High-dimensional and infinite-dimensional hyperbolic crosses and their applications in approximation and uncertainty quantification Dinh D˜ ung Vietnam National University, Hanoi, Vietnam Workshop on Information-Based Complexity and Stochastic Computation September 15 – 19, 2014, ICERM, Brown University
October 2, 2014
Dinh D˜ ung (VNU, Hanoi)
Hyperbolic cross
October 2, 2014
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This talk is based in the recent joint works: 1 DD and T. Ullrich, N-Widths and ε-dimensions for high-dimensional approximations, Foundations Comp. Math. 13 (2013), 965-1003. 2 A. Chernov and DD, New explicit-in-dimension estimates for the cardinality of high-dimensional hyperbolic crosses and approximation of functions having mixed smoothness, (2014) http://arxiv.org/abs/1309.5170. 3 DD and M. Griebel, Hyperbolic cross approximation in infinite dimensions and applications in sPDEs, Manuscript (2014).
Dinh D˜ ung (VNU, Hanoi)
Hyperbolic cross
October 2, 2014
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The “curse of dimensionality”
There has been a great interest in solving numerical problems involving functions of big number s of variables. By classical methods as usually, the computation cost grows exponentially in s. We suffer the “curse of dimensionality” [Bellmann,1957] (“Dimensionality” is referred to the number s of variables). A way to get rid of it is to assume that mixed derivatives of functions are bounded, then to apply hyperbolic cross (HC) approximation.
Dinh D˜ ung (VNU, Hanoi)
Hyperbolic cross
October 2, 2014
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Infinite-dimensional approximation
The efficient approximation of a function of infinitely many variables is important in applications in physics, finance, engineering and statistics. It arises in UQ, computational finance and computational physics and is encountered for stochastic or parametric PDEs. Attempt: Apply infinite-dimensional HC to construct a linear approximation method to the solution of stochastic or parametric PDEs.
Dinh D˜ ung (VNU, Hanoi)
Hyperbolic cross
October 2, 2014
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Classical hyperbolic crosses
Classical HCs Γ(s, T ) are a domain of frequencies of trigonometric polynomials used for approximations of periodic functions having mixed derivative. They are given by Γ(s, T ) :=
s
k∈Z :
s Y
max(|ki |, 1) ≤ T .
i=1
Their cardinality is estimated as C (s) T logs−1 T ≤ |Γ(s, T )| ≤ C 0 (s) T logs−1 T , where |G | denotes the cardinality of G .
Dinh D˜ ung (VNU, Hanoi)
Hyperbolic cross
October 2, 2014
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n-Widths and ε-dimensions Kolmogorov n-widths: dn (W , H) :=
inf
sup inf kf − g kH .
{Ln linear subspaces, dim Ln ≤n} f ∈W g ∈Ln
ε-dimension is the inverse of dn (W , H): nε (W , H) := inf{n : ∃Ln : dim Ln ≤ n, sup inf kf − g kH ≤ ε}. f ∈W g ∈Ln
nε (W , H) is the necessary dimension of linear subspace for approximation of functions from W with accuracy ε. From the computational view it is more convenient to study nε (W , H) since it is directly related to the computation cost. Dinh D˜ ung (VNU, Hanoi)
Hyperbolic cross
October 2, 2014
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High-dimensional approach Sobolev space of mixed smoothness α ∈ N kf k2H α mix
X
=
|k|∞ ≤α
∂ |k|1 f
2
, ks
∂x1k1 · · · ∂xs
2
|k|∞ := max ki . 1≤i≤s
α is the unit ball in H α . Umix mix
Traditional estimations [Babenko 1960]: α A(α, s) ε−1/α | log ε|s−1 ≤ nε (Umix , L2 ) ≤ A0 (α, s) ε−1/α | log ε|s−1 .
Our goal: to compute A(α, s), A0 (α, s) explicitly in s. The basis for estimation of nε : Reduction to computation of cardinality of the associated HCs: α |Γ(s, 1/ε)| − 1 ≤ nε (Umix , L2 ) ≤ |Γ(s, 1/ε)| Dinh D˜ ung (VNU, Hanoi)
Hyperbolic cross
October 2, 2014
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Plan of our talk
High-dimensional HC approximation for two models of mixed smoothness: Dyadic version [DD&Ullrich 2013]; Korobov version [Chernov&DD 2014].
Infinite-dimensional HC approximation for two models of regularity: Korobov version [DD&Griebel 2014]; Analytic version [DD&Griebel 2014].
Application of infinite-dimensional HC approximation in stochastic or parametric PDEs.
Dinh D˜ ung (VNU, Hanoi)
Hyperbolic cross
October 2, 2014
8 / 42
Dyadic version: decomposition in frequency domain L2 (Ts ) is the space of periodic functions onR the torus Ts := [0, 1]s equipped with the inner product (f , g ) := Ts f (x)g (x) dx. Q Let ek (x) := sj=1 e 2πi kj xj . For m ∈ Zs+ and f ∈ L2 (Ts ), define the operator: P ˆ δm (f ) := m := {k ∈ Zs : b2mi −1 c ≤ |ki | < 2mi }, k∈m f (k)ek , where fˆ(k) is the kth Fourier coefficient. P 2 Based on Parseval’s identity kf k22 = m∈Zs+ kδm (f )k2 , we define the α space Hmix of mixed smoothness α: kf k2H α mix
Dinh D˜ ung (VNU, Hanoi)
:=
X
2α|m|1
2
kδm (f )k22
m∈Zs+
< ∞,
|m|1 :=
s X
mj .
j=0
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October 2, 2014
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Dyadic version: Fourier HC approximation Step HCs are formed from dyadic boxes m : [ s Gstep (s, n) := m : m ∈ Z+ , |m|1 ≤ n . V s (n) – the trigonometric polynomials with frequencies in Gstep (s, n). Linear Fourier operator: X
Sn (f ) :=
fˆ(k) ek .
k∈Gstep (s,n) α be the unit ball in H α . For n ∈ N, Let Umix mix
sup α f ∈Umix
inf
g ∈V s (n)
kf − g k2 ≤
sup kf − Sn (f )k2 ≤ 2−n ;
α f ∈Umix
We have dim V s (n) = |Gstep (s, n)|. Dinh D˜ ung (VNU, Hanoi)
Hyperbolic cross
October 2, 2014
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Dyadic version: nε and the cardinality of HCs Estimation of nε is reduced to estimation of |Gstep (s, n)| for ε = 2−n : α |Gstep (s, n)| − 1 ≤ nε (Umix , L2 (Ts )) ≤ |Gstep (s, n)|,
For any n ∈ Z+ , n n+s −1 n+1 n + s − 1 2 ≤ |Gstep (s, n)| ≤ 2 . s −1 s −1
Theorem (DD&Ullrich 2014) Let α > 0. Then we have for any 0 < ε ≤ 2−αs , α(s − 1) −(s−1) α , L (Ts )) nε (Umix 1 2 [α(s − 1)]−(s−1) ≤ −1/α ≤ 4 . 2 2e ε | log ε|s−1 The ratio decays exponentially fast in s. Dinh D˜ ung (VNU, Hanoi)
Hyperbolic cross
October 2, 2014
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Dyadic version [DD&Ullrich 2013] Estimates in this manner have been proven also α , H γ (Ts )) in energy norm of Sobolev space H γ (Ts ). for nε (Umix In the dyadic version, we can prove lower and upper bounds for α , L (Ts )) only for very small ε ≤ 2−αs . nε (Umix 2 α involve a The reason: The step HC approximation for the class Umix whole dyadic block X δk (f ) := fˆ(m)em m∈k
with the cardinality |k | ≥ 2s . Let us consider another model of mixed smoothness: Korobov-type mixed smoothness. Dinh D˜ ung (VNU, Hanoi)
Hyperbolic cross
October 2, 2014
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A modification of Korobov space Ksr For r > 0 and k ∈ Zs , define the scalar λ(k) by λ(k) :=
s Y
λ(kj ),
λ(kj ) := (1 + |kj |),
j=1
Korobov function: κrs :=
X
λ(k)−r ek .
k∈Zs
Korobov space
Ksr : Ksr := {f : f = κrs ∗ g , g ∈ L2 (Ts )}
with the norm kf kKsr := kg k2 . Dinh D˜ ung (VNU, Hanoi)
Hyperbolic cross
October 2, 2014
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Hyperbolic crosses for Ksr The symmetric continuous HC: G (s, T ) :=
s
k∈Z :
s Y
(|ki | + 1) ≤ T .
i=1
Usr is the unit ball in Ksr . Using Fourier approximation by trigonometric polynomials with frequencies in HC G (s, T ) we have |G (s, T )| − 1 ≤ nε (Usr , L2 (Ts )) ≤ |G (s, T )|, T = ε−1/r .
⇒ Estimation of nε (Usr , L2 (Ts )) is reduced to estimation of |G (s, T )|. Dinh D˜ ung (VNU, Hanoi)
Hyperbolic cross
October 2, 2014
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New estimates for the cardinality of HCs Theorem (Chernov&DD 2014) For T ≥ 1, |G (s, T )|
(3/2)s , |G (s, T )| >
2s T (ln T − s ln(3/2))s (s − 1)! (ln T − s ln(3/2) + s)
For ε > 0, |G (s, T )| − 1 ≤ nε (Usr , L2 (Ts )) ≤ |G (s, T )|, T = ε−1/r .
Dinh D˜ ung (VNU, Hanoi)
Hyperbolic cross
October 2, 2014
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New bounds for nε (Usr , L2 (Ts )) ⇒
Theorem (Chernov&DD 2014) Let r > 0, s ≥ 2. Then we have for every ε ∈ (0, 1], nε (Usr , L2 (Ts )) ≤
2s ε−1/r (ln ε−1/r + s ln 2)s , (s − 1)! ln ε−1/r + s ln 2 + s − 1
and for every ε ∈ (0, [3/2]−sr ), nε (Usr , L2 (Ts )) ≥
2s ε−1/r (ln ε−1/r − s ln(3/2))s − 1 (s − 1)! (ln ε−1/r − s ln(3/2) + s)
In traditional estimations, ε−1/r | log ε|(s−1)/r is a priori split from constants which are a function of dimension parameter s. ⇒ Any high-dimensional estimate based on them leads to a rougher bound. Dinh D˜ ung (VNU, Hanoi)
Hyperbolic cross
October 2, 2014
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Related results
[K¨ uhn, Sickel & Ullrich 2014] have established upper and lower bounds explicit in s for large n and small n (preasymptotics), for the approximation number α an (Is : Hmix → L2 (Ts ))
(given in the talk by Winfried Sickel yesterday).
Dinh D˜ ung (VNU, Hanoi)
Hyperbolic cross
October 2, 2014
17 / 42
Infinite-dimensional HC approximation Infinite tensor product probability measure Let I := [−1, 1]. Let dµ∞ be the tensor product infinite tensor product measure on I∞ of the univariate uniform probability measures on I: O1 dµ∞ (y) := dyj . 2 j∈Z
The sigma algebra Σ for dµ∞ is generated by the finite rectangles Y Ij , Ij ⊂ I, j∈N
where only a finite number of the Ij are different from I. Let dµn = dx be the uniform probability measure on the n-dimensional torus Tn := [0, 1]n . We define O dµ(x, y) := dµn (x) dµ∞ (y). Dinh D˜ ung (VNU, Hanoi)
Hyperbolic cross
October 2, 2014
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Space L2 (Tn ⊗ I∞ , dµ)
L2 (Tn ⊗ I∞ , dµ) is the Hilbert space of functions on Tn ⊗ I∞ : Z (f , g ) := f (x, y)g (x, y) dµ(x, y). I∞
The norm in L2 (Tn ⊗ I∞ , dµ) is defined as kf k := (f , f )1/2 . L2 (Tn ⊗ I∞ , dµ) = L2 (Tn , dµn ) ⊗ L2 (I∞ , dµ∞ ).
Dinh D˜ ung (VNU, Hanoi)
Hyperbolic cross
October 2, 2014
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Bochner space H := H b (Tn ) ⊗ L2 (I∞ , dµ∞ ) For b ≥ 0, H := H b (Tn ) ⊗ L2 (I∞ , dµ∞ ), H b (Tn ) is the Sobolev space of smoothness b. ⇒ H is a Bochner space: H = L2 (I∞ , dµ∞ ; H b (Tn )) – the set of all functions f : I∞ → H b (Tn ) such that Z 2 kf (·, yk2H b (Tn ) dµ∞ (y) < ∞. kf kH := I∞
For b = 0, H = L2 (Tn ⊗ I∞ , dµ) = L2 (Tn , dµn ) ⊗ L2 (I∞ , dµ∞ ). Dinh D˜ ung (VNU, Hanoi)
Hyperbolic cross
October 2, 2014
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Infinite-dimensional Legendre polynomials Let {Lk }∞ k=0 be the family of univariate orthonormal Legendre polynomials in L2 (I, 21 dx), i.e. Z 1 Lk (y )Ls (y )dy = δks . 2 I Z∞ – the set of all sequences k = (kj )∞ j=1 with kj ∈ Z; ∞ : k ≥ 0, j = 1, 2, ..., supp(k) is finite}. Z∞ j +∗ := {k ∈ Z
For s ∈ Z∞ +∗ , we define Ls (y) :=
Y
Lsj (yj ).
j∈supp(s)
(Ls )s∈Z∞ is an orthonormal basic of L2 (I∞ , dµ). +∗ Dinh D˜ ung (VNU, Hanoi)
Hyperbolic cross
October 2, 2014
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Infinite-dimensional mixed polynomials
Let {ek }k∈Zn be the orthonormal trigonometric basis in L2 (Tn , dx). For (k, s) ∈ Zn ⊗ Z∞ +∗ , we define h(k,s) (x, y) := ek Ls . (h(k,s) )(k,s)∈Zn ⊗Z∞ is an orthonormal basis of L2 (Tn ⊗ I∞ , dµ). +∗ For every f ∈ L2 (Tn ⊗ I∞ , dµ), X f = f(k,s) h(k,s) ,
f(k,s) = (f , h(k,s) ).
(k,s)∈Zn ⊗Z∞ +∗
Dinh D˜ ung (VNU, Hanoi)
Hyperbolic cross
October 2, 2014
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Infinite-variate Korobov-type space ∞ with r > 0. Let n ∈ Z+ , a > 0 and r = (rj )∞ j j=1 ∈ I
For a (k, s) ∈ Zn ⊗ Z∞ +∗ , we define λa,n,r (k, s) := max (1 + |kj |)a 1≤j≤n
∞ Y (1 + sj )rj . j=1
The Korobov-type space K a,r (Tn ⊗ I∞ ) is the set of all functions f ∈ L2 (Tn ⊗ I∞ , dµ) such that X f = λa,n,r (k, s)g(k,s) h(k,s) , g ∈ L2 (Tn ⊗ I∞ , dµ). (k,s)∈Zn ⊗Z∞ +∗
The norm of K a,r (Tn ⊗ I∞ ) is defined by kf kK a,r (Tn ⊗I∞ ) := kg k. Dinh D˜ ung (VNU, Hanoi)
Hyperbolic cross
October 2, 2014
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Infinite-variate Korobov-type space For n = 0, K a,r (Tn ⊗ I∞ ) := K r (I∞ ). K a,r (Tn ⊗ I∞ ) = H a (Tn ) ⊗ K r (I∞ ). The subspace K a,r (Tn ⊗ Is ) in K a,r (Tn ⊗ I∞ ) is the set of all functions f ∈ L2 (Tn ⊗ I∞ , dµ) such that X f = λa,n,r (k, s)g(k,s) h(k,s) , (k,s)∈Zn ⊗Z∞ +∗ : supp(k)⊂{1,··· ,s}
with g ∈ L2 (Tn ⊗ I∞ , dµ).
Dinh D˜ ung (VNU, Hanoi)
Hyperbolic cross
October 2, 2014
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Infinite-dimensional HC approximation The infinite-dimensional HC with different weights: G (T ) := (k, s) ∈ Zn ⊗ Z∞ +∗ : λa−b,n,r (k, s) ≤ T (a > b ≥ 0). Let P(T ) be the HC subspace of polynomials g of the form X g= g(k,s) h(k,s) . (k,s)∈G (T )
dim P(T ) = |G (T )|. The HC operator ST : H → P(T ) X ST (f ) :=
f(k,s) h(k,s) .
(k,s)∈G (T )
Dinh D˜ ung (VNU, Hanoi)
Hyperbolic cross
October 2, 2014
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Infinite-dimensional HC approximation H := H b (Tn ) ⊗ L2 (I∞ , dµ∞ ); U a,r is the unit ball in K a,r (Tn ⊗ I∞ ); Usa,r is the unit ball in K a,r (Tn ⊗ Is ). For arbitrary T ≥ 1, sup
inf
f ∈U a,r g ∈P(T )
kf − g kH = sup kf − ST (f )kH ≤ T −1 f ∈U a,r
For ε ∈ (0, 1], |H(1/ε)| − 1 ≤ nε (U a,r , H)) ≤ |H(1/ε)|
Dinh D˜ ung (VNU, Hanoi)
Hyperbolic cross
October 2, 2014
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Cardinality of infinite-dimensional HCs Theorem (DD&Griebel 2014) ∞ with Let n ∈ N, 0 ≤ b < a < r /n and r = (rj )∞ j=1 ∈ I
0 < r = rn+1 = · · · = rn+ν+1 < rn+ν+2 ≤ rn+ν+3 ≤ · · · . Assume that there holds the condition M :=
∞ X j=ν+2
1 nrj /(a − b) − 1
−(nrj /(a−b)−1) 3 < ∞. 2
Then we have for every T ≥ 1, bT n/(a−b) c ≤ |G (T )| ≤ C (a, n, r) T n/(a−b) ,
(1)
where C (a, b, n, r) := e M 3n 1 + Dinh D˜ ung (VNU, Hanoi)
1 rn/(a−b)−1 Hyperbolic cross
ν 3 rn/(a−b)−1 2
.
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Estimates of ε-dimensions
Theorem (DD&Griebel 2014) Let nε := nε (U a,r , H), nε (s) := nε (Usa,r , H)). Under the assumptions and notation of the previous theorem, we have for every ε ∈ (0, 1], bε−n/(a−b) c − 1 ≤ nε (s) ≤ nε ≤ C (a, n, r) ε−n/(a−b) .
Dinh D˜ ung (VNU, Hanoi)
Hyperbolic cross
October 2, 2014
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Estimates of ε-dimensions Remarks The terms C (a, b, n, r) is independent of s when s may be very large. The main component ε−n/(a−b) depends on ε and a, b, n, only. The restriction on r ∞ X j=ν+2
1 n rj /(a − b) − 1
−(n rj /(a−b)−1) 3 < ∞ 2
is moderate. It is satisfied if a, b, n are fixed and the subsequence (rj )∞ j=ν+2 is mildly increasing say as an arithmetic progression. The problem of nε (Usa,r , L2 (Tn ⊗ Rs , dµ)) is strongly polynomially tractable with respect to large s. Dinh D˜ ung (VNU, Hanoi)
Hyperbolic cross
October 2, 2014
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Infinite-variate space Aa,r (Tn ⊗ I∞ ) ∞ with r > 0. For a Let n ∈ Z+ , a > 0 and r = (rj )∞ j j=1 ∈ I n ∞ (k, s) ∈ Z ⊗ Z+∗ , we define a
ρa,n,r (k, s) := max (1 + |kj |) exp(r, s), 1≤j≤n
(r, s) :=
∞ X
rj s j .
j=1
The space Aa,r (Tn ⊗ I∞ ) is the set of all functions f ∈ L2 (Tn ⊗ I∞ , dµ) such that X f = ρa,n,r (k, s)g(k,s) h(k,s) , g ∈ L2 (Tn ⊗ I∞ , dµ). (k,s)∈Zn ⊗Z∞ +∗
The norm of Aa,r (Tn ⊗ I∞ ) is defined by kf kAa,r (Tn ⊗I∞ ) := kg k. Dinh D˜ ung (VNU, Hanoi)
Hyperbolic cross
October 2, 2014
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Infinite-variate space Aa,r (Tn ⊗ I∞ ) = H a (Tn ) ⊗ Ar (I∞ )
For n = 0, K a,r (Tn ⊗ I∞ ) := Ar (I∞ ). Aa,r (Tn ⊗ I∞ ) = H a (Tn ) ⊗ Ar (I∞ ). ⇒ Aa,r (Tn ⊗ I∞ ) is a subspace of H for a > b.
Dinh D˜ ung (VNU, Hanoi)
Hyperbolic cross
October 2, 2014
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HC approximation in Bochner space The infinite-dimensional exp. HC with different weights: E (T ) := (k, s) ∈ Zn ⊗ Z∞ +∗ : ρa−b,n,r (k, s) ≤ T , (a > b). Let E(T ) be the HC subspace of polynomials g of the form X g= g(k,s) h(k,s) . (k,s)∈E (T )
The HC operator: PT (f ) :=
X
f(k,s) h(k,s) .
(k,s)∈E (T )
Dinh D˜ ung (VNU, Hanoi)
Hyperbolic cross
October 2, 2014
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HC approximation in Bochner space H := H b (Tn ) ⊗ L2 (I∞ , dµ∞ ). B a,r is the unit ball in Aa,r (Tn ⊗ I∞ ) = H a (Tn ) ⊗ Ar (I∞ ). Let a > b ≥ 0. For arbitrary T ≥ 1, sup
inf
g ∈E(T )
f ∈B a,r
kf − g kH = sup kf − PT (f )kH ≤ T −1 . f ∈B a,r
For ε ∈ (0, 1], |E (1/ε)| − 1 ≤ nε (B a,r , H) ≤ |E (1/ε)|.
Dinh D˜ ung (VNU, Hanoi)
Hyperbolic cross
October 2, 2014
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Cardinality of infinite-dimensional HCs Theorem (DD&Griebel 2014) ∞ with r > 0. Assume that Let n ∈ N, a > b ≥ 0 and r = (rj )∞ j j=1 ∈ I there holds the condition P∞ nr /(a−b) j − 1)−1 < ∞. j=1 (e
Then we have for every T ≥ 1, bT n/(a−b) c ≤ |E (T )| ≤ D(a, b, n, r) T n/(a−b) ,
(2)
where D(a, b, n, r) := 32n exp
Dinh D˜ ung (VNU, Hanoi)
hP
∞ nrj /(a−b) j=1 (e
Hyperbolic cross
i − 1)−1 .
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ε-dimensions in Bochner space
Theorem (DD&Griebel 2014) Under the assumptions and notation of the previous theorem, we have for every ε ∈ (0, 1], bε−n/(a−b) c − 1 ≤ nε (B a,r , H)) ≤ D(a, b, n, r) ε−n/(a−b) .
Related papers: Liberating the dimension for function approximation [Wasilkowski,Wozniakowski 2011], [Wasilkowski 2012].
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Application in Elliptic sPDEs (periodic model) Let D := (0, 1)n ⊂ Rn . Consider the following parametric (stochastic) periodic elliptic problem − ∇x (a(x, y)∇x u(x, y)) = f (x, y) in D,
u|∂D = 0,
(3)
where the diffusion coefficients a(x, y) are functions 1-periodic ∞ functions in x, and parameters y = (yj )∞ j=1 ∈ Y := I , and f (·, y) is a fixed 1-periodic function in L2 (D). In a typical case, a(x, y) has the following expansion a(x, y) := ¯a(x) +
∞ X
yj ψj (x, y),
(4)
j=1
where ¯a is 1-periodic, ¯a ∈ L∞ (D) and (ψj )∞ j=1 ⊂ L∞ (D) with 1-periodic ψj . A choice for (ψj )∞ unen-Lo`eve basis where ¯a is j=1 in sPDEs is the Karh´ the average of a and yj is pairwise decorrelated random variables. Dinh D˜ ung (VNU, Hanoi)
Hyperbolic cross
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Papers on sPDEs
[Nobile,Tempone,Webster 2008] Sparse grid collcation method. [Cohen,DeVore,Schwab 2010], [Hoang,Schwab 2014] N-term Galerkin approximations. [Beck,Nobile,Tammelli,Tempone 2012, 2013] Polynomial approximation by Galerkin and collocation methods.
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Application in Elliptic sPDEs (periodic model) The solution u is living in the Bochner spase H0 := L2 (I∞ , dµ; V ),
V := H0b (Tn ) (b = 0, 1).
Depending on the properties of the diffusion function a(x, y) and the function f (x, y), we have higher regularity of u in both, x and y. We may assume that (K) u ∈ H a (Tn ) ⊗ K r (I∞ ) = K a,r (Tn ⊗ I∞ ) (mixed smoothness), or (A) u ∈ H a (Tn ) ⊗ Ar (I∞ ) = Aa,r (Tn ⊗ I∞ ) (analyticity) for some a > b (a = 1, 2), r ∈ R∞ + satisfying the assumptions of the above theorems on the infinite-dimensional HCs G (T ) and E (T ). With some natural restrictions (for instance, kψj kL∞ (D) ≤ C j −s , s > 1, in K-L expansion), u has analytic regularity in y [Cohen,DeVore,Schwab 2010] ⇒ the assumption (A) is quite reasonable. Dinh D˜ ung (VNU, Hanoi)
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Linear approximation to the solution u of elliptic sPDEs Let a > b ≥ 0. For arbitrary T ≥ 1, (K)
ku − ST (u)kL2 (I∞ ,dµ;V ) ≤ kukH a (Tn )⊗K r (I∞ ) T −1
(A)
ku − PT (u)kL2 (I∞ ,dµ;V ) ≤ kukH a (Tn )⊗Ar (I∞ ) T −1
and
The cardinality index sets of the associated HCs are bounded by (K)
|G (T )| ≤ C (a, b, n, r)T n/(a−b) ,
and (A) |E (T )| ≤ D(a, b, n, r)T n/(a−b) . Dinh D˜ ung (VNU, Hanoi)
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Linear approximation to the solution u of elliptic sPDEs Case (K) N := |G (T )| ⇒ rank ST = N ⇒ ST as a linear operator of rank N: LN := ST , for which ku − LN (u)kL2 (I∞ ,dµ;V ) ≤ C ∗ kukH a (Tn )⊗K r (I∞ ) N −(a−b)/n where C ∗ := C (a, b, n, r)(a−b)/n . Case (A) N := |E (T )| ⇒ rank PT = N ⇒ PT as a linear operator of rank N: ΛN := PT , for which ku − ΛN (u)kL2 (I∞ ,dµ;V ) ≤ D ∗ kukH a (Tn )⊗Ar (I∞ ) N −(a−b)/n where D ∗ := D(a, b, n, r)(a−b)/n . Dinh D˜ ung (VNU, Hanoi)
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Conclusion
We have shown that, under our assumptions and for linear information, the bounds are completely free of the dimension. In any case, the stochastic part has disappeared from the complexities and only appears in the constants. The analysis for standard information still needs to be done.
Dinh D˜ ung (VNU, Hanoi)
Hyperbolic cross
October 2, 2014
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Thank you for your attention!
Dinh D˜ ung (VNU, Hanoi)
Hyperbolic cross
October 2, 2014
42 / 42
