Quasi-optimal and adaptive sparse grids with control variates for PDEs with random diffusion coefficient

Transcription

1 Quasi-optimal and adaptive sparse grids with control variates for PDEs with random diffusion coefficient F. Nobile, L. Tamellini, R. Tempone, F. Tesei CSQI - MATHICSE, EPFL, Switzerland Dipartimento di Matematica, Università di Pavia, Italy SRI UQ Center, KAUST, Saudi Arabia UQAW 2016 KAUST, January 5, 2016 Lorenzo Tamellini (UNIPV) January 5, / 21

2 Outline 1 The lognormal Darcy problem (with rough coefficients) 2 Optimized sparse grids 3 Dimension-adaptive algorithms 4 Numerical results - part I 5 Monte Carlo Control Variate 6 Numerical results - part II Lorenzo Tamellini (UNIPV) January 5, / 21

3 The uncertain Darcy problem Find a pressure p : D Γ R, such that { (e γν p) = f in D, +B.C. (see plot on the right). u = outward flux from the right-hand boundary. ( e γ p ) n = 0 D p=1 p=0 ( e γ p ) n = 0 Lorenzo Tamellini (UNIPV) January 5, / 21

4 The uncertain Darcy problem Find a pressure p(x, y) : D Γ R, such that ϱ-a.e.: { (e γν(x,y) p(x, y)) = f (x) x D, +B.C. (see plot on the right). u(y) = outward flux is a random fun. appr. e.g. E[u] ( e γ p ) n = 0 D p=1 p=0 ( e γ p ) n = 0 γ ν = rand. field with tensor Matérn covariance expanded by Kar.-Loève γ ν (x, y) = σ y k γ k,ν φ k (x), y i N (0, 1) i.i.d. k=1 Lorenzo Tamellini (UNIPV) January 5, / 21

5 The uncertain Darcy problem Find a pressure p(x, y) : D Γ R, such that ϱ-a.e.: { (e γν(x,y) p(x, y)) = f (x) x D, +B.C. (see plot on the right). u(y) = outward flux is a random fun. appr. e.g. E[u] ( e γ p ) n = 0 D p=1 p=0 ( e γ p ) n = 0 γ ν = rand. field with tensor Matérn covariance expanded by Kar.-Loève γ ν (x, y) = σ y k γ k,ν φ k (x), y i N (0, 1) i.i.d. k=1 The smoothness of the realizations of γ (hence the decay of γ k,ν ) depends on ν [0.5, ]: ν = 0.5 γ ν = exponential cov. fun. (C 0,s, s < 1/2 realizations) ν = γ ν = Gaussian cov. fun. (C realizations) Lorenzo Tamellini (UNIPV) January 5, / 21

6 Hierarchical representation of a sparse grid S I [u](y) = i I c i T m(i) [u](y) is a sparse grid interp. = S I [u](y) = T [15 1] [u](y) + T [1 15] [u](y) + T [5 5] [u](y) Q I [u] = c i E [ T m(i) [u] ] = i I i I c i y j T m(i) ω j u(y j ) is a s.g. quadrature Admissibility condition for I: i I, i e j I if i j > 1. Collocation pts: y i N (0, 1) Gauss Hermite, Genz Keister, gen. Leja Lorenzo Tamellini (UNIPV) January 5, / 21

7 Hierarchical representation of a sparse grid S I [u](y) = i I c i T m(i) [u](y) = i I m(i) [u](y) is a sparse grid interp. Un m(in) [u](y n ) is an interpolant along y n over m(i n ) points N T m(i) [u](y) = Un m(in) [u](y) is a tensor interpolant n=1 Un m(in) [u] Un m(in 1) [u] is an interpolation detail N m(i) [u](y) = [u] U m(in 1) [u]) is a hierarchical surplus Q I [u] = i I n=1 (U m(in) n c i E [ T m(i) [u] ] = i I n c i y j T m(i) ω j u(y j ) is a s.g. quadrature Admissibility condition for I: i I, i e j I if i j > 1. Collocation pts: y i N (0, 1) Gauss Hermite, Genz Keister, gen. Leja Lorenzo Tamellini (UNIPV) January 5, / 21

8 Optimized (knapsack) sparse grids S I [u](y) = i I m(i) [u](y). What terms m(i) to include in the sum? 1 How much will the approx. improve if I add m(i) [u]? E(i) 2 How many new problem solves if I add m(i) [u]? W (i) Use as I the set of m(i) [u] with the highest ratio E(i) W (i) (i.e. a knapsack problem solved by the Dantzig algorithm more ) Note: E(i) depends on the norm used to measure the improvement. We show next three different strategies, two a-posteriori and one a-priori Lorenzo Tamellini (UNIPV) January 5, / 21

9 A-posteriori (adaptive) strategies Gerstner & Griebel, Multi-index set I curr. idx 1 Sparse grid set Given i = 1, I = {i} and R = repeat: 1 Add to R the neighbors of i feasible wrt to I 2 Compute S I B [u] 3 find the index j R with the highest profit (estimated a-posteriori ) 4 set i = j and move it from R to I NB: slight changes needed for non-nested points and unbounded Γ, see later and poster. Lorenzo Tamellini (UNIPV) January 5, / 21

10 A-posteriori (adaptive) strategies Gerstner & Griebel, Multi-index set I reduced margin curr. idx Sparse grid set Given i = 1, I = {i} and R = repeat: 1 Add to R the neighbors of i feasible wrt to I 2 Compute S I B [u] 3 find the index j R with the highest profit (estimated a-posteriori ) 4 set i = j and move it from R to I NB: slight changes needed for non-nested points and unbounded Γ, see later and poster. Lorenzo Tamellini (UNIPV) January 5, / 21

11 A-posteriori (adaptive) strategies Gerstner & Griebel, Multi-index set I reduced margin curr. idx Sparse grid set Given i = 1, I = {i} and R = repeat: 1 Add to R the neighbors of i feasible wrt to I 2 Compute S I B [u] 3 find the index j R with the highest profit (estimated a-posteriori ) 4 set i = j and move it from R to I NB: slight changes needed for non-nested points and unbounded Γ, see later and poster. Lorenzo Tamellini (UNIPV) January 5, / 21

12 A-posteriori (adaptive) strategies Gerstner & Griebel, Multi-index set I reduced margin curr. idx Sparse grid set Given i = 1, I = {i} and R = repeat: 1 Add to R the neighbors of i feasible wrt to I 2 Compute S I B [u] 3 find the index j R with the highest profit (estimated a-posteriori ) 4 set i = j and move it from R to I NB: slight changes needed for non-nested points and unbounded Γ, see later and poster. Lorenzo Tamellini (UNIPV) January 5, / 21

13 A-posteriori (adaptive) strategies Gerstner & Griebel, Multi-index set I reduced margin curr. idx Sparse grid set Given i = 1, I = {i} and R = repeat: 1 Add to R the neighbors of i feasible wrt to I 2 Compute S I B [u] 3 find the index j R with the highest profit (estimated a-posteriori ) 4 set i = j and move it from R to I NB: slight changes needed for non-nested points and unbounded Γ, see later and poster. Lorenzo Tamellini (UNIPV) January 5, / 21

14 A-posteriori (adaptive) strategies Gerstner & Griebel, Multi-index set I reduced margin curr. idx Sparse grid set Given i = 1, I = {i} and R = repeat: 1 Add to R the neighbors of i feasible wrt to I 2 Compute S I B [u] 3 find the index j R with the highest profit (estimated a-posteriori ) 4 set i = j and move it from R to I NB: slight changes needed for non-nested points and unbounded Γ, see later and poster. Lorenzo Tamellini (UNIPV) January 5, / 21

15 Method 1: Quadrature-adaptive sparse grids Here we consider E(i) E [ m(i) [u] ]. { N ( 1 n=1 m(in ) m(i n 1) ) Genz Keister W (i) = N n=1 m(i n) Gauss Hermite or alternatively W (i) = 1. 2 E(i) = Q I {i} [u] Q I [u] for any I admissible st I {i} is admissible. Note: The algorithm above is not bound to use nested points! Hence we can use Gauss Hermite points, whose m( ) grows much slower than Genz Keister nodes. Lorenzo Tamellini (UNIPV) January 5, / 21

16 Method 2: Interpolation-adaptive sparse grids Here we consider E(i) max y Γ ( m(i) [u](y) π(y) ). Note that π(y) > 0 needs not be equal to ϱ normal (y). { N ( 1 n=1 m(in ) m(i n 1) ) for Genz Keister W (i) = N n=1 m(i n) for Gauss Hermite or alternatively W (i) = 1. 2 If using Genz Keister (nested) knots, ( the sparse grid is interpolatory, hence u(y) E(i) = max y pts(si {i} )\pts(s I ) S m I [u](y) ) π(y). If using Gauss Hermite (non-nested) knots, the sparse grid is not interpolatory, hence ( ) E(i) = max y pts(t m(i) ) SI {i} m [u](y) Sm I [u](y) π(y). So again, we can use the algorithm with non-nested nodes too Lorenzo Tamellini (UNIPV) January 5, / 21

17 A-priori strategy Method 3: Quasi-optimal sparse grids more Here we consider E(i) m(i) [u] L 2 ϱ (Γ). By linking m(i) [u] to the spectral (Hermite) expansion of u, we can get computable estimates of E(i), before actually running the sparse grid algorithm (see next slide). W (i) can also be (easily) estimated (see next slide). { } E(i) Then, just compute the set I k = W (i) > ɛ k Lorenzo Tamellini (UNIPV) January 5, / 21

18 Method 3: Quasi-optimal sparse grids Here we consider E(i) m(i) [u] L 2 ϱ (Γ). Let H p (y) be the Hermite pol. of degree p n wrt y n. 1 Let m(i n ) = i n for Gauss Hermite pts (non-nested), m(i n ) = 1, 3, 9, 19, 35 for Genz Keister { (nested). Then let N ( n=1 m(in ) m(i n 1) ) Genz Keister W (i) = N n=1 m(i n) Gauss Hermite N 2 E(i) = C e n=1 gnm(in 1) N n=1 m(in 1) B(i), with B(i) = m(i) [H m(i 1) ] L 2 ϱ (Γ) 3 For a suff. large universe U N N compute the profit of each m(i), P(i) = E(i)/ W (i) 4 sort decreasingly P(i) 5 build the sparse grid with the m(i) with largest profits In order to compute the profits we need to assess the rates g n. This can be performed numerically with a comp. cost linear in N ( a priori/a posteriori approach ) Lorenzo Tamellini (UNIPV) January 5, / 21 more

19 Method 3: Quasi-optimal sparse grids Here we consider E(i) m(i) [u] L 2 ϱ (Γ). Let H p (y) be the Hermite pol. of degree p n wrt y n. 1 Let m(i n ) = i n for Gauss Hermite pts (non-nested), m(i n ) = 1, 3, 9, 19, 35 for Genz Keister { (nested). Then let N ( n=1 m(in ) m(i n 1) ) Genz Keister W (i) = N n=1 m(i n) Gauss Hermite N 2 E(i) = C e n=1 gnm(in 1) N n=1 m(in 1) B(i), with B(i) = m(i) [H m(i 1) ] L 2 ϱ (Γ) 3 For a decreasing sequence ɛ 1, ɛ 2... compute the sets { } E(i) I k = W (i) > ɛ k more In order to compute the profits we need to assess the rates g n. This can be performed numerically with a comp. cost linear in N ( a priori/a posteriori approach ) Lorenzo Tamellini (UNIPV) January 5, / 21

20 In order to compute the profits we need to assess the rates g n. This can be performed numerically with a comp. cost linear in N ( a priori/a posteriori approach ) Lorenzo Tamellini (UNIPV) January 5, / 21 Method 3: Quasi-optimal sparse grids Here we consider E(i) m(i) [u] L 2 ϱ (Γ). Let H p (y) be the Hermite pol. of degree p n wrt y n. 1 Let m(i n ) = i n for Gauss Hermite pts (non-nested), m(i n ) = 1, 3, 9, 19, 35 for Genz Keister { (nested). Then let N ( n=1 m(in ) m(i n 1) ) Genz Keister W (i) = N n=1 m(i n) Gauss Hermite N 2 E(i) = C e n=1 gnm(in 1) N n=1 m(in 1) B(i), with B(i) = m(i) [H m(i 1) ] L 2 ϱ (Γ) 3 For a decreasing sequence ɛ 1, ɛ 2... compute the sets e N n=1 gnm(in 1) N B(i) n=1 m(in 1) I k = > ɛ k W more

21 In order to compute the profits we need to assess the rates g n. This can be performed numerically with a comp. cost linear in N ( a priori/a posteriori approach ) Lorenzo Tamellini (UNIPV) January 5, / 21 Method 3: Quasi-optimal sparse grids Here we consider E(i) m(i) [u] L 2 ϱ (Γ). Let H p (y) be the Hermite pol. of degree p n wrt y n. 1 Let m(i n ) = i n for Gauss Hermite pts (non-nested), m(i n ) = 1, 3, 9, 19, 35 for Genz Keister { (nested). Then let N ( n=1 m(in ) m(i n 1) ) Genz Keister W (i) = N n=1 m(i n) Gauss Hermite N 2 E(i) = C e n=1 gnm(in 1) N n=1 m(in 1) B(i), with B(i) = m(i) [H m(i 1) ] L 2 ϱ (Γ) 3 For a decreasing sequence ɛ 1, ɛ 2... compute the sets { ( ) } > log ɛ k I k = log e N n=1 gnm(in 1) B(i) N n=1 m(in 1) W more

22 In order to compute the profits we need to assess the rates g n. This can be performed numerically with a comp. cost linear in N ( a priori/a posteriori approach ) Lorenzo Tamellini (UNIPV) January 5, / 21 Method 3: Quasi-optimal sparse grids Here we consider E(i) m(i) [u] L 2 ϱ (Γ). Let H p (y) be the Hermite pol. of degree p n wrt y n. 1 Let m(i n ) = i n for Gauss Hermite pts (non-nested), m(i n ) = 1, 3, 9, 19, 35 for Genz Keister { (nested). Then let N ( n=1 m(in ) m(i n 1) ) Genz Keister W (i) = N n=1 m(i n) Gauss Hermite N 2 E(i) = C e n=1 gnm(in 1) N n=1 m(in 1) B(i), with B(i) = m(i) [H m(i 1) ] L 2 ϱ (Γ) 3 For an integer sequence w 1, w 2... compute the sets { N I k = g n m(i n 1) log B(i) + log } m(i n 1) + log W (i) < w k n=1 more

23 Convergence theorem for quasi-optimal sparse grids If the profits satisfy the weighted summability condition ( ) 1/τ P τ (i) W (i) = CP (τ) < i N N for some 0 < τ < 1, then the knapsack sparse grid approximation of u satisfies u S I [u] W 1 1/τ C P (τ). where W is the number of collocation points used to build S I [u]. Sketch of proof: Let {Q j } j N+ = j k=1 W k and { P k } k N+ = E 1 E 1 E 1,,..., E 2 E 2 E 2,,.... W 1 W 1 W 1 W 2 W 2 W 2 }{{}}{{} W 1 times W 2 times u SI(w) m [u] i / I(w) m(i) u = j>w E j = k>q Pk w. Use Stechkin Lemma: ( k>q Pk w Qw 1/τ+1 P k>0 k τ ) 1/τ 1/τ+1 ( = Q w k>0 Pτ k W ) 1/τ k It is quite easy to extend the proof for cases where I is not admissible. Lorenzo Tamellini (UNIPV) January 5, / 21

24 Convergence theorem for quasi-optimal sparse grids If the profits satisfy the weighted summability condition ( ) 1/τ P τ (i) W (i) = CP (τ) < i N N for some 0 < τ < 1, then the knapsack sparse grid approximation of u satisfies u S I [u] W 1 1/τ C P (τ). where W is the number of collocation points used to build S I [u]. Ex.: assume for a moment y 1,..., y N uniform random variables, sparse grids with Clenshaw Curtis points, Legendre coeff. decay as C(N)e N n=1 gnm(in 1). Then the theorem above implies that ( u S I [u] C 1 (N) exp C 2 N N ) W Sketch of proof: observe that in this case P are τ-summable for 0 < τ < 1; then, optimize the error estimate w.r.t. τ. Lorenzo Tamellini (UNIPV) January 5, / 21

25 To refine or to add random var.? Dimension-adaptivity Problem: exploring reduced margin in high-dimensional spaces / computing all rates g n beforehand is too expensive. Lorenzo Tamellini (UNIPV) January 5, / 21

26 To refine or to add random var.? Dimension-adaptivity Problem: exploring reduced margin in high-dimensional spaces / computing all rates g n beforehand is too expensive. Assume that the Karhunen Loève expansion γ = σ N k=1 y kγ k φ k introduces a weak ordering of random variables, i.e. there exists B 1 (buffer) s.t. y n+b is guaranteed to be less important than y n. Lorenzo Tamellini (UNIPV) January 5, / 21

27 To refine or to add random var.? Dimension-adaptivity Problem: exploring reduced margin in high-dimensional spaces / computing all rates g n beforehand is too expensive. Assume that the Karhunen Loève expansion γ = σ N k=1 y kγ k φ k introduces a weak ordering of random variables, i.e. there exists B 1 (buffer) s.t. y n+b is guaranteed to be less important than y n. Idea: Add random variables gradually (balance refinement and addition of variables). Note that this means we don t need to truncate a-priori the Karhunen Loève expansion of γ! Lorenzo Tamellini (UNIPV) January 5, / 21

28 To refine or to add random var.? Dimension-adaptivity Problem: exploring reduced margin in high-dimensional spaces / computing all rates g n beforehand is too expensive. Assume that the Karhunen Loève expansion γ = σ N k=1 y kγ k φ k introduces a weak ordering of random variables, i.e. there exists B 1 (buffer) s.t. y n+b is guaranteed to be less important than y n. Idea: Add random variables gradually (balance refinement and addition of variables). Note that this means we don t need to truncate a-priori the Karhunen Loève expansion of γ! Define a random variable y n as activated if min i I i n > 1. Lorenzo Tamellini (UNIPV) January 5, / 21

29 To refine or to add random var.? Dimension-adaptivity Problem: exploring reduced margin in high-dimensional spaces / computing all rates g n beforehand is too expensive. Assume that the Karhunen Loève expansion γ = σ N k=1 y kγ k φ k introduces a weak ordering of random variables, i.e. there exists B 1 (buffer) s.t. y n+b is guaranteed to be less important than y n. Idea: Add random variables gradually (balance refinement and addition of variables). Note that this means we don t need to truncate a-priori the Karhunen Loève expansion of γ! Define a random variable y n as activated if min i I i n > 1. A simple dimension-adaptive algorithm 1 start the adaptive/quasi-optimal algorithm using B rand. var. 2 As soon as one of these buffer variables gets activated, add a new rand. var. to the approximaton (in the quasi optimal setting, this means computing the corresponding g n ). Lorenzo Tamellini (UNIPV) January 5, / 21

30 The uncertain Darcy problem results 1 Field data: σ = 1, corr. length L c = 0.5, ν = 2.5 (C 2 realizations) error MC error AD D AD D/NP ADNN D ADNN D/NP 1(1) 3(2) 1(1) 2(2) work 4(3) 4(3) 8(3) 7(3) 11(4) 17(4) 11(3) 17(4) 27(4) 38(4) 23(4) 35(4) adaptive schemes with activated variables error MC error OPT OPT NN AD WLinf/NP ADNN WLinf/NP AD D ADNN D work adaptive and quasi-optimal schemes moderate number of rand. vars. needed quasi-optimal plateau due to computation of g quasi-optimal and adaptive schemes have similar convergence nested and non-nested points have similar convergence convergence robust wrt. type of points and E[ ]/L -driven adaptation similar results for adaptive sparse interpolation Lorenzo Tamellini (UNIPV) January 5, / 21

31 Case ν = 0.5: Monte Carlo Control Variate Realizations of γ are non-differentiable KL eigenval decay very slow sparse grids may be non-effective. Remedy: use sparse grids as control var. (preconditioner) for MC 1 Consider a smoothed field γ ɛ, such that Q I [u ɛ ] E[u ɛ ] quickly. smoothed field, ɛ = 1/2 4 smoothed field ɛ = 1/2 6 non-smoothed field, ɛ = 0 Lorenzo Tamellini (UNIPV) January 5, / 21

32 Case ν = 0.5: Monte Carlo Control Variate Realizations of γ are non-differentiable KL eigenval decay very slow sparse grids may be non-effective. Remedy: use sparse grids as control var. (preconditioner) for MC 1 Consider a smoothed field γ ɛ, such that Q I [u ɛ ] E[u ɛ ] quickly. 2 Define u CV = u u ɛ + Q I [u ɛ ]. There holds E[u CV ] = E[u], Var(u CV ) = Var(u) + Var(u ɛ ) 2cov(u, u ɛ ) Thus, the smaller ɛ, the smaller the MC error, but slower the convergence Q I [u ɛ ] E[u ɛ ]. Lorenzo Tamellini (UNIPV) January 5, / 21

33 Case ν = 0.5: Monte Carlo Control Variate Realizations of γ are non-differentiable KL eigenval decay very slow sparse grids may be non-effective. Remedy: use sparse grids as control var. (preconditioner) for MC 1 Consider a smoothed field γ ɛ, such that Q I [u ɛ ] E[u ɛ ] quickly. 2 Define u CV = u u ɛ + Q I [u ɛ ]. There holds E[u CV ] = E[u], Var(u CV ) = Var(u) + Var(u ɛ ) 2cov(u, u ɛ ) Thus, the smaller ɛ, the smaller the MC error, but slower the convergence Q I [u ɛ ] E[u ɛ ]. 3 Set E[u CV ] 1 M M u CV (ω i ) = i=1 1 M M (u(ω i ) u ɛ (ω i )) + Q m I [u ɛ ]. M can be chosen balancing either the works or the errors of MC and sparse grids. i=1 Lorenzo Tamellini (UNIPV) January 5, / 21

34 The uncertain Darcy problem results 2 Field data: σ = 1, corr. length L c = 0.5, ν = 0.5 error MC error OPT OPT NN AD D ADNN D AD WLinf/NP ADNN WLinf/NP work MCCV error for adaptive and quasi-optimal sparse grids. 30 r.v. activated. error (1) MC error 10(1) 11(1) 11(1) OPT NN, ε=0 OPT NN, ε=2 5 27(2) 32(2) 79(3) 30(2) 45(3) 71(3) 242(4) 80(4) 113(4) 571(4) 224(4) OPT NN ε= (5) work Sparse grid component of the error for different values of ɛ. The performance deteriorates as ɛ 0 Lorenzo Tamellini (UNIPV) January 5, / 21

35 Conclusions 1 General framework for quasi-optimal and adaptive sparse grids schemes; 2 The schemes can be applied to the lognormal case, also with non-nested points; 3 The dimension-adaptive implementation allows to work without a-priori truncation of the random field; 4 A Monte Carlo Control Variate can be used to improve results in the rough case (exponential covariance). Lorenzo Tamellini (UNIPV) January 5, / 21

36 Thank you for your attention! (see also poster) Lorenzo Tamellini (UNIPV) January 5, / 21

37 Bibliography F. Nobile, L. Tamellini, F. Tesei and R. Tempone. An adaptive sparse grid algorithm for elliptic PDEs with lognormal diffusion coefficient Sparse Grids and Applications 2014, Springer (to appear). Also available as MATHICSE report 4/2015 F. Tesei Numerical Approximation of Flows in Random Porous Media Ph.D. Thesis Nr. 6860, EPFL, to appear on F. Tesei and F. Nobile. A Multi Level Monte Carlo Method with Control Variate for elliptic PDEs with log-normal coefficients. Stochastic Partial Differential Equations: Analysis and Computations, 3(3), 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 (available online) J. Beck, F. Nobile, L. Tamellini, and R. Tempone. A Quasi-optimal Sparse Grids Procedure for Groundwater Flows. Selected papers from the ICOSAHOM 12 conference. J. Beck, F. Nobile, L. Tamellini, and R. Tempone. On the optimal polynomial approximation of stochastic PDEs by Galerkin and Collocation methods Math. Mod. Methods Appl. Sci. (M3AS) 22(9), 2012 Lorenzo Tamellini (UNIPV) January 5, / 21

38 Matérn covariance cov γν (x, x ) = σ 2 2 i=1 ( 2ν xi x i /L ) ν ( c Kν 2ν xi x i /L c) G(ν)2 ν 1 Here K ν is the modified Bessel function of the second kind and G the Gamma function. back Lorenzo Tamellini (UNIPV) January 5, / 21

39 Knapsack approach S I [u](y) = m(i) [u](y). What terms m(i) to include in the sum? i I [ Given that E [u S I [u]] = E i I [u]] m(i) i I E [ m(i) [u] ], Lorenzo Tamellini (UNIPV) January 5, / 21

40 Knapsack approach S I [u](y) = i I m(i) [u](y). What terms m(i) to include in the sum? Given that 1 Set E [ m(i) [u] ] E(i) [ E [u S I [u]] = E i I [u]] m(i) W (i) = work of m(i) [u] x i {0, 1}, x i = 1 i I. i I E(i) Lorenzo Tamellini (UNIPV) January 5, / 21

41 Knapsack approach S I [u](y) = m(i) [u](y). What terms m(i) to include in the sum? i I [ Given that E [u S I [u]] = E i I [u]] m(i) i I E(i) 1 Set E [ m(i) [u] ] E(i) W (i) = work of m(i) [u] x i {0, 1}, x i = 1 i I. 2 Consider max i N N + E(i)x i s.t. i N N + W (i)x i W max and x i {0, 1} Lorenzo Tamellini (UNIPV) January 5, / 21

42 Knapsack approach S I [u](y) = m(i) [u](y). What terms m(i) to include in the sum? i I [ Given that E [u S I [u]] = E i I [u]] m(i) i I E(i) 1 Set E [ m(i) [u] ] E(i) W (i) = work of m(i) [u] x i {0, 1}, x i = 1 i I. 2 Consider max i N N + E(i)x i s.t. i N N + W (i)x i W max and x i {0, 1} 3 Solve by Dantzig Algorithm (knapsack problem approach) 1 compute the profit of each m(i) [u], P(i) = E(i)/ W (i) 2 sort m(i) [u] decreasing in profits 3 set x i = 1 until W max is reached back Lorenzo Tamellini (UNIPV) January 5, / 21

43 Quasi-optimal sparse grids back E(i) = m(i) [u] = m(i)[ ] u q H q = u q m(i) [H q ] q N N q N N Next, by construction m(i) [H q ] = 0 for polynomials such that n : q n < m(i n 1). By triangular inequality we get E(i) u q H 1 (D) m(i) [H q ]. L 2 ρ (Γ) q m(i 1) By assuming that the summation is dominated by the first term, we get E(i) B(i) u H m(i 1) 1 (D), B(i) = m(i) [H m(i 1) ] L 2 ρ (Γ) and we use the Heuristic N u H m(i 1) 1 (D) e n=1 gnm(in 1) N n=1 m(in 1) Lorenzo Tamellini (UNIPV) January 5, / 21