i/ I m(i) [u] i/ I m(i) [u] One can use a knapsack problem-approach [Griebel-Knapek 09, Gerstner-Griebel 03, Bungartz-Griebel 04] to select the best I: for each multiindex i: Estimated error contribution (how much error decreases if i is added to I) E(i) m(i) [u] V Estimated work contribution (how much the work, i.e. number of evaluations, increases if i is added to I) W (i) such that W (i) W (I) where W (I) is the total number of points in the sparse grid F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10, i I

24 Polynomial approximation by sparse grid collocation Quasi optimal sparse grid construction Quasi-optimal sparse grid construction S I u = i I m(i) [u] = u S I u = i/ I m(i) [u] i/ I m(i) [u] One can use a knapsack problem-approach [Griebel-Knapek 09, Gerstner-Griebel 03, Bungartz-Griebel 04] to select the best I: for each multiindex i: Estimated error contribution (how much error decreases if i is added to I) E(i) m(i) [u] V Estimated work contribution (how much the work, i.e. number of evaluations, increases if i is added to I) W (i) such that W (i) W (I) where W (I) is the total number of points in the sparse grid F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10, i I

25 Polynomial approximation by sparse grid collocation Quasi optimal sparse grid construction Quasi-optimal sparse grid construction S I u = i I m(i) [u] = u S I u = i/ I m(i) [u] i/ I m(i) [u] One can use a knapsack problem-approach [Griebel-Knapek 09, Gerstner-Griebel 03, Bungartz-Griebel 04] to select the best I: for each multiindex i: Estimated error contribution (how much error decreases if i is added to I) E(i) m(i) [u] V Estimated work contribution (how much the work, i.e. number of evaluations, increases if i is added to I) W (i) such that W (i) W (I) where W (I) is the total number of points in the sparse grid F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10, i I

26 Polynomial approximation by sparse grid collocation Quasi optimal sparse grid construction Quasi-optimal sparse grid construction S I u = i I m(i) [u] = u S I u = i/ I m(i) [u] i/ I m(i) [u] One can use a knapsack problem-approach [Griebel-Knapek 09, Gerstner-Griebel 03, Bungartz-Griebel 04] to select the best I: for each multiindex i: Estimated error contribution (how much error decreases if i is added to I) E(i) m(i) [u] V Estimated work contribution (how much the work, i.e. number of evaluations, increases if i is added to I) W (i) such that W (i) W (I) where W (I) is the total number of points in the sparse grid F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10, i I

27 Polynomial approximation by sparse grid collocation Quasi optimal sparse grid construction Quasi-optimal sparse grid construction S I u = i I m(i) [u] = u S I u = i/ I m(i) [u] i/ I m(i) [u] One can use a knapsack problem-approach [Griebel-Knapek 09, Gerstner-Griebel 03, Bungartz-Griebel 04] to select the best I: for each multiindex i: Estimated error contribution (how much error decreases if i is added to I) E(i) m(i) [u] V Estimated work contribution (how much the work, i.e. number of evaluations, increases if i is added to I) W (i) such that W (i) W (I) where W (I) is the total number of points in the sparse grid F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10, i I

28 Polynomial approximation by sparse grid collocation Quasi optimal sparse grid construction Quasi-optimal sparse grid construction Then estimate the profit of each i as P(i) = E(i) W (i) and build the sparse grid using the set I M of the M indices with the largest estimated profit. I M := {i N N P(i) P ord M } where {Pj ord } j is the ordered sequence of profits. If the set I M is not downward closed (lower), take the smallest lower set Ĩ M I M. This is equivalent to consider the modified profits P(i) = max P(j). (see [Chkifa-Cohen-DeVore-Schwab M2AN 13]) j>i F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

29 Polynomial approximation by sparse grid collocation Quasi optimal sparse grid construction Quasi-optimal sparse grid construction Then estimate the profit of each i as P(i) = E(i) W (i) and build the sparse grid using the set I M of the M indices with the largest estimated profit. I M := {i N N P(i) P ord M } where {Pj ord } j is the ordered sequence of profits. If the set I M is not downward closed (lower), take the smallest lower set Ĩ M I M. This is equivalent to consider the modified profits P(i) = max P(j). (see [Chkifa-Cohen-DeVore-Schwab M2AN 13]) j>i F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

30 Polynomial approximation by sparse grid collocation Quasi optimal sparse grid construction A posteriori approach [Gerstner-Griebel 03, Klimke, PhD 06]. Given a set Λ, explore all the neighbor multi-indices (margin) and pick up those corresponding to the largest profits. A priori approach [Back-Nobile-Tamellini-Tempone 11]. Whenever possible, use a-priori/a-posteriori information to build the optimal set. Avoids the exploration cost that can be expensive in high dimension. The estimate of E(i) can be related to the decay of the coefficients of the gpc expansion u = p u pψ p of the solution onto an orthonormal polynomial basis (Legendre, Chebyshev, Hermite,...) and to the Lebesgue constant of the interpolation scheme. F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

31 Polynomial approximation by sparse grid collocation Quasi optimal sparse grid construction A posteriori approach [Gerstner-Griebel 03, Klimke, PhD 06]. Given a set Λ, explore all the neighbor multi-indices (margin) and pick up those corresponding to the largest profits. A priori approach [Back-Nobile-Tamellini-Tempone 11]. Whenever possible, use a-priori/a-posteriori information to build the optimal set. Avoids the exploration cost that can be expensive in high dimension. The estimate of E(i) can be related to the decay of the coefficients of the gpc expansion u = p u pψ p of the solution onto an orthonormal polynomial basis (Legendre, Chebyshev, Hermite,...) and to the Lebesgue constant of the interpolation scheme. F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

32 Polynomial approximation by sparse grid collocation General convergence result Quasi optimal sparse grid construction Theorem [Tamellini PhD thesis 12], [Nobile-Tamellini-Tempone, in preparation] Let SĨM u be the quasi-optimal sparse grid approximation and WĨM total number of points in the sparse grid. the ( If C(τ) := P(i) ) 1 i N τ τ W (i) N < for some τ < 1 Then u SĨM u L 2 ρ (Γ,V ) C(τ)W 1 1 τ Ĩ M F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

33 Proof Polynomial approximation by sparse grid collocation Quasi optimal sparse grid construction The proof uses Stechkin lemma: given a non-negative descreasing sequence {a k } k, then k=n+1 a k N 1 1 τ ( k=1 a τ k ) 1 τ, 0 < τ < 1. ordered repeated sequence of profits {ˆP k } k = { P 1,..., P }{{} 1, P 2,..., P 2,...}. }{{} W 1 times W 2 times Let W M = M j=1 W j and observe that W M WĨM. Then u SĨM u L 2 ρ (Γ,V ) E k ˆP k k=m+1 k=w M +1 [apply Stechkin] {ˆP k } k l τ W 1 1 τ M C(τ)W 1 1 τ Ĩ M F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

34 Proof Polynomial approximation by sparse grid collocation Quasi optimal sparse grid construction The proof uses Stechkin lemma: given a non-negative descreasing sequence {a k } k, then k=n+1 a k N 1 1 τ ( k=1 a τ k ) 1 τ, 0 < τ < 1. ordered repeated sequence of profits {ˆP k } k = { P 1,..., P }{{} 1, P 2,..., P 2,...}. }{{} W 1 times W 2 times Let W M = M j=1 W j and observe that W M WĨM. Then u SĨM u L 2 ρ (Γ,V ) E k ˆP k k=m+1 k=w M +1 [apply Stechkin] {ˆP k } k l τ W 1 1 τ M C(τ)W 1 1 τ Ĩ M F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

35 Proof Polynomial approximation by sparse grid collocation Quasi optimal sparse grid construction The proof uses Stechkin lemma: given a non-negative descreasing sequence {a k } k, then k=n+1 a k N 1 1 τ ( k=1 a τ k ) 1 τ, 0 < τ < 1. ordered repeated sequence of profits {ˆP k } k = { P 1,..., P }{{} 1, P 2,..., P 2,...}. }{{} W 1 times W 2 times Let W M = M j=1 W j and observe that W M WĨM. Then u SĨM u L 2 ρ (Γ,V ) E k ˆP k k=m+1 k=w M +1 [apply Stechkin] {ˆP k } k l τ W 1 1 τ M C(τ)W 1 1 τ Ĩ M F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

36 Proof Polynomial approximation by sparse grid collocation Quasi optimal sparse grid construction The proof uses Stechkin lemma: given a non-negative descreasing sequence {a k } k, then k=n+1 a k N 1 1 τ ( k=1 a τ k ) 1 τ, 0 < τ < 1. ordered repeated sequence of profits {ˆP k } k = { P 1,..., P }{{} 1, P 2,..., P 2,...}. }{{} W 1 times W 2 times Let W M = M j=1 W j and observe that W M WĨM. Then u SĨM u L 2 ρ (Γ,V ) E k ˆP k k=m+1 k=w M +1 [apply Stechkin] {ˆP k } k l τ W 1 1 τ M C(τ)W 1 1 τ Ĩ M F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

37 Polynomial approximation by sparse grid collocation How to estimate W and E Quasi optimal sparse grid construction 1 W (i): number of new points in N n=1 m(in) n Count all points in U m(i 1) 1 U m(i N) N (non-nested case) or just the extra points added (nested case) 2 E(i): use expansion on a suitable basis u = p u pψ p (Legendre, Chebyshev,...) and relate E(i) with the decay of the Fourier coefficients u p. E.g. for Chebyshev expansion ( ψ p = 1) E(i) 2 L m(i) p m(i 1) u p V, where L m(i) = N n=1 L m(i n) and L m(in) := Un m(in) L(C 0,L 2 is the ρ) Lebesgue constant from C 0 to L 2 ρ. F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

38 Polynomial approximation by sparse grid collocation How to estimate W and E Quasi optimal sparse grid construction 1 W (i): number of new points in N n=1 m(in) n Count all points in U m(i 1) 1 U m(i N) N (non-nested case) or just the extra points added (nested case) 2 E(i): use expansion on a suitable basis u = p u pψ p (Legendre, Chebyshev,...) and relate E(i) with the decay of the Fourier coefficients u p. E.g. for Chebyshev expansion ( ψ p = 1) E(i) 2 L m(i) p m(i 1) u p V, where L m(i) = N n=1 L m(i n) and L m(in) := Un m(in) L(C 0,L 2 is the ρ) Lebesgue constant from C 0 to L 2 ρ. F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

39 Polynomial approximation by sparse grid collocation Convergence result for elliptic PDEs with random inclusions Elliptic equation with random inclusions Y i U[a i, b i ], independent Γ = N i=1 [a i, b i ] u(y) : Γ H 1 0 (D) The solution u(y) : Γ V H 1 0 (D) is analytic in a polydisk in the complex plane C N The solution can be expanded in Chebyshev series u(y) = p u pψ p (y) with ψ p 1. Estimates on Chebyshev coefficients are available [Babuska-Nobile-Tempone 07] [Nobile-Tamellini-Tempone 13] u p V Ce N n=1 g i p i F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

40 Polynomial approximation by sparse grid collocation Convergence result for elliptic PDEs with random inclusions Elliptic equation with random inclusions Y i U[a i, b i ], independent Γ = N i=1 [a i, b i ] u(y) : Γ H 1 0 (D) The solution u(y) : Γ V H 1 0 (D) is analytic in a polydisk in the complex plane C N The solution can be expanded in Chebyshev series u(y) = p u pψ p (y) with ψ p 1. Estimates on Chebyshev coefficients are available [Babuska-Nobile-Tempone 07] [Nobile-Tamellini-Tempone 13] u p V Ce N n=1 g i p i F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

41 Polynomial approximation by sparse grid collocation Convergence result for elliptic PDEs with random inclusions Quasi-optimal sparse grid construction using (nested) Clenshaw-Curtis points m(i) = 2 i (doubling the points). W (i) N n=1 2in 2 Decay of Chebyshev coefficients: u p V Ce g N n=1 pn Lebesgue constant: L(i) 2 log(i + 1) + 1 π Error estimate: E(i) = C N n=1 e ĝm(in 1) Profit estimate: P(i) = Ce ĝ N n=1 m(in 1) Theorem [Nobile-Tamellini-Tempone 13] ( i N N P(i) τ W (i) ) 1 τ < for all 0 < τ < 1 = u SĨM u L 2 ρ (Γ,V ) C(τ)W 1 1 τ Ĩ M, 0 < τ < 1 By optimizing with respect to τ one can get the estimate u SĨM u L 2 ρ (Γ,V ) C 1 (N) exp{ NC 2 W 1 N ĨM } F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

42 Polynomial approximation by sparse grid collocation Convergence result for elliptic PDEs with random inclusions Quasi-optimal sparse grid construction using (nested) Clenshaw-Curtis points m(i) = 2 i (doubling the points). W (i) N n=1 2in 2 Decay of Chebyshev coefficients: u p V Ce g N n=1 pn Lebesgue constant: L(i) 2 log(i + 1) + 1 π Error estimate: E(i) = C N n=1 e ĝm(in 1) Profit estimate: P(i) = Ce ĝ N n=1 m(in 1) Theorem [Nobile-Tamellini-Tempone 13] ( i N N P(i) τ W (i) ) 1 τ < for all 0 < τ < 1 = u SĨM u L 2 ρ (Γ,V ) C(τ)W 1 1 τ Ĩ M, 0 < τ < 1 By optimizing with respect to τ one can get the estimate u SĨM u L 2 ρ (Γ,V ) C 1 (N) exp{ NC 2 W 1 N ĨM } F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

43 Polynomial approximation by sparse grid collocation Convergence result for elliptic PDEs with random inclusions Quasi-optimal sparse grid construction using (nested) Clenshaw-Curtis points m(i) = 2 i (doubling the points). W (i) N n=1 2in 2 Decay of Chebyshev coefficients: u p V Ce g N n=1 pn Lebesgue constant: L(i) 2 log(i + 1) + 1 π Error estimate: E(i) = C N n=1 e ĝm(in 1) Profit estimate: P(i) = Ce ĝ N n=1 m(in 1) Theorem [Nobile-Tamellini-Tempone 13] ( i N N P(i) τ W (i) ) 1 τ < for all 0 < τ < 1 = u SĨM u L 2 ρ (Γ,V ) C(τ)W 1 1 τ Ĩ M, 0 < τ < 1 By optimizing with respect to τ one can get the estimate u SĨM u L 2 ρ (Γ,V ) C 1 (N) exp{ NC 2 W 1 N ĨM } F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

44 Polynomial approximation by sparse grid collocation Convergence result for elliptic PDEs with random inclusions Quasi-optimal sparse grid construction using (non-nested) Legendre points m(i) = i (adding 1 point at the time). W (i) = N n=1 i n Decay of Chebyshev coefficients: u p V Ce g N n=1 pn Lebesgue constant: L(i) = 1. Error estimate: E(i) = C N n=1 e ĝm(in 1) Profit estimate: P(i) = Ce ĝ N n=1 m(in 1) Theorem [Nobile-Tamellini-Tempone 13] ( i N N P(i) τ W (i) ) 1 τ < for all 0 < τ < 1 = u SĨM u L 2 ρ (Γ,V ) C(τ)W 1 1 τ Ĩ M, 0 < τ < 1 By optimizing with respect to τ one can get the estimate u SĨM u L 2 ρ (Γ,V ) C 1 (N) exp{ N C 2 W 1 2N Ĩ M } F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

45 Polynomial approximation by sparse grid collocation Convergence result for elliptic PDEs with random inclusions Quasi-optimal sparse grid construction using (non-nested) Legendre points m(i) = i (adding 1 point at the time). W (i) = N n=1 i n Decay of Chebyshev coefficients: u p V Ce g N n=1 pn Lebesgue constant: L(i) = 1. Error estimate: E(i) = C N n=1 e ĝm(in 1) Profit estimate: P(i) = Ce ĝ N n=1 m(in 1) Theorem [Nobile-Tamellini-Tempone 13] ( i N N P(i) τ W (i) ) 1 τ < for all 0 < τ < 1 = u SĨM u L 2 ρ (Γ,V ) C(τ)W 1 1 τ Ĩ M, 0 < τ < 1 By optimizing with respect to τ one can get the estimate u SĨM u L 2 ρ (Γ,V ) C 1 (N) exp{ N C 2 W 1 2N Ĩ M } F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

46 Polynomial approximation by sparse grid collocation Quasi optimal sparse grid in practice Convergence result for elliptic PDEs with random inclusions We have the theoretical bound u p V Ce N n=1 gnpn, hence, in particular, for p = e j = (0,..., 0, 1, 0,..., 0), u ej V Ce g j p j which implies that a 1D polynomial interpolation in the variable y j only converges exponentially with rate g j. The rates g j can be estimated numerically by 1D analyses. (increase the polynomial degree in one variable at the time and fit the convergence rate). Once the rates g j are available, we build the quasi optimal index set based on the profit estimate P(i) = E(i) W (i) N n=1 L m(in)e g in m(in 1) W (i n ) F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

47 Polynomial approximation by sparse grid collocation Quasi optimal sparse grid in practice Convergence result for elliptic PDEs with random inclusions We have the theoretical bound u p V Ce N n=1 gnpn, hence, in particular, for p = e j = (0,..., 0, 1, 0,..., 0), u ej V Ce g j p j which implies that a 1D polynomial interpolation in the variable y j only converges exponentially with rate g j. The rates g j can be estimated numerically by 1D analyses. (increase the polynomial degree in one variable at the time and fit the convergence rate). Once the rates g j are available, we build the quasi optimal index set based on the profit estimate P(i) = E(i) W (i) N n=1 L m(in)e g in m(in 1) W (i n ) F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

48 Polynomial approximation by sparse grid collocation Quasi optimal sparse grid in practice Convergence result for elliptic PDEs with random inclusions We have the theoretical bound u p V Ce N n=1 gnpn, hence, in particular, for p = e j = (0,..., 0, 1, 0,..., 0), u ej V Ce g j p j which implies that a 1D polynomial interpolation in the variable y j only converges exponentially with rate g j. The rates g j can be estimated numerically by 1D analyses. (increase the polynomial degree in one variable at the time and fit the convergence rate). Once the rates g j are available, we build the quasi optimal index set based on the profit estimate P(i) = E(i) W (i) N n=1 L m(in)e g in m(in 1) W (i n ) F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

49 Polynomial approximation by sparse grid collocation Quasi optimal sparse grid in practice Convergence result for elliptic PDEs with random inclusions When working with random fields instread of inclusion problmes a good estimate for the Legendre/Chebyshev coefficients is (see [Bech-N.-Tamellini-Tempone M3AS 12, Cohen-DeVore-Schwab FoCM 10]) u p V C p! N p! exp{ g n p n }. The rates g n can be estimated again by 1D inexpensive analyses. n=1 F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

50 Polynomial approximation by sparse grid collocation Quasi optimal sparse grid in practice Convergence result for elliptic PDEs with random inclusions When working with random fields instread of inclusion problmes a good estimate for the Legendre/Chebyshev coefficients is (see [Bech-N.-Tamellini-Tempone M3AS 12, Cohen-DeVore-Schwab FoCM 10]) u p V C p! N p! exp{ g n p n }. The rates g n can be estimated again by 1D inexpensive analyses. n=1 F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

51 Polynomial approximation by sparse grid collocation Numerical results Anisotropic test case 4 random inclusions y F x Conductivity coefficient: matrix k=1 circular inclusions: k Ωi 1 + γ i U( 0.99, 0.99) 2 iid uniform random variables γ 1,2,3,4 = 1, 0.06, , mean std F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

52 Polynomial approximation by sparse grid collocation Numerical results Anisotropic test case 4 random inclusions convergence plot for ψ(u) S I ψ(u) L 2 ρ (Γ) versus # pts sparse grid aniso 4D nested bmt CC OPT KL SM asm C W aniso 4D nonnested OPT bmt GL OPT NN atd C W 2 ε ε work work Figure: Results for the anisotropic problem. Left: (nested) CC points Right: (non-nested) Gauss-Legendre points F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

53 Polynomial approximation by sparse grid collocation Numerical results aniso nested case, err ~ exp( N W 1/N ) N=2 N=4 N= aniso non nested case, err ~ exp( N W 1/2N ) N=2 N=4 N= err err N W 1/N N W 1/2N Figure: Results for the anisotropic problem. Optimal sparse grids and their predicted convergence rates. Top: Nested CC points. Bottom: Non-nested Gauss Legendre points. F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

54 Polynomial approximation by sparse grid collocation Numerical results Numerical test - 1D stationary lognormal field We approximate γ as γ(y, x) µ(x) + σa 0 y 0 + σ with y i N (0, 1), i.i.d. k=1 Given the Fourier series σ 2 e z 2 LC 2 L = 1, D = [0, L] 2. a(y, x) u(y, x) = 0 u = 1 on x = 0, h = 0 on x = 1 no flux otherwise a(x, y) = e γ(x,y), µ γ (x) = 0, Cov γ (x, x ) = σ 2 e x 1 x LC K ( π ) ( π )] a k [y 2k 1 cos L kx 1 + y 2k sin L kx 1 = k=0 c k cos ( π L kz), a k = c k. F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

55 Polynomial approximation by sparse grid collocation Numerical results (0,1) ( a p ) n = 0 (1,1) p=1 p=0 (0,0) (1,0) ( a p ) n = 0 Quantity [ of interest: effective permeability E[Φ(u)], with ] L Φ = k(, x) u(,x) dx 0 x Convergence: E[Φ(SĨM u)] E[Φ(u)] We compare Monte Carlo estimate with quasi-optimal sparse grids based on Gauss-Hermite-Patterson points (nested Gauss-Hermite) Estimate of Hermite coefficients decay: for the simpler problem a(y) u(y, x) = f, a(y) = e b 0+ N n=1 ynbn, we have u i V = C bin n. in! Heuristic: use the same ansatz u i V C N n=1 e gnin in! estimate the rates g n numerically. F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10, but

56 Polynomial approximation by sparse grid collocation Numerical results (0,1) ( a p ) n = 0 (1,1) p=1 p=0 (0,0) (1,0) ( a p ) n = 0 Quantity [ of interest: effective permeability E[Φ(u)], with ] L Φ = k(, x) u(,x) dx 0 x Convergence: E[Φ(SĨM u)] E[Φ(u)] We compare Monte Carlo estimate with quasi-optimal sparse grids based on Gauss-Hermite-Patterson points (nested Gauss-Hermite) Estimate of Hermite coefficients decay: for the simpler problem a(y) u(y, x) = f, a(y) = e b 0+ N n=1 ynbn, we have u i V = C bin n. in! Heuristic: use the same ansatz u i V C N n=1 e gnin in! estimate the rates g n numerically. F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10, but

57 Polynomial approximation by sparse grid collocation Numerical results (0,1) ( a p ) n = 0 (1,1) p=1 p=0 (0,0) (1,0) ( a p ) n = 0 Quantity [ of interest: effective permeability E[Φ(u)], with ] L Φ = k(, x) u(,x) dx 0 x Convergence: E[Φ(SĨM u)] E[Φ(u)] We compare Monte Carlo estimate with quasi-optimal sparse grids based on Gauss-Hermite-Patterson points (nested Gauss-Hermite) Estimate of Hermite coefficients decay: for the simpler problem a(y) u(y, x) = f, a(y) = e b 0+ N n=1 ynbn, we have u i V = C bin n. in! Heuristic: use the same ansatz u i V C N n=1 e gnin in! estimate the rates g n numerically. F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10, but

58 Polynomial approximation by sparse grid collocation Numerical results Correlation length: LC = 0.2, Std: σ = 0.3 (c.o.v. 30%) (1) (1) 14(2) (2) 18(3) sparse grid, N=11 sparse grid, N=21 sparse grid, N=33 1/M 1/M 0.5 1/M (a) Convergence of quasi-optimal sparse grid approximations OPT sparse grid, N=11 OPT sparse grid, N=21 OPT sparse grid, N=33 SM sparse grid, N=11 SM sparse grid, N=21 SM sparse grid, N=33 21(3) 23(4) 26(5) 29(5) (b) Convergence with respect to the reference solution with N = 33 r.vs. The quasi optimal construction automatically adds new variables when needed. No need to truncate a-priori the random field A quasi-optimal sparse grids procedure for groundwater flows by J. Beck, F. Nobile, L. Tamellini and R. Tempone. To appear, LNCSE, Springer, F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

59 Outline Discrete L 2 projection using random evaluations 1 Introduction PDEs with random parameters 2 Polynomial approximation by sparse grid collocation Quasi optimal sparse grid construction Convergence result for elliptic PDEs with random inclusions Numerical results 3 Discrete L 2 projection using random evaluations Convergence analysis Numerical results 4 Conclusions F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

60 Discrete L 2 projection using random evaluations Discrete L 2 projection using random evaluations Another way to compute a polynomial approximation by collocation (alternative to sparse grids) consists in doing a discrete least square approx. using random evaluations (see e.g. [Burkardt-Eldred 2009, Hosder-Walters et al. 2010, Eldred 2011, Blatman-Sudret 2008,...]) 1 Generate M random i.i.d. samples y k ρ(y)dy, k = 1,..., M 2 Compute the corresponding solutions u k = u(y (k) ) 3 Find the discrete least square approximation P M,ω Λ u P Λ (Γ) V P M,ω Λ u = argmin v P Λ (Γ) V 1 M M u k v(y (k) ) 2 V k=1 Two relevant questions For a given set Λ, how many samples should one use? What is the accuracy of the random discrete least square approx.? F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

61 Discrete L 2 projection using random evaluations Discrete L 2 projection using random evaluations Another way to compute a polynomial approximation by collocation (alternative to sparse grids) consists in doing a discrete least square approx. using random evaluations (see e.g. [Burkardt-Eldred 2009, Hosder-Walters et al. 2010, Eldred 2011, Blatman-Sudret 2008,...]) 1 Generate M random i.i.d. samples y k ρ(y)dy, k = 1,..., M 2 Compute the corresponding solutions u k = u(y (k) ) 3 Find the discrete least square approximation P M,ω Λ u P Λ (Γ) V P M,ω Λ u = argmin v P Λ (Γ) V 1 M M u k v(y (k) ) 2 V k=1 Two relevant questions For a given set Λ, how many samples should one use? What is the accuracy of the random discrete least square approx.? F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

62 Discrete L 2 projection using random evaluations Discrete L 2 projection using random evaluations Another way to compute a polynomial approximation by collocation (alternative to sparse grids) consists in doing a discrete least square approx. using random evaluations (see e.g. [Burkardt-Eldred 2009, Hosder-Walters et al. 2010, Eldred 2011, Blatman-Sudret 2008,...]) 1 Generate M random i.i.d. samples y k ρ(y)dy, k = 1,..., M 2 Compute the corresponding solutions u k = u(y (k) ) 3 Find the discrete least square approximation P M,ω Λ u P Λ (Γ) V P M,ω Λ u = argmin v P Λ (Γ) V 1 M M u k v(y (k) ) 2 V k=1 Two relevant questions For a given set Λ, how many samples should one use? What is the accuracy of the random discrete least square approx.? F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

63 Discrete L 2 projection using random evaluations Convergence analysis Error analysis continuous norm: v 2 L 2 ρ V = Γ v(y) 2 V ρ(y)dy discrete norm: v 2 M,V = 1 M M i=1 v(y i) 2 V v 2 L 2 ρ V Define the constant C ω (M, Λ) := sup v PΛ (Γ) V v 2 M,V Theorem [Migliorati-Nobile-von Schwerin-Tempone 11] 1 C ω (M, Λ) 1 almost surely when M 2 u P M,ω Λ u L 2 ρ V (1 + C ω (M, Λ)) inf v P Λ (Γ) V u v L (Γ,V ) Proof: for any v P Λ V : u P M,ω Λ u L 2 ρ V u v L 2 ρ V + v P M,ω Λ u L 2 ρ V u v L 2 ρ V + C ω (M, Λ) v P M,ω Λ u M,V u v L 2 ρ V + C ω (M, Λ) u v M,V F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

64 Discrete L 2 projection using random evaluations Convergence analysis Error analysis continuous norm: v 2 L 2 ρ V = Γ v(y) 2 V ρ(y)dy discrete norm: v 2 M,V = 1 M M i=1 v(y i) 2 V v 2 L 2 ρ V Define the constant C ω (M, Λ) := sup v PΛ (Γ) V v 2 M,V Theorem [Migliorati-Nobile-von Schwerin-Tempone 11] 1 C ω (M, Λ) 1 almost surely when M 2 u P M,ω Λ u L 2 ρ V (1 + C ω (M, Λ)) inf v P Λ (Γ) V u v L (Γ,V ) Proof: for any v P Λ V : u P M,ω Λ u L 2 ρ V u v L 2 ρ V + v P M,ω Λ u L 2 ρ V u v L 2 ρ V + C ω (M, Λ) v P M,ω Λ u M,V u v L 2 ρ V + C ω (M, Λ) u v M,V F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

65 Discrete L 2 projection using random evaluations Convergence analysis Error analysis continuous norm: v 2 L 2 ρ V = Γ v(y) 2 V ρ(y)dy discrete norm: v 2 M,V = 1 M M i=1 v(y i) 2 V v 2 L 2 ρ V Define the constant C ω (M, Λ) := sup v PΛ (Γ) V v 2 M,V Theorem [Migliorati-Nobile-von Schwerin-Tempone 11] 1 C ω (M, Λ) 1 almost surely when M 2 u P M,ω Λ u L 2 ρ V (1 + C ω (M, Λ)) inf v P Λ (Γ) V u v L (Γ,V ) Proof: for any v P Λ V : u P M,ω Λ u L 2 ρ V u v L 2 ρ V + v P M,ω Λ u L 2 ρ V u v L 2 ρ V + C ω (M, Λ) v P M,ω Λ u M,V u v L 2 ρ V + C ω (M, Λ) u v M,V F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

66 Discrete L 2 projection using random evaluations Convergence analysis Error analysis continuous norm: v 2 L 2 ρ V = Γ v(y) 2 V ρ(y)dy discrete norm: v 2 M,V = 1 M M i=1 v(y i) 2 V v 2 L 2 ρ V Define the constant C ω (M, Λ) := sup v PΛ (Γ) V v 2 M,V Theorem [Migliorati-Nobile-von Schwerin-Tempone 11] 1 C ω (M, Λ) 1 almost surely when M 2 u P M,ω Λ u L 2 ρ V (1 + C ω (M, Λ)) inf v P Λ (Γ) V u v L (Γ,V ) Proof: for any v P Λ V : u P M,ω Λ u L 2 ρ V u v L 2 ρ V + v P M,ω Λ u L 2 ρ V u v L 2 ρ V + C ω (M, Λ) v P M,ω Λ u M,V u v L 2 ρ V + C ω (M, Λ) u v M,V F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

67 Discrete L 2 projection using random evaluations Convergence analysis Error analysis continuous norm: v 2 L 2 ρ V = Γ v(y) 2 V ρ(y)dy discrete norm: v 2 M,V = 1 M M i=1 v(y i) 2 V v 2 L 2 ρ V Define the constant C ω (M, Λ) := sup v PΛ (Γ) V v 2 M,V Theorem [Migliorati-Nobile-von Schwerin-Tempone 11] 1 C ω (M, Λ) 1 almost surely when M 2 u P M,ω Λ u L 2 ρ V (1 + C ω (M, Λ)) inf v P Λ (Γ) V u v L (Γ,V ) Proof: for any v P Λ V : u P M,ω Λ u L 2 ρ V u v L 2 ρ V + v P M,ω Λ u L 2 ρ V u v L 2 ρ V + C ω (M, Λ) v P M,ω Λ u M,V u v L 2 ρ V + C ω (M, Λ) u v M,V F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

68 Discrete L 2 projection using random evaluations Convergence analysis First estimate of C ω (M, Λ) monov. uniform r.v. In this case P Λ = P w = polynomial space of degree w. Theorem [Migliorati-Nobile-von Schwerin-Tempone, FoCM 13] For any α (0, 1), let M be such that M 3 log((m + 1)/α) = 4 3 w 2 Then, with probability at least 1 α it holds C ω (M, Λ) 3 log((m + 1)/α) and u P w,ω Λ u L 2 ρ V ( log M + 1 α ) inf v P w u v L (Γ,V ) F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

69 Discrete L 2 projection using random evaluations Convergence analysis First estimate of C ω (M, Λ) monov. uniform r.v. In this case P Λ = P w = polynomial space of degree w. Theorem [Migliorati-Nobile-von Schwerin-Tempone, FoCM 13] For any α (0, 1), let M be such that M 3 log((m + 1)/α) = 4 3 w 2 Then, with probability at least 1 α it holds C ω (M, Λ) 3 log((m + 1)/α) and u P w,ω Λ u L 2 ρ V ( log M + 1 α ) inf v P w u v L (Γ,V ) F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

70 Discrete L 2 projection using random evaluations Convergence analysis A more general result multivar. uniform r.vs Theorem [Chkifa-Cohen-Migliorati-Nobile-Tempone 13] For any set Λ N N downward closed and any δ, γ > 0, if M log M 1 + γ (#Λ) 2, β δ = δ + (1 δ) log(1 δ) β δ then with probability at least 1 2M γ it holds and (1 δ) v 2 L 2 ρ V v 2 M,V (1 + δ) v 2 L 2 ρ V 1 u P M,ω Λ u L 2 ρ V (1 + 1 δ ) inf u v L v P Λ V (Γ,V ) F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

71 Discrete L 2 projection using random evaluations Convergence analysis Improvements can be obtained by sampling from a different distribution ˆρ. Let us consider the weighted least square approx. P M,ω Λ u = argmin v P Λ (Γ) V 1 M M k=1 ρ(y (k) ) ˆρ(y (k) ) u k v(y (k) ) 2 V where the sample {y (k) } k is drawn from the distribution ˆρ(y)dy. For instance: In the case ρ(y) = ˆρ(y) = Chebyshev distribution in [ 1, 1] N, then the relation M min{2 N #Λ, (#Λ) log(3) log(2) } is enough to guarantee optimal convergence [Chkifa-Cohen-Migliorati-Nobile-Tempone 13] In the case ρ(y)=uniform distribution in [ 1, 1] N and ˆρ(y)=Chebyshev distribution in [ 1, 1] N, then, the relation M C N #Λ guarantees optimal convergence [Rauhut-Ward 2012]. F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

72 Discrete L 2 projection using random evaluations Convergence analysis Improvements can be obtained by sampling from a different distribution ˆρ. Let us consider the weighted least square approx. P M,ω Λ u = argmin v P Λ (Γ) V 1 M M k=1 ρ(y (k) ) ˆρ(y (k) ) u k v(y (k) ) 2 V where the sample {y (k) } k is drawn from the distribution ˆρ(y)dy. For instance: In the case ρ(y) = ˆρ(y) = Chebyshev distribution in [ 1, 1] N, then the relation M min{2 N #Λ, (#Λ) log(3) log(2) } is enough to guarantee optimal convergence [Chkifa-Cohen-Migliorati-Nobile-Tempone 13] In the case ρ(y)=uniform distribution in [ 1, 1] N and ˆρ(y)=Chebyshev distribution in [ 1, 1] N, then, the relation M C N #Λ guarantees optimal convergence [Rauhut-Ward 2012]. F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

73 Discrete L 2 projection using random evaluations Convergence analysis Improvements can be obtained by sampling from a different distribution ˆρ. Let us consider the weighted least square approx. P M,ω Λ u = argmin v P Λ (Γ) V 1 M M k=1 ρ(y (k) ) ˆρ(y (k) ) u k v(y (k) ) 2 V where the sample {y (k) } k is drawn from the distribution ˆρ(y)dy. For instance: In the case ρ(y) = ˆρ(y) = Chebyshev distribution in [ 1, 1] N, then the relation M min{2 N #Λ, (#Λ) log(3) log(2) } is enough to guarantee optimal convergence [Chkifa-Cohen-Migliorati-Nobile-Tempone 13] In the case ρ(y)=uniform distribution in [ 1, 1] N and ˆρ(y)=Chebyshev distribution in [ 1, 1] N, then, the relation M C N #Λ guarantees optimal convergence [Rauhut-Ward 2012]. F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

74 Discrete L 2 projection using random evaluations Numerical results Some numerical examples 1D function Condition number of D T D c=2 c=3 c=5 c=10 c=20 Condition number, N=1, M=c #Λ Condition number, N=1, M=c #Λ 2 c=0.5 c=1 c=1.5 c=2 c=3 D M(ω) ) D M(ω) ) T cond(d M(ω) T cond(d M(ω) w M = c #Λ w M = c (#Λ) 2 F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,

75 Discrete L 2 projection using random evaluations Numerical results Some numerical examples 1D function Approximation of the meromorphic function φ(y) = y Error φ(y)=1/(1+0.5y), N=1, M=c #Λ Error φ(y)=1/(1+0.5y), N=1, M=c #Λ φ Π w M(ω) φ cv c=2 c=3 c=5 c=10 c= w w M = c #Λ M = c (#Λ) 2 error with respect to polynomial degree. φ Π w M(ω) φ cv c=0.5 c=1 c=1.5 c=2 c=3 F. Nobile (EPFL) Collocation methods for random PDEs KAUST, January 6-10,