Tractability of Multivariate Integration in Hermite Spaces Friedrich - - PowerPoint PPT Presentation

Tractability of Multivariate Integration in Hermite Spaces

Friedrich Pillichshammer1

JKU Linz/Austria

Joint work with

• Ch. Irrgeher, P. Kritzer and G. Leobacher (JKU Linz)

1Supported by the Austrian Science Fund (FWF), Project F5509-N26. Friedrich Pillichshammer (JKU Linz/Austria) Multivariate Integration in Hermite Spaces 1 / 25

Multivariate integration over Rs – linear algorithms

We study the numerical approximation of integrals Is(f ) =

• Rs f (x)ϕs(x) dx,

where ϕs is the density of the s-dimensional standard Gaussian measure, and f ∈ H(K) (RKHS) with norm · K. We use linear algorithms An,s(f ) =

n

• k=1

αkf (tk) for αk ∈ R and tk ∈ Rs. Linear algorithms are optimal (Smolyak 1965, Bakhvalov 1971)

Friedrich Pillichshammer (JKU Linz/Austria) Multivariate Integration in Hermite Spaces 2 / 25

Multivariate integration over Rs – worst-case setting

Worst-case error: e(An,s, K) = sup

f ∈H(K) f K ≤1

|Is(f ) − An,s(f )|. nth minimal worst-case error: e(n, s) = inf

An,s e(An,s, K).

Initial error: For n = 0, we approximate Is(f ) by zero, and e(0, s) = Is for all s ∈ N. Information complexity: For ε ∈ (0, 1), n(ε, s) = min{n : e(n, s) ≤ ε e(0, s)}.

Friedrich Pillichshammer (JKU Linz/Austria) Multivariate Integration in Hermite Spaces 3 / 25

Smoothness of the problems

We study problems with infinite smoothness. It is natural to demand more

◮ of the nth minimal errors e(n, s) and ◮ of the information complexity n(ε, s)

than for those cases where we only have finite smoothness. For problems with unbounded smoothness we are interested in

• btaining (uniform) exponential convergence of the minimal errors.

Well studied: Korobov spaces of periodic functions over [0, 1]s with infinite smoothness (Dick, Kritzer, Larcher, Wo´ zniakowski)

Friedrich Pillichshammer (JKU Linz/Austria) Multivariate Integration in Hermite Spaces 4 / 25

Exponential convergence

Definition

Exponential convergence (EXP) if ∃ q ∈ (0, 1) and functions p, C, C1 : N → (0, ∞) such that e(n, s) ≤ C(s) q (n/C1(s)) p(s) for all s, n ∈ N. The largest possible rate of EXP is p∗(s) = sup

• p > 0 : ∃ C, C1 > 0 s.t. ∀n ∈ N : e(n, s) ≤ Cq(n/C1)p

.

Friedrich Pillichshammer (JKU Linz/Austria) Multivariate Integration in Hermite Spaces 5 / 25

Uniform exponential convergence

Definition

Uniform exponential convergence (UEXP) if ∃ q ∈ (0, 1), ∃ p > 0 and functions C, C1 : N → (0, ∞) such that e(n, s) ≤ C(s) q (n/C1(s)) p for all s, n ∈ N. The largest rate of UEXP is p∗ = sup

• p > 0 : ∃C, C1 : N → (0, ∞) s.t.

∀n, s ∈ N : e(n, s) ≤ C(s)q(n/C1(s))p .

Friedrich Pillichshammer (JKU Linz/Austria) Multivariate Integration in Hermite Spaces 6 / 25

Exponential convergence

UEXP implies n(ε, s) ≤

• C1(s)

log C(s) + log ε−1 log q−1 1/p for all s ∈ N, ε ∈ (0, 1). With respect to ε → 0, we need O

• log ε−11/p

function values to reduce the initial error by a factor of ε.

Friedrich Pillichshammer (JKU Linz/Austria) Multivariate Integration in Hermite Spaces 7 / 25

EC-tractability

(a) Exponential Convergence-Weak Tractability (EC-WT) if lim

s+ε−1→∞

log n(ε, s) s + log ε−1 = 0. (b) Exponential Convergence-Polynomial Tractability (EC-PT) if ∃ c, τ1, τ2 > 0 such that n(ε, s) ≤ c s τ1 (1 + log ε−1) τ2 for all s ∈ N, ε ∈ (0, 1). (c) Exponential Convergence-Strong Polynomial Tractability (EC-SPT) if ∃ c, τ > 0 such that n(ε, s) ≤ c (1 + log ε−1) τ for all s ∈ N, ε ∈ (0, 1). The exponent τ ∗ of EC-SPT is the infimum of τ for which EC-SPT holds, i.e., τ ∗ = inf{τ ≥ 0 : ∃c > 0 s.t. n(ε, s) ≤ c(1 + log ε−1)τ ∀s, ε}.

Friedrich Pillichshammer (JKU Linz/Austria) Multivariate Integration in Hermite Spaces 8 / 25

EC-tractability

Proposition

1 EC-SPT ⇒ EC-PT ⇒ UEXP 2 EC-WT ⇒ limn→∞ nαe(n, s) = 0 for all α > 0 3 If we have UEXP (e(n, s) ≤ C(s) q(n/C1(s))p), then: ◮ C(s) = exp(exp(o(s))) and C1(s) = exp(o(s)) ⇒ EC-WT ◮ C(s) = exp(O(sτ)) and C1(s) = O(sη) ⇒ EC-PT ◮ C(s) = O(1) and C1(s) = O(1) ⇒ EC-SPT Friedrich Pillichshammer (JKU Linz/Austria) Multivariate Integration in Hermite Spaces 9 / 25

Hermite polynomials

Univariate Hermite polynomials Hk(x) = (−1)k √ k! e

x2 2

dk dxk e− x2

2

for k ∈ N0, x ∈ R E.g. H0(x) = 1, H1(x) = x, H2(x) = x2

√ 2 − 1 √ 2, H3(x) = x3 √ 3 −

• 3

2x

Multivariate Hermite polynomials Hk(x) =

s

• j=1

Hkj(xj) for k ∈ Ns

0, x ∈ Rs

(Hk)k∈Ns

0 is an ONB of L2(Rs, ϕs)

Hermite expansion of f ∈ L2(Rs, ϕs): f (x) ∼

• k∈Ns
• f (k)Hk(x)

with kth Hermite coefficient f (k) =

• Rs f (x)Hk(x)ϕs(x) dx

Friedrich Pillichshammer (JKU Linz/Austria) Multivariate Integration in Hermite Spaces 10 / 25

Hermite spaces

Let r : Ns

0 → R+ be a summable function.

Define a kernel Kr(x, y) =

• k∈Ns

r(k)Hk(x)Hk(y) for x, y ∈ Rs, and a inner product f , gKr :=

• k∈Ns

1 r(k)

• f (k)

g(k). Let f 2

Kr = f , f Kr .

We call the RKHS H(Kr) a Hermite space.

Friedrich Pillichshammer (JKU Linz/Austria) Multivariate Integration in Hermite Spaces 11 / 25

The Hermite space H(Ks,a,b,ω)

Let a = {aj}j≥1 and b = {bj}j≥1, where we assume that 1 ≤ a1 ≤ a2 ≤ a3 ≤ . . . and 1 ≤ b1 ≤ b2 ≤ b3 ≤ . . . . Fix ω ∈ (0, 1). For a vector k = (k1, . . . , ks) ∈ Ns

0, consider

r(k) = ω

s

j=1 ajk bj j .

We modify the notation for the kernel function to Ks,a,b,ω(x, y) :=

• k∈Ns

ω

s

j=1 ajk bj j Hk(x)Hk(y).

Proposition

f ∈ H(Ks,a,b,ω) ⇒ f is analytic R[x] ⊂ H(Ks,a,b,ω) f (x) = exp(λ · x) belongs to H(Ks,a,b,ω) for suitable a, b e(0, s) = 1

Friedrich Pillichshammer (JKU Linz/Austria) Multivariate Integration in Hermite Spaces 12 / 25

The main result

Main Theorem

1 EXP holds for all a and b and

p∗(s) = 1/B(s) with B(s) :=

s

• j=1

b−1

j

.

2 Let B := ∞

j=1 b−1 j

. Then B < ∞ ⇔ UEXP ⇔ EC-PT ⇔ EC-SPT and p∗ = 1/B and the exponent τ ∗ of EC-SPT is B.

3 EC-WT ⇒ limj→∞ aj2bj = ∞ 4 limj→∞ aj = ∞ ⇒ EC-WT Friedrich Pillichshammer (JKU Linz/Austria) Multivariate Integration in Hermite Spaces 13 / 25

Remarks on the theorem

We always have EXP, independent of a and b. The best rate p∗(s) is 1/B(s), which decreases for growing s. A necessary and sufficient condition for UEXP, EC-PT and EC-SPT is that B =

∞

• j=1

b−1

j

< ∞ with no extra conditions on a and ω. The best rate p∗ is 1/B < 1. Small B implies a large p∗. a and ω have no influence on UEXP, EC-PT and EC-SPT. There is a gap between the necessary and sufficient condition for EC-WT. The results for EXP, UEXP, EC-PT and EC-SPT are constructive.

Friedrich Pillichshammer (JKU Linz/Austria) Multivariate Integration in Hermite Spaces 14 / 25

Gauss-Hermite rules – one-dimensional case

A Gauss-Hermite rule of order n is a linear integration rule An of the form An(f ) =

n

• i=1

αif (xi) that is exact for all p ∈ R[x] with deg(p) < 2n, i.e.

• R

p(x)ϕ(x) dx = An(p) ∀p ∈ R[x] with deg(p) < 2n. The nodes x1, . . . , xn ∈ R are exactly the zeros of the nth Hermite polynomial Hn and the weights are given by αi = 1 nH2

n−1(xi).

Friedrich Pillichshammer (JKU Linz/Austria) Multivariate Integration in Hermite Spaces 15 / 25

Gauss-Hermite rules – multivariate case

For j = 1, 2, . . . , s let A(j)

mj (f ) = mj

• i=1

α(j)

i f (x(j) i

) Let n = m1m2 · · · ms and set An,s = A(1)

m1 ⊗ · · · ⊗ A(s) ms ,

i.e., for f ∈ H(Ks,a,b,ω) An,s(f ) =

m1

• i1=1

. . .

ms

• is=1

α(1)

i1 · · · α(s) is f (x(1) i1 , . . . , x(s) is ).

Proposition

e2(An,s, Ks,a,b,ω) ≤ −1 +

s

• j=1
• 1 + ωaj(2mj)bj

√ 8π 1 − ω2

• .

Friedrich Pillichshammer (JKU Linz/Austria) Multivariate Integration in Hermite Spaces 16 / 25

Gauss-Hermite rules – multivariate case

Theorem

For s ∈ N, let B(s) := s

j=1 b−1 j

. For ε ∈ (0, 1), let m = max

j=1,2,...,s

       1 aj log √

8π 1−ω2 s log(1+ε2)

• log ω−1

 

B(s) 

    . and define mj := ⌊m1/(B(s)·bj)⌋. Then e(An,s, Ks,a,b,ω) ≤ ε and n(ε, s) ≪s log B(s)

• 1 + 1

ε

• .

This implies EXP.

Friedrich Pillichshammer (JKU Linz/Austria) Multivariate Integration in Hermite Spaces 17 / 25

Gauss-Hermite rules – multivariate case

Theorem

Assume that B = ∞

j=1 b−1 j

< ∞. For ε ∈ (0, 1) define mj =        log √

8π 1−ω2 π2 6 j2 log(1+ε2)

• aj2bj log ω−1

 

1/bj

    . Then e(An,s, Ks,a,b,ω) ≤ ε and n(ε, s) ≪δ log B+δ

• 1 + 1

ε

• .

This implies EC-SPT with τ ∗ at most B and also UEXP.

Friedrich Pillichshammer (JKU Linz/Austria) Multivariate Integration in Hermite Spaces 18 / 25

Finite smoothness

For k ∈ N0 define β0(k) = 1 and for τ ∈ N define βτ(k) =

• if 0 ≤ k < τ,

k! (k−τ)!

if k ≥ τ. For α ∈ N we define rα(k) = α

• τ=0

βτ(k) −1 ≍α 1 kα . For k = (k1, . . . , ks) ∈ Ns

0 let

r(k) = rα(k) =

s

• j=1

rα(kj).

Friedrich Pillichshammer (JKU Linz/Austria) Multivariate Integration in Hermite Spaces 19 / 25

Finite smoothness

For s ∈ N define the kernel function Ks,α(x, y) =

• k∈Ns

rα(k)Hk(x)Hk(y) for x, y ∈ Rs and inner product f , gKs,α =

• k∈Ns

1 rα(k)

• f (k)

g(k). The inner product can also be written as f , gKs,α =

• τ∈{0,...,α}s
• Rs

∂τf ∂xτ (x)∂τg ∂xτ (x) ϕs(x) dx. We call H(Ks,α) the Hermite space of smoothness α.

Friedrich Pillichshammer (JKU Linz/Austria) Multivariate Integration in Hermite Spaces 20 / 25

Integration in H(Ks,α)

Theorem (Irrgeher and Leobacher 2014)

There exist QMC rules Qn,s(f ) = 1

n

n

i=1 f (xi) such that

e(Qn,s, Ks,α) ≪s 1 √n.

Friedrich Pillichshammer (JKU Linz/Austria) Multivariate Integration in Hermite Spaces 21 / 25

A Smolyak algorithm based on Gauss-Hermite rules

For i ∈ N0 let A2i(f ) be one-dimensional Gauss-Hermite rules of order 2i. Define ∆i = A1 if i = 0, A2i − A2i−1 if i ∈ N. The Smolyak algorithm based on Gauss-Hermite rules is defined as Aq,s =

∞

• i1,...,is =0

i1+···+is ≤q

s

• j=1

∆ij. Aq,s is linear and requires n =

q

• t=q−s+1

2t t + s − 1 s − 1

• function evaluations. Hence q ≍s log n.

Friedrich Pillichshammer (JKU Linz/Austria) Multivariate Integration in Hermite Spaces 22 / 25

A Smolyak algorithm based on Gauss-Hermite rules

Theorem

e(Aq,s, Ks,α) ≪s,α (log n)s−1 n(α−1)/2 For α > 2 this improves the existence result for QMC rules. For 1 ≤ α ≤ 2 the QMC result is better.

Theorem (Dick, 2014)

For any linear quadrature rule An,s we have e(An,s, Ks,1) ≫s 1 n.

Friedrich Pillichshammer (JKU Linz/Austria) Multivariate Integration in Hermite Spaces 23 / 25

Summary

For infinite smoothness:

◮ exponential convergence holds always ◮ if and only if condition for UEXP, EC-PT and EC-SPT ◮ gap between the sufficient and necessary condition for EC-WT

For finite smoothness α:

◮ convergence of order O

• 1

√n

• for QMC, and O
• (log n)s−1

n(α−1)/2

• for Smolyak

rules based on Gauss-Hermite

Open question

What is the exact convergence order for e(n, s) in H(Ks,α) and what are the

• ptimal algorithms?

◮ tractability not yet studied Friedrich Pillichshammer (JKU Linz/Austria) Multivariate Integration in Hermite Spaces 24 / 25

Announcement

10th IMACS Seminar on Monte Carlo Methods July 6-10, 2015 Linz, Austria www.mcm2015.jku.at

Friedrich Pillichshammer (JKU Linz/Austria) Multivariate Integration in Hermite Spaces 25 / 25