HERMITE SPECTRAL METHOD WITH HYPERBOLIC CROSS APPROXIMATIONS TO HIGH-DIMENSIONAL PARABOLIC PDES∗

APPROXIMATIONS TO HIGH-DIMENSIONAL PARABOLIC PDES

XUE LUO^† AND STEPHEN S.-T. YAU^‡

Dedicated to Professor Peter Caines on the occasion of his 68th birthday

Abstract. It is well-known that sparse grid algorithm has been widely accepted as an efficient tool to overcome the “curse of dimensionality” in some degree. In this note, we first give the error estimate of hyperbolic cross (HC) approximations with generalized Hermite functions. The exponential convergence in both regular and optimized hyperbolic cross approximations has been shown. Moreover, the error estimate of Hermite spectral method to high- dimensional linear parabolic PDEs with HC approximations has been investigated in the properly weighted Korobov spaces. The numerical result verifies the exponential convergence of this approach.

Key words. hyperbolic cross, Hermite spectral method, high-dimensional parabolic PDEs, convergence rate AMS subject classifications. 65N35, 65N22, 35K10

1. Introduction. Our study is motivated by solving the conditional density function of the states of certain nonlinear filtering. The conditional density function satisfies a linear parabolic PDE, which comes from the robust Duncan-Mortensen-Zakai equation after some exponential transforma- tion, see [18], [30]. We need to solve this equation inR^d, since the states lived in the whole space, where d is the number of the states. Moreover, the real-time solution is expected in the filtering problems, so it is natural to adopt the spectral methods. Among the existing literature, the Hermite and Laguerre spectral methods are the commonly used approaches based on orthogonal polynomials in infinite interval, referring to [7], [29]. Although the Hermite spectral method (HSM) appears to be a natural choice, it is not commonly used as Chebyshev and Fourier spectral method, due to its poor resolution (see [8]) and the lack of fast algorithm for the transformation (see [3]). However, it is shown in [2] that an appropriately chosen scaling factor could greatly improve the resolution. Some further investigations on the scaling factor can be found in [28] and also in Chapter 7, [24]. Moreover, recently a guideline of choosing the suitable scaling factors for Gaussian/super-Gaussian functions is described in [19], as well as the application of HSM to 1-dim forward Kolmogorov equation.

Nevertheless, the number of the states is generally greater than one. Taking the target tracking problem in 3-dim as an example, there are at least six states involved in this system (three for position, three for velocity). That is, we need to solve a linear parabolic PDE inR⁶. Naively, if we implement the spectral method with tensor product formulation and assume the firstN modes need to be computed in each direction, then the total amount of the computation is N⁶. Even if with moderately smallN, it is still not within the reasonable computing capacity. This is the so-called

“curse of dimensionality”. An efficient tool to reduce this effect is the sparse grids approximations from Smolyak’s algorithm [27], which is based on a hierarchy of one-dimensional quadrature. It has a potential to obtain higher rates of convergence than many existing methods, under certain regularity conditions. For example, the convergence rate of Monte Carlo simulations are O(N⁻¹²) withN sample points, while the sparse grids from [27] achievesO(N^−r(logN)^(d−1)(r+1)), under the condition that the function has bounded mix derivatives of orderr. The studies of sparse grids start from the basis functions in the physical spaces: piecewise linear multiscale bases [5], wavelets [5], [22].

In the recent one decade, the hyperbolic cross (HC) approximation in the frequency space has also been investigated with various basis functions: Fourier series [10], [12], polynomial approximations generated from the Chebyshev-Gauss-Lobatto points [1], Jacobi polynomials [25].

Although the regular hyperbolic cross (RHC) approximation (2.23) reduces the effect of the

“curse of dimensionality” in some degree, the convergence rate is still deteriorated slowly with the

∗Received by the editors ***; accepted for publication (in revised form) ***; published electronically ***.

†Department of Electronic Engineering, Tsinghua University, Beijing, P. R. China, 100084 and School of Mathe- matics and Systems Science, Beijing University of Aeronautics and Astronautics (Beihang University), Beijing, P. R.

China, 100083 ([email protected], [email protected]).

‡Department of Mathematical Sciences, Tsinghua University, Beijing, P. R. China, 100084 ([email protected]). The work of this author was supported by start-up fund from Tsinghua University.

1

dimension increasing (noting the term (logN)^(d−1)(r+1) in the previous paragraph). To completely break the “curse of dimensionality”, the optimized hyperbolic cross (OHC) approximation (2.38) is introduced in [12]. It has been shown in [17] that the convergence rate of the OHC approximation with γ ∈ (0,1) (see definition in (2.37)) with Fourier series is of O(N^−r) in our notation, where the dimension enters the constant in front. The first purpose of this paper is to establish the error estimate for the HC approximations with the generalized Hermite functions in the weighted Korobov spaces K^m_α,β(R^d), see (2.25). In particular, we obtain the following results for the RHC/OHC approximation with the generalized Hermite functions.

Theorem 1.1. For any u∈ K^m_α,β(R^d),0≤l < m, (and0< γ≤ _m^l,) inf

U_N∈XN(or X_N,γ)||u−UN||_Kl

α,β(R^d)(or W_α,β^l (R^d))≤CN^l−m2 |u|_K^m

α,β(R^d), ∀0≤l < m,

where C is some constant depending on α, l,m and d (or γ),XN (or XN,γ) is defined in (2.23) (or (2.38)), W_α,β^l (R^d) and K^l_α,β(R^d) are the Sobolev-type spaces (2.18) and the weighted Korobov spaces (2.25), respectively.

We follow the error analysis developed in [25] to show Theorem 1.1. But it is necessary to point out that there is a gap in the proof of Theorem 2.3, [25]. We circumvent this by more delicate analysis.

We are also interested in the dimensional adaptive HC approximation. The following error estimate is obtained with respect to the dependence of dimensions.

Theorem 1.2 (TheoremB.1). For any u∈ K^m_α,β(R^d), for 0< l≤m, we have inf

UN∈XN

|u−UN|_Wl

α,β(R^d).|α|^l−m_∞

N₁^l−m+N

1−γ d−d1−γ(l−m) 2

¹₂

|u|_Km α,β(R^d),

whereXN is defined in (B.2),γ is in the definition of OHC (2.37), andN1,d1,N2 are clarified in (B.1).

To avoid the distraction of our main results, we leave the detailed proof of this theorem in Appendix B.

The second purpose of this paper is to study the application of the Galerkin-type HSM with the HC approximation to high-dimensional linear parabolic PDEs. The error estimates in appropriate weighted Korobov spaces are investigated under various conditions (cf. conditions (C₁)-(C₆) in section 3). There also exist rich literatures of the applications of sparse grids algorithm to solve equations. It has already been successfully applied to problems from the integral equations [14], to interpolation and approximation [16], to the stochastic differential equations [23], [20], to high dimensional integration problems from physics and finance [9], and to the solutions to elliptic PDEs, [31], [26]. As to the parabolic PDEs, they are treated with a wavelet-based sparse grid discretization in [21]. Besides the finite element approaches, they are also handled with finite differences on sparse grids [11] and finite volumn schemes [15]. Griebel and Oeltz [13] proposed a space-time sparse grid technique, where the tensor product of one-dimensional multilevel basis in time and a proper multilevel basis in space have been employed. To our best knowledge, it is the first time in this paper that the Galerkin HSM with sparse grids algorithm is applied to parabolic PDEs, and the error estimates are obtained in the appropriate spaces.

Theorem 1.3. Assume that conditions(C1)-(C3)are satisfied, and the solution to the equation (3.1)u∈L^∞(0, T;K^m_α,β(R^d))∩L²(0, T;K^m_α,β(R^d)), form >1. LetuN be the approximate solution obtained by HSM (3.3), then

||u−u_N||(t).c^∗N^1−m2 ,

wherec^∗ depends onα, the norms of L²(0, T;K_α,β^m (R^d))andL^∞(0, T;K^m_α,β(R^d)).

Theorem 1.4. Assume that conditions(C3)-(C6)are satisfied and the solution to the equation (3.1)u∈L²(0, T;K_α,β^m (R^d)), for some integerm >max{|γ|1,|δ|1+ 1} (γ,δ are two parameters in condition(C6)), anduN is the approximate solution obtained by HSM (3.3), then

||u−u_N||(t).c^]N^{max{|γ|1,|δ|2 1 +1}−m},

wherec^] depends onα,T and the norm of L²(0, T;K^m_α,β(R^d)).

The paper is organized as following. The error analysis of the HC approximations with gener- alized Hermite functions is in section 2. Section 3 is devoted to the error estimate of HSM with HC approximation applying to linear parabolic PDE in suitable spaces under certain conditions. Finally, in section 4, the numerical experiment has been included to verify the exponential convergence of the HSM with the HC approximation to PDE. In the appendices, the error analysis of the full grid approximation and the dimensional adaptive HC approximation with generalized Hermite function are illustrated in detail.

2. Hyperbolic cross approximation with generalized Hermite functions.

2.1. Notations. Let us first clarify the notations to be used throughout the paper.

LetR(resp.,N) denote all the real numbers (resp., natural numbers), and letN0=N∪ {0}.

For anyd∈N, we use boldface lowercase letters to denote d-dimensional multi-indices and vectors, e.g.,k= (k₁, k₂, . . . , k_d)∈N^d0 andα= (α₁, α₂, . . . , α_d)∈R^d.

Let 1= (1,1, . . . ,1) ∈N^d, and let e_i = (0, . . . ,1, . . . ,0) be thei^th unit vector in R^d. For any scalars∈R, we define the componentwise operations:

α±k=(α₁±k₁, . . . , α_d±k_d), α±s:=α±s1= (α₁±s, . . . , α_d±s), 1

α = 1

α1

, . . . , 1 αd

, α^k=α^k₁¹· · ·α^k_d^d, and

α≥k⇔αj≥kj, ∀1≤j≤d; α≥s⇔αj≥s, ∀1≤j≤d.

The frequently used norms are denoted as

|k|1=

d

X

j=1

kj; |k|_∞= max

1≤j≤dkj; |k|mix =

d

Y

j=1

¯kj,

where ¯k_j= max{1, kj}.

Given a multivariate functionu(x), we denote, thek^th mixed partial derivative by

∂_x^ku= ∂^|k|1u

∂x^k₁¹· · ·∂x^k_d^d =∂_x^k1

1· · ·∂_x^kd

du.

In particular, we denote∂_x^su=∂_x^s1u=∂(s,s,...,s)

x u.

LetL²(R^d) be the Lebesgue space inR^d, equipped with the norm|| · ||= R

R^d| · |²dx¹₂ and the scalar producth·,·i.

We follow the convention in the asymptotic analysis, a ∼ b means that there exist some constantsC₁, C₂>0 such thatC₁a≤b≤C₂a;a.bmeans that there exists some constant C₃>0 such thata≤C₃b; N1 means thatN is sufficiently large.

We denoteC as some generic positive constant, which may vary from line to line.

2.2. Generalized Hermite functions and its properties. Recall that the univariate phys- ical Hermite polynomials Hn(x) are given by Hn(x) = (−1)ⁿe^x2∂ⁿ_xe^−x2, n ≥0. Two well-known and useful facts of Hermite polynomials are the mutually orthogonality with respect to the weight w(x) =e^−x2 and the three-term recurrence, i.e.,

H0≡1; H1(x) = 2x; and Hn+1(x) = 2xHn(x)−2nH_n−1(x).

(2.1)

It is studied in [28] that the scaling and translating factors are crucial to the resolution of Hermite functions. And the necessity of the translating factor is discussed in [19]. Let us define the generalized Hermite functions as

H^α,β_n (x) = α

2ⁿn!√ π

¹₂

H_n(α(x−β))e^{−12α2(x−β)2}, (2.2)

for n≥ 0, whereα > 0 is the scaling factor, and β ∈R is the translating factor. It is readily to derive the following properties for (2.2):

The{H^α,β_n }n∈N0 forms an orthonormal basis of L²(R), i.e.

Z

R

H^α,β_n (x)H^α,β_m (x)dx=δnm, (2.3)

whereδ_nm is the Kronecker function.

H^α,β_n (x) is then^th eigenfunction of the following Strum-Liouville problem e^{12α2(x−β)2}∂x(e^{−α2(x−β)2}∂x(e^{12α2(x−β)2}u(x))) +λnu(x) = 0, (2.4)

with the corresponding eigenvalueλn= 2α²n.

By convention,H^α,β_n ≡0, forn <0. Forn≥0, the three-term recurrence is inherited from the Hermite polynomials:

2α²(x−β)H^α,β_n (x) =p

λnH^α,β_n−1(x) +p

λn+1H^α,β_n+1(x).

(2.5)

The derivative ofH^α,β_n (x) is explicitly expressed, namely

∂xH_n^α,β(x) = 1 2

pλnH^α,β_n−1(x)−1 2

pλn+1H^α,β_n+1(x).

(2.6)

LetDx=∂_x+α²(x−β). Then D^k_xH^α,β_n (x) =√

µn,kH^α,β_n−k(x), ∀n≥k≥1, (2.7)

where

µn,k =

k−1

Y

j=0

λ_n−j = 2^kα^2kn!

(n−k)!, ∀n≥k≥1.

(2.8)

The orthogonality of{D^k_xH^α,β_n (x)}_n∈N0 holds, i.e., Z

R

D^k_xH^α,β_n (x)D^k_xH^α,β_m (x)dx=µn,kδnm. (2.9)

For notational convenience, we extendµn,k in (2.8) for alln, k∈N0: µn,k =

(1, ifn≥k, k= 0, 0, ifk > n≥0.

(2.10)

Now we define the d-dimensional tensorial generalized Hermite functions as H^α,β_n (x) =

d

Y

j=1

H^α_n^j_j^,βj(xj),

for α>0, β∈ R^d and x∈R^d. It verifies readily that the properties (2.7)-(2.9) can be extended correspondingly to multivariate generalized Hermite functions. LetD^k_x=D^k_x¹

1· · · D_x^kd

d, then D^k_xH^α,β_n =√

µ_n,kH^α,β_n−k; (2.11)

and

Z

R^d

D^k_xH^α,β_n (x)D^k_xH^α,β_m (x)dx=µn,kδnm, (2.12)

forα>0,β∈R^d, where

µn,k=

d

Y

j=1

µnj,kj and δnm=

d

Y

j=1

δnjmj. (2.13)

Here,µ·,·is defined in (2.8) and (2.10), andδnmis the tensorial Kronecker function.

The generalized Hermite functions{H^α,β_n (x)}_n∈Nd

0 form an orthonormal basis ofL²(R^d). That is, for any functionu∈L²(R^d), it can be written in the form

u(x) =X

n≥0

ˆ

u^α,β_n H^α,β_n (x), with uˆ^α,β_n = Z

R^d

u(x)H^α,β_n (x)dx.

(2.14)

Hence, we haveD^k_xu(x) =P

n≥kuˆ^α,β_n D^k_xH^α,β_n (x). Furthermore,

||D^k_xu||²=X

n≥k

µn,k|ˆu^α,β_n |^{2 (2.10)}= X

n∈N^d₀

µn,k|ˆu^α,β_n |². (2.15)

2.3. Multivariate orthogonal projection and approximations. In this section, we aim to arrive at some typical error esitmate of the form

inf

UN∈XN

||u−UN||l.N^−c(l,r)||u||r,

where c(l, r) is some positive constant depending onl andr, || · ||l is the norm of some functional space, l indicates the regularity of the function in some sense, andX_N is an approximation space.

In this paper,X_N is defined as

X_N^α,β= span{H^α,β_n : n∈Ω_N}, (2.16)

whereΩN ⊂N^d0is some index set. With different choices ofΩN, it yields full grid, RHC, OHC, etc..

Let us denote the orthogonal projection operator P_N^α,β : L²(R^d) → X_N^α,β, i.e., for any u ∈ L²(R^d),

h(u−P_N^α,βu), vi= 0, ∀v∈X_N^α,β. Or equivalently,

P_N^α,βu(x) = X

n∈ΩN

ˆ

u^α,β_n H^α,β_n (x).

(2.17)

We shall estimate how close the projected functionP_N^α,βuis tou, with respect to various index setsΩN and norms.

2.3.1. Appoximations on the full grid. The index setΩN corresponding to the d-dimensional full tensor grid is

ΩN ={n∈N^d0: |n|_∞≤N}.

AndX_N^α,β is defined in (2.16). Let us define the Sobolev-type space as

W_α,β^m (R^d) ={u: D^k_xu∈L²(R^d),0≤ |k|₁≤m}, ∀m∈N0, (2.18)

equipped with the norm and seminorm

||u||_W^m

α,β(R^d)=



 X

0≤|k|1≤m

D^k_xu

2





1 2

, (2.19)

|u|_W^m

α,β(R^d)=





d

X

j=1

D^m_xju

2





1 2

. (2.20)

It is clear thatW_α,β⁰ (R^d) =L²(R^d), and

|u|²_Wm α,β(R^d)

(2.15)

=

d

X

j=1

X

n∈N^d0

µ_nj,m ˆu^α,β_n

2. (2.21)

Theorem 2.1. Given u∈ W_α,β^m (R^d), we have for any0≤l≤m,

P_N^α,βu−u

W_α,β^l (R^d).|α|^l−m_∞ N^l−m2 |u|_W^m

α,β(R^d), (2.22)

forN 1. Furthermore,

P_N^α,βu−u _Wl

α,β(R^d).C_α,l,mN^l−m2 |u|_W^m

α,β(R^d),

whereC_α,l,mis some constant depending onα,l andm. Since the proof of this theorem is similar to that in [25], and to avoid the distraction of our main results, we put the proof in Appendix A.

It is clear that the convergence rate deteriorates rapidly with respect to the cardinality of the full grid. That is,

P_N^α,βu−u _Wl

α,β(R^d).C_α,l,mM^l−m2d |u|_Wm α,β(R^d), whereM = card(Ω_N) = (N+ 1)^d.

2.3.2. RHC approximation. As we mentioned in the introduction, the HC approximation is an efficient tool to overcome the “curse of dimensionality” in some degree. The index set of RHC approximation is Ω_N ={n∈N^d0 : |n|mix ≤N}. It is known that the cardinality ofΩ_N is O(N(lnN)^d−1) [12]. Correspondingly, the finite dimensional subspaceX_N^α,β is

X_N^α,β= span{H^α,β_n : |n|mix≤N}.

(2.23)

Let the orthogonal projection operator P_N^α,β : L²(R^d) → X_N^α,β be defined before. Denote the k−complement ofΩN by

Ω^c_N,k:={n∈N^d0 : |n|mix> N andn≥k}, ∀k∈N^d0. (2.24)

We define the Koborov-type space as

K_α,β^r (R^d) ={u: D^k_xu∈L²(R^d), 0≤ |k|_∞≤r}, ∀m∈N^d0, (2.25)

equipped with the norm and seminorm

||u||_K^r

α,β(R^d)=



 X

0≤|k|∞≤r

D^k_xu

2





1 2

, (2.26)

|u|_K^r

α,β(R^d)=



 X

|k|∞=r

D_x^ku

2





1 2

. (2.27)

Remark 2.1. It is easy to see from the definitions that K_α,β⁰ (R^d) = L²(R^d) andW_α,β^dl (R^d)⊂ K_α,β^l (R^d)⊂ W_α,β^l (R^d).

Theorem 2.2. Given u∈ K^m_α,β(R^d), for 0≤l≤m, we have

D_x^l

P_N^α,βu−u

≤Cα,l,m,dN^|l|∞−m2 |u|_Km α,β(R^d),

where Cα,l,m,d is some constant depending on α, l, m andd, for N 1 (more precisely, at least N > m^d). In particular, ifα=1, then

C1,l,m,d= 2^|l|∞−mm(2d−1)m−|l|₁−(d−1)|l|∞.

Proof. From (2.17), (2.15), we have

D^l_x(P_N^α,βu−u)

2

= X

n∈Ω^c_N

µ_n,l ˆu^α,β_n

2= X

n∈Ω^c_N,m

µ_n,l ˆu^α,β_n

2+ X

n∈Ω^c_N,l\Ω^c_N,m

µ_n,l ˆu^α,β_n

2

:=II1+II2. ForII1:

II1≤ max

n∈Ω^c_N,m

µn,l

µn,m

X

n∈Ω^c_N,m

µn,m

ˆu^α,β_n

2.

With the facts that µn,l

µ_n,m =2^|l|1−dm

d

Y

j=1

α^2(l_j ^j−m)

d

Y

j=1

1

(n_j−l_j)· · ·(n_j−m+ 1)

=2^|l|1−dm

d

Y

j=1

α^2(l_j ^j−m)

d

Y

j=1

n^l_j^j−m

d

Y

j=1

1− l_j

nj

−1

· · ·

1−m−1 nj

−1

(2.24)

≤ 2^|l|1−dm

d

Y

j=1

α^2(l_j ^j−m)N^|l|∞−m

d

Y

j=1

1− lj

n_j −1

· · ·

1−m−1 n_j

−1

(2.28) and

max

n∈Ω^c_N,m







d

Y

j=1

1− l_j

nj

−1

· · ·

1−m−1 nj

−1





≤ max

n∈Ω^c_N,m







d

Y

j=1

1−m−1 nj

lj−m





≤

d

Y

j=1

m^m−lj =m^dm−|l|1, (2.29)

we arrive that

II1≤m 2

dm−|l|1 d

Y

j=1

α^2(l_j ^j−m)N^|l|∞−m

D_x^m·1u

2. (2.30)

ForII2: The index setΩ^c_N,l\Ω^c_N,m is

Ω^c_N,l\Ω^c_N,m={n∈N^d0: |n|_mix > N andn≥l, ∃j, such thatn_j < m}.

Let us divide the index 1≤j≤dinto two parts

N :={j: l_j ≤n_j < m,1≤j≤d}, N^c :={j: n_j≥m,1≤j ≤d}.

(2.31)

It is easy to see that neitherN norN^c is empty set. We denote

˜ µ_n,l,m=



 Y

j∈N

µ_nj,lj



 Y

i∈N^c

µ_ni,m

!

:=µ_n,k, (2.32)

wherekis a d-dimensional index consisting oflj forj∈ N andmforj ∈ N^c. Now, we treatII2 as

II2 (2.32)

≤ max

n∈Ω^c_N,l\Ω^c_N,m

µn,l

µn,k

X

n∈Ω^c_N,l\Ω^c_N,m

µn,k

ˆu^α,β_n

2≤ max

n∈Ω^c_N,l\Ω^c_N,m

µn,l

µn,k

|u|²_Km α,β(R^d), (2.33)

since|k|∞=m. It remains to estimate the maximum in (2.33).

µ_n,l µn,k

=2^|l|1−|k|1 Y

j∈N^c

α^2(l_j ^j−m) 1

(nj−lj)· · ·(nj−m+ 1)

=2^|l|1−|k|1 Y

j∈N^c

α^2(l_j ^j−m) Y

j∈N^c

n^l_j^j−m Y

j∈N^c

1− l_j

nj

−1

· · ·

1−m−1 nj

−1

. (2.34)

Observe thatj∈ N^c impliesnj ≥m >l≥0. That is,nj ≥1. Hence, ¯nj =nj, for allj= 1,· · ·, d.

In view of|n|mix> N, we deduce that Y

j∈N^c

¯

nj> N Q

j∈Nn¯_j > N Q

j∈Nm. With the same estimate in (2.29) and the fact that

2^|l|1−|k|1 = 2^{Pj∈Nc(lj−m)}≤2^|l|∞−m, (2.35)

it yields that

max

n∈Ω^c_N,l\Ω^c_N,m

µn,l

µ_n,k

≤C_α,l,m2^|l|∞−mm(2d−1)m−|l|1−(d−1)|l|∞N^|l|∞−m, (2.36)

whereCα,l,mdenotes some constant depending onα,landm. The desired result follows immediately from (2.30), (2.33) and (2.36).

Corollary 2.3.

P_N^α,βu−u

K^l_α,β(R^d)≤Cα,l,m,dN^l−m2 |u|_K^m

α,β(R^d), ∀0≤l≤m, whereCα,l,m,d is some constant depending on α,l,mandd.

Remark 2.2. Recall that M = card(ΩN) =O(N(lnN)^d−1)≤CN^1+(d−1), for arbitrary small >0. Then

P_N^α,βu−u

K^l_α,β(R^d)≤Cα,l,m,dM

l−m

2(1+(d−1))|u|_K^m

α,β(R^d), ∀0≤l≤m,

whereCα,l,m,d is some constant depending on α,l,m andd. It is clear to see that the convergence rate deteriorates slightly with increasingd.

2.3.3. OHC approximation. In order to completely break the curse of dimensionality, we consider the index set introduced in [12]

Ω_N,γ:={n∈N^d0: |n|_mix|n|^−γ_∞ ≤N^1−γ}, −∞ ≤γ <1.

(2.37)

The cardinality ofΩN,γ isO(N), forγ∈(0,1), where the dependence of dimension is in the big-O, see [12]. The family of spaces are defined as

X_N,γ^α,β:= span{H^α,β_n : n∈ΩN,γ}.

(2.38)

Remark 2.3. In particular, we have X_N,0^α,β=X_N^α,βin RHC (2.23), andX_N,−∞^α,β = span{H^α,β_n :

|n|_∞ ≤N}, i.e. the full grid. We denote the projection operator asP_N,γ^α,β : L²(R^d)→X_N,γ^α,β. In this case, thek−complement of index set ofΩN,γ is

Ω^c_N,γ,k={n∈N^d0: n∈Ω^c_N,γ andn≥k}, ∀k∈N^d0. (2.39)

Although [25] obtains the similar result for Jacobi polynomials as Theorem 2.4 below, we believe that there is a gap in their error analysis of OHC, namely Theorem 2.3, [25]. We circumvent it with more delicate analysis.

Theorem 2.4. For any u∈ K^m_α,β(R^d),d≥2, and0≤ |l|1< m,

D^l_x

P_N,γ^α,βu−u

≤Cα,l,m,d,γ|u|_K^m

α,β(R^d)







N^|l|12−m, if 0< γ≤ |l|1

m N^{(1−γ)[|l|d−1−γ1−(d−1)m]}, if |l|1

m ≤γ <1, (2.40)

whereCα,l,m,d,γ is some constant depending on α,l,m,d andγ. In particular, if α=1, then

C_1,l,m,d,γ =m^dm−|l|1







2^|l|∞−mm(d−1)(γm−|l|1 )

1−γ , if 0< γ≤ |l|₁ m 2^|l|1−dm, if |l|1

m ≤γ <1.

Proof. As argued in the proof of Theorem 2.2, we arrive

D^l_x

P_N,γ^α,βu−u

2

≤ max

n∈Ω^c_N,γ,m

µn,l

µn,m

X

n∈Ω^c_N,γ,m

µ_n,m ˆu^α,β_n

2

+ max

n∈Ω^c_N,γ,l\Ω^c_N,γ,m

µ_n,l µ˜n,l,m

X

n∈Ω^c_N,γ,l\Ω^c_N,γ,m

˜ µ_n,l,m

ˆu^α,β_n

2

:=III1+III2, (2.41)

where ˜µ_n,l,m is defined as in (2.32). To estimateIII₁, like in (2.28), we have µn,l

µ_n,m =2^|l|1−dm

d

Y

j=1

α^2(l_j ^j−m)

d

Y

j=1

1− lj

n_j −1

· · ·

1−m−1 n_j

−1 d

Y

j=1

n^l_j^j−m

:=D1 d

Y

j=1

n^l_j^j−m. (2.42)

The estimate of max_n∈Ω^c

N,γ,mD1is followed by the similar argument in (2.29), i.e., max

n∈Ω^c_N,γ,mD₁≤m 2

dm−|l|1 d

Y

j=1

α^2(l_j ^j−m). (2.43)

Notice that for anyn∈Ω^c_N,γ,

|n|mix|n|^−γ_∞ > N^1−γ ⇒

|n|^γ_∞

|n|mix

1−γ¹

< 1 (2.44) N

and furthermore, ifn∈Ω^c_N,γ,m,

|n|∞

|n|mix

≤ 1

m^d−1. (2.45)

Moreover,

|n|^d−γ_∞ ≥ |n|_mix|n|^−γ_∞ > N^1−γ ⇒ |n|_∞> N^1−γd−γ. (2.46)

Let us estimate the product on the right-hand side of (2.42):

d

Y

j=1

n^l_j^j−m=





d

Y

j=1

n^l_j^j









d

Y

j=1

nj





−m

≤





d

Y

j=1

|n|^l_∞^j



|n|^−m_mix =|n|^|l|_∞¹|n|^−m_mix. (2.47)

If 0< γ≤ ^|l|_m¹, then

n∈Ωmax^c_N,γ,m

d

Y

j=1

n^l_j^j−m

(2.47)

≤ max

n∈Ω^c_N,γ,m





 |n|^γ_∞

|n|mix

^m−|l|_1−γ¹

|n|∞

|n|mix

^|l|1_1−γ^−γm







(2.44),(2.45)

< m(d−1)(γm−|l|1 )

1−γ N^|l|1−m. (2.48)

Otherwise, if ^|l|_m¹ ≤γ <1, then

max

n∈Ω^c_N,γ,m d

Y

j=1

n^l_j^j−m

(2.47)

≤ max

n∈Ω^c_N,γ,m

|n|^γ_∞

|n|mix

^m

|n|^|l|_∞^1−γm

(2.44),(2.46)

≤ N^{1−γd−γ(|l|1}−γm)−(1−γ)m. (2.49)

Combine (2.43), (2.48) and (2.49), the first term on the right-hand side of (2.41) has the upper bound

III₁≤m 2

^dm−|l|1 d

Y

j=1

α^2(l_j ^j−m)

D^m·1_x u

2







m(d−1)(γm−|l|1 )

1−γ N^|l|1−m, if 0< γ≤ |l|1

m N^{1−γd−γ(|l|1}−γm)−(1−γ)m, if |l|1

m ≤γ <1.

(2.50)

Next, we considerIII2. DefineN andN^c as in (2.31). Like in (2.33), we obtain that III₂≤ max

n∈Ω^c_N,γ,l\Ω^c_N,γ,m

µ_n,l µn,k

|u|²_Km α,β(R^d). (2.51)

We need to estimate the maximum similarly as in (2.34):

µn,l

µ_n,k = 2^|l|1−|k|1 Y

j∈N^c

α^2(l_j ^j−m) Y

j∈N^c

n^l_j^j−m Y

j∈N^c

1− lj

n_j ⁻¹

· · ·

1−m−1 n_j

⁻¹ :=D2

Y

j∈N^c

n^l_j^j−m. (2.52)

Similar argument as in (2.29) yields that max

n∈Ω^c_N,γ,l\Ω^c_N,γ,mD2≤2^|l|1−|k|1 Y

j∈N^c

α^2(l_j ^j−m)m^dm−|^l˜|₁, (2.53)

where

˜l= (l₁,· · · , l_d) =

(lj, ifj∈ N^c 0, otherwise.

(2.54)

And it is verified that Y

j∈N^c

n^l_j^j−m≤



 Y

j∈N^c

|n|˜ ^l_∞^j







 Y

j∈N^c

n_j





−m

=|n|˜ |^˜l|₁

∞ |n|˜ ^−m_mix ≤ |˜n|^|l|_∞¹|˜n|^−m_mix, (2.55)

where ˜nis defined similarly as ˜lin (2.54). With similar argument as in (2.44), we deduce that for anyn∈Ω^c_N,γ,

N^1−γ <|n|mix|n|^−γ_∞ ≤m^d−1|n|˜ mix|n|˜ ^−γ_∞ ⇒

|n|˜ ^γ_∞

|n|˜ mix

_1−γ¹

< m^1−γd−1N⁻¹. (2.56)

And similarly as in (2.45), we have for anyn∈Ω^c_N,γ,m,

|˜n|∞

|n|˜ mix

≤ 1

m^d−2, (2.57)

and

N^{1−γ (2.57)}< m^d−1|n|˜ _mix|n|˜ ^−γ_∞ ≤m^d−1|˜n|^d−1−γ_∞ ⇒ |n|˜ _∞>

N^1−γ m^d−1

d−1−γ¹

. (2.58)

If 0< γ≤ ^|l|_m¹, then

n∈Ω^c_N,γ,lmax\Ω^c_N,γ,m

Y

j∈N^c

n^l_j^j−m

(2.55)

< max

n∈Ω^c_N,γ,l\Ω^c_N,γ,m





 |˜n|^γ_∞

|n|˜ mix

^m−|l|_1−γ¹

|n|˜ ∞

|n|˜ mix

^|l|_1−γ^1−γm







(2.56),(2.57)

≤ m^1−γ1 {[(γ+1)d−(2γ+1)]m−(2d−3)|l|1}N^|l|1−m. (2.59)

Otherwise, if ^|l|_m¹ ≤γ <1, then max

n∈Ω^c_N,γ,l\Ω^c_N,γ,m

Y

j∈N^c

n^l_j^j−m(2.55)< max

n∈Ω^c_N,γ,l\Ω^c_N,γ,m

|n|˜ ^γ_∞

|n|˜ _mix ^m

|n|˜ ^|l|_∞^1−γm

(2.56),(2.58)

≤ m^(d−1)

h

m−^|l|_d−1−γ^1−γmi

N^{(1−γ)[|l|d−1−γ1−(d−1)m]}. (2.60)

Combine (2.34), (2.51), (2.53), (2.59) and (2.60), we arrive III2≤2^|l|∞−m Y

j∈N^c

α^2(l_j ^j−m)m^dm−|l|1|u|²_Km α,β(R^d)

(2.61)







m^1−γ1 {[(γ+1)d−(2γ+1)]m−(2d−3)|l|1}N^|l|1−m, if 0< γ≤|l|₁ m m^(d−1)

hm−^|l|_d−1−γ^1−γmi

N

(1−γ)[|l|1−(d−1)m]

d−1−γ , if |l|1

m ≤γ <1.

Therefore, the desired result follows immediately from (2.50) and (2.61).

Corollary 2.5. For any u∈ K^m_α,β(R^d),0≤l < m, and0< γ≤_m^l,

P_N,γ^α,βu−u _Wl

α,β(R^d)

≤Cα,l,m,d,γN^l−m2 |u|_Km α,β(R^d). whereCα,l,m,d,γ is some constant depending on α,l,m,d andγ.

Remark 2.4. Due to the fact thatM = card(Ω_N,γ) =O(N)≤CN, we obtain that

P_N,γ^α,βu−u

W^l_α,β(R^d)≤Cα,l,m,d,γM^l−m2 |u|_K^m

α,β(R^d).

whereCα,l,m,d,γ is some constant depending onα,l,m,dandγ. It is clear to see that the conver- gence rate does not deteriorate with respect todanymore. The effect of the dimension goes into the constant in front.

3. Application to linear parabolic PDE. In this section, we shall study the Galerkin HSM with the HC approximation applying to high dimensional linear parabolic PDE. Let us consider the linear parabolic PDE of the general form:

(∂tu(x) +Lu(x) =f(x, t), x∈R^d, t∈[0, T] u(x,0) =u₀(x),

(3.1) where

Lu=−∇ ·(A∇u) +b· ∇u+cu, (3.2)

withA= (aij)^d_i,j=1 : R^d7→R^d×d,b= (bi)^d_i=1: R^d 7→R^d andc: R^d 7→R. The aim of HSM is to finduN ∈X, such that

h∂_tu_N, ϕi − A(u_N, ϕ) =hf, ϕi, ∀ϕ∈X, (3.3)

whereX is some approximate space, andA(u, v) is a bilinear form given by A(u, v) =

Z

R^d

(∇u)^TA∇v+vb· ∇u+cuv dx.

(3.4)

In our content,X could be chosen asX_N^α,β, X_N,γ^α,βin the previous section.

To guarantee the existence and regularity of the solution to (3.1), we assume that (C₁) The bilinear form is continuous, i.e., there is a constantC >0 such that

|A(u, v)| ≤C||u||_H1

0(R^d)||v||_H1

0(R^d), ∀u, v∈H₀¹(R^d).

(3.5)

(C₂) The bilinear form is coercive, i.e., there exists somec >0 such that A(u, u)≥c||u||²_H1

0(R^d), ∀u∈H₀¹(R^d).

(3.6)

(C₃) The coefficientsa_ij,b_i andc are smooth.

Here,H₀¹(R^d) denotes the normal Sobolev space with the functions decaying to zero at infinity. More generally,H₀^m(R^d) is defined as, for anyu∈H^m(R^d), it satisfies|u| →0, as|x| → ∞ and

||u||²_Hm(R^d)= X

0≤|k|1≤m

∂_x^ku

2<∞.

(3.7)

Let us first show some relation between the Sobolev-type spaceW_α,β^l (R^d) (see (2.18)) and the normal Sobolev spaceH^l(R^d).

Lemma 3.1. Foru∈ W_α,β^|k|1+|r|1(R^d), for anyr,k∈N^d0, we have

x^r∂_x^ku .

d

Y

i=1

α^−r_i ⁱ

!

|k+r|_mix¹² · ||u||_W|k|1 +|r|1 α,β (R^d).

Proof. For clarity, we show it holds ford= 1 in detail.

x^r∂_x^ku

2=

∞

X

n=0

ˆ

u^α,β_n x^r∂_x^kH^α,β_n (x)

2

(2.5),(2.6)

= α^−2r

∞

X

n=0

ˆ u^α,β_n

k+r

X

i=−(k+r)

ηn,iH^α,β_n+i(x)

2

, (3.8)

where, for eachn, ηn,i is a product ofk+rfactors of

±

√λ_n+i 2

or ^β₂ with−(k+r)≤i≤k+r.

Notice that

λ_n+i∼λ_n+j, (3.9)