# Finite element and wavelet approximation of a parabolic stochastic

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```					 Finite element and wavelet approximation of a
parabolic stochastic partial differential equation
Department of Mathematical Sciences
Chalmers University of Technology
¨
and Goteborg University
¨
Goteborg, Sweden

EFEF, Marseille, 2007 – p. 1/2
Outline

Parabolic stochastic partial differential equation with additive noise
 du − ∆u dt = dW,       x ∈ D, t > 0


u = 0,                 x ∈ ∂D, t > 0


u(0) = u0

Abstract framework
Mild solution
Finite element approximation
Error estimates
Approximation of the noise
Survey of the literature
Wavelet multiresolution

EFEF, Marseille, 2007 – p. 2/2
Co-workers

Yubin Yan
Mihaly Kovacs
Li Bin
Asta Hall
Fredrik Lindgren

EFEF, Marseille, 2007 – p. 3/2
Abstract framework

du + Au dt = dW,            t>0
u(0) = u0

H = L2 (D), · , (·, ·), D ⊂ Rd , bounded domain
1
A = −∆, D(A) = H 2 (D) ∩ H0 (D)
u(t), H-valued random process on probability space (Ω, F, P)
W (t), H-valued Wiener process
E(t) = e−tA , analytic semigroup generated by −A

Mild solution (stochastic convolution):
t
u(t) = E(t)u0 +           E(t − s) dW (s),   t≥0
0

EFEF, Marseille, 2007 – p. 4/2
Wiener process

∞                                               ∞
1/2                                              1/2
W (t) =            γl βl (t)el ,             W (t, x, ω) =          γl     βl (t, ω)el (x)
l=1                                                l=1

Qel = γl el ,        γl > 0,          {el } ON basis
covariance operator Q : H → H, self-adjoint, positive deﬁnite,
bounded, linear operator, cov(W (t)) = tQ
βl (t), independent identically distributed, real-valued, Brownian
motions

Two important cases:
Tr(Q) < ∞: W (t) is an H-valued Wiener process
∞                         2       ∞                       ∞
1/2
E         γl       βl (t)el       =         γl Eβl (t)2 = t         γl = t Tr(Q) < ∞
l=1                               l=1                     l=1

Q = I: W (t) is not H-valued, since Tr(I) = ∞, “white noise”

EFEF, Marseille, 2007 – p. 5/2
Stochastic integral

t
u(t) = E(t)u0 +                     E(t − s) dW (s),          t≥0
0

t
We can deﬁne the stochastic integral                                  B(s) dW (s)
0
if B(s)Q1/2 is a Hilbert-Schmidt operator on H
Isometry property
t                        2            t
E               B(s) dW (s)              =E           B(s)Q1/2        2
HS   ds
0                                     0

Hilbert-Schmidt operator B, B                       HS      <∞
2                ∞               2
B   HS   =           l=1   Bϕl           , {ϕl } arbitrary orthonormal basis in H

Da Prato and Zabczyk, Stochastic Equations in Inﬁnite Dimensions

EFEF, Marseille, 2007 – p. 6/2
Regularity

|v|β = Aβ/2 v ,           ˙
H β = D(Aβ/2 ),               β∈R
2
v         ˙
L2 (Ω,H β )
= E(|v|2 ) =
β                     |Aβ/2 v|2 dx dP(ω),       β∈R
Ω    D

Theorem 1. If         A(β−1)/2 Q1/2         HS   < ∞ for some β ≥ 0, then
u(t)         ˙
L2 (Ω,H β )   ≤C        u0          ˙
L2 (Ω,H β )   + A(β−1)/2 Q1/2       HS

Remark:                           ˙                   ˙
W (t) ∈ L2 (Ω, H −(1−β) ) ⊂ L2 (Ω, H −1 )

Two cases:

If Q1/2        2
HS   = Tr(Q) < ∞, then β = 1
∂2
If Q = I, d = 1, A =             − ∂x2 ,   then A(β−1)/2       HS   < ∞ for β < 1/2
−(1−β)
A(β−1)/2       2
HS   =        j λj           ≈    j   j −(1−β)2/d < ∞ iff d = 1, β < 1/2

EFEF, Marseille, 2007 – p. 7/2
The ﬁnite element method

triangulations {Th }0<h<1 , mesh size h
ﬁnite element spaces {Sh }0<h<1
1       ˙
Sh ⊂ H0 (D) = H 1
Sh continuous piecewise linear functions
Ah : Sh → Sh , discrete Laplacian, (Ah ψ, χ) = (∇ψ, ∇χ), ∀χ ∈ Sh
Ph : L2 → Sh , orthogonal projection, (Ph f, χ) = (f, χ), ∀χ ∈ Sh

uh (t) ∈ Sh ,   uh (0) = Ph u0
duh + Ah uh dt = Ph dW

More rigorously, with Eh (t) = e−tAh ,

 uh (t) ∈ Sh , uh (0) = Ph u0

t
 uh (t) = Eh (t)Ph u0 +
                                  Eh (t − s)Ph dW (s)
0

EFEF, Marseille, 2007 – p. 8/2
Error estimates for the deterministic problem

ut + Au = 0,            t>0                uh,t + Ah uh = 0,     t>0
u(0) = v                                   uh (0) = Ph v
u(t) = E(t)v                               uh (t) = Eh (t)Ph v

Denote
Fh (t)v = Eh (t)Ph v − E(t)v,                       |v|β = Aβ/2 v

We have, for 0 ≤ β ≤ 2,

Fh (t)v ≤ Chβ |v|β , t ≥ 0
t                      1/2
2
Fh (s)v       ds         ≤ Chβ |v|β−1 , t ≥ 0
0

V. Thomée, Galerkin Finite Element Methods for Parabolic Problems

EFEF, Marseille, 2007 – p. 9/2
Strong convergence in L2 norm

Theorem 2. If    A(β−1)/2 Q1/2      HS   < ∞ for some β ∈ [0, 2], then
uh (t) − u(t)   L2 (Ω,H)   ≤ Chβ    u0          ˙
L2 (Ω,H β )   + A(β−1)/2 Q1/2   HS

2
Recall:     uh (t) − u(t)    L2 (Ω,H)    = E( uh (t) − u(t) 2 )

Two cases:

If Q1/2     2
HS   = Tr(Q) < ∞, then the convergence rate is O(h).
∂2
If Q = I, d = 1, A =      − ∂x2 ,   then the rate is almost O(h1/2 ).

No result for Q = I, d ≥ 2.

EFEF, Marseille, 2007 – p. 10/2
Strong convergence: proof

t
u(t) = E(t)u0 +              E(t − s) dW (s)
0
t
uh (t) = Eh (t)Ph u0 +                 Eh (t − s) Ph dW (s)
0

Fh (t) = Eh (t)Ph − E(t)
t
uh (t) − u(t) = Fh (t)u0 +                      Fh (t − s) dW (s) = e1 (t) + e2 (t)
0

Fh (t)u0 ≤ Chβ |u0 |β                 (deterministic error estimate)

=⇒      e1 (t)   L2 (Ω,H)     ≤ Chβ u0                    ˙
L2 (Ω,H β )

EFEF, Marseille, 2007 – p. 11/2
Strong convergence: proof

               t                                             t
                                       2
E
                   B(s) dW (s)             =E                    B(s)Q1/2   2
ds (isometry)
                                                                            HS
0                                             0
          t                           1/2

                             2
                   Fh (s)v       ds                ≤ Chβ |v|β−1 , with v = Q1/2 ϕl (deterministic)
0
=⇒
t                            2                t
2
e2 (t)      L2 (Ω,H)      =E                    Fh (t − s) dW (s)            =                Fh (t − s)Q1/2    2
HS     ds
0                                             0
∞            t                                                   ∞
=                         Fh (t − s)Q1/2 ϕl        2
ds ≤ C              h2β |Q1/2 ϕl |2
β−1
l=1       0                                                      l=1
∞
= Ch2β                      A(β−1)/2 Q1/2 ϕl           2
= Ch2β A(β−1)/2 Q1/2                2
HS
l=1

If Tr(Q) < ∞, we may choose β = 1, otherwise β < 1.

EFEF, Marseille, 2007 – p. 12/2
Time discretization

du + Au dt = dW ,                 t>0
u(0) = u0

The implicit Euler method: k = ∆t, tn = nk, ∆W n = W (tn ) − W (tn−1 )

U n ∈ Sh ,   U 0 = Ph u0
U n − U n−1 + kAh U n = Ph ∆W n ,

U n = Ekh U n−1 + Ekh Ph ∆W n ,             Ekh = (I + kAh )−1

n
n−j+1
n
U n = Ekh Ph u0 +             Ekh    Ph ∆W j
j=1

tn
u(tn ) = E(tn )u0 +                E(tn − s) dW (s)
0

EFEF, Marseille, 2007 – p. 13/2
Error estimates for deterministic parabolic problem

n
Denote Fn = Ekh Ph − E(tn )
We have the following estimates for 0 ≤ β ≤ 2:

Fn v ≤ C(k β/2 + hβ )|v|β
n                1/2
2
k         Fj v             ≤ C(k β/2 + hβ )|v|β−1
j=1

EFEF, Marseille, 2007 – p. 14/2
Strong convergence

Theorem 3. If A(β−1)/2 Q1/2         HS   < ∞ for some β ∈ [0, 2], then, with
en = U n − u(tn ),

en   L2 (Ω,H)   ≤ C(k β/2 + hβ )    u0         ˙
L2 (Ω,H β )   + A(β−1)/2 Q1/2   HS

J. Printems (2001) (only time-discretization)
Y. Yan, BIT (2004), SIAM J. Numer. Anal (2005)

EFEF, Marseille, 2007 – p. 15/2
Approximating the noise

No result for Q = I, d ≥ 2.
Increments ∆W j are not directly computable: inﬁnite series with
unknown eigenfunctions el .

EFEF, Marseille, 2007 – p. 16/2
Approximating the noise: white noise in 1-D

Q = I,   d = 1,        D = (0, 1)
∂u        ∂2W
SPDE:        + Au =
∂t        ∂x∂t
Piecewise constant approximation:
tj+1        xi+1                              tj+1    xi+1
∂2W      1                                ∂2W           1
:=                                        dx dt =                              dW
∂x∂t    ∆x∆t       tj         xi          ∂x∂t         ∆x∆t        tj       xi

tj+1        xi+1
1
ηij := √                              dW ∈ N (0, 1)
∆x∆t      tj          xi

N   M
∂2W           ∂2W            1                             √
(x, t) ≈      (x, t) =                             ηij ∆x∆t χ[xi ,xi+1 ] (x) χ[tj ,tj+1 ] (t)
∂x∂t          ∂x∂t          ∆x∆t              i=0 j=0

∂u
ˆ        ∂2W
PDE:           u
+ Aˆ =
∂t        ∂x∂t

EFEF, Marseille, 2007 – p. 17/2
Approximating the noise: white noise in 1-D

∂u
ˆ        ∂2W
PDE:           u
+ Aˆ =
∂t        ∂x∂t
ˆ
Finite element or ﬁnite difference approximation U :
1/p
ˆ
E|Uij − u(xi , tj )|p            ≤ C ∆t1/4 + ∆x1/2

Gyöngy (1999)
Allen, Novosel, and Zhang (1998)
Davie and Gaines (2000) (also lower bounds)
Walsh (2005)
Kossioris and Zouraris

Proof technique:
1                               t       1
∂2W
u(x, t) =           G(x, y, t)u0 (y) dy +                   G(x, y, t − s)      (y, s) dy ds
0                               0       0                      ∂x∂t

EFEF, Marseille, 2007 – p. 18/2
Approximating the noise: colored noise in rectangle

A(β−1)/2 Q1/2   HS   < ∞,       D = (0, 1)d
Explicit eigenfunctions:
Ael = λl el
Qel = γl el
know: λl ≈ l2/d ,     assume: γl ≈ l−α
∞                 ∞
A(β−1)/2 Q1/2   2
HS   =         λβ−1 γl ≈
l                l(β−1)d/2−α < ∞
l=1               l=1

if α > 1 − (β − 1)d/2

EFEF, Marseille, 2007 – p. 19/2
Approximating the noise: colored noise in rectangle

A(β−1)/2 Q1/2      HS   < ∞,     D = (0, 1)d ,       explicit eigenfunctions

1. Spectral Galerkin approximation in x, difference method in t
N                                     N
1/2
W N (t) =         γj βj (t)ej ,   uN (x, t) =           ˆ
uj (t)ej (x)
j=1                                   j=1

Shardlow (1999)
Lord and Rougemont (2003)
Müller-Gronbach and Ritter (2004), (2005) (lower bounds)

EFEF, Marseille, 2007 – p. 20/2
Approximating the noise: colored noise in rectangle

A(β−1)/2 Q1/2        HS   < ∞,        D = (0, 1)d ,   explicit eigenfunctions

2. Finite element approximation in x, difference method in t
J
1/2
W J (t) =           γj βj (t)ej
j=1

duJ + Ah uJ dt = Ph dW J
h       h
1/2
E   uJ (t)
h       − uh (t)     2
≤ Chβ A(β−1)/2 Q1/2    HS

if J ≥ Nh = dim(Sh )

Du and Zhang (2002) (d = 1)
Yan (2004)

EFEF, Marseille, 2007 – p. 21/2
Approx. noise: colored noise in general domain

Wavelet multiresolution may provide a solution.

Initial attempt:      d = 1,   Haar basis
φ = φ0,0 = χ[0,1] ,      ψ(x) = φ(2x) − φ(2x − 1),        ψj,k (x) = 2j/2 ψ(2j x − k)
2j −1, ∞
{φ0,0 , ψj,k }k=0, j=0     is an ON basis in H = L2 ((0, 1)).

∞ 2j −1
1/2                                1/2
W (t) = γ0,0 β0,0 (t)φ0,0 (x) +             γj,k βj,k (t)ψj,k (x)
j=0 k=0

J 2j −1
1/2                                1/2
W J (t) = γ0,0 β0,0 (t)φ0,0 (x) +             γj,k βj,k (t)ψj,k (x)
j=0 k=0

EFEF, Marseille, 2007 – p. 22/2
Wavelet multiresolution

J 2j −1
1/2                                1/2
W J (t) = γ0,0 β0,0 (t)φ0,0 (x) +             γj,k βj,k (t)ψj,k (x)
j=0 k=0

Framework of Yan is directly applicable. For example:

When γj,k = 1 we have white noise and the same piecewise constant
approximation as Allen, Novosel and Zhang. The rate is O(h1/2 ). (Li Bin)

For general domains D with d > 1 we may have to give up orthogonality.

EFEF, Marseille, 2007 – p. 23/2
Biorthogonal expansion

Frame: a countable set {φj }j∈J such that

2
a f       ≤         |(f, φj )|2 ≤ b f   2

j∈J

Then there is a dual frame {φ∗ }j∈J such that
j

f=          (f, φ∗ )φj
j
j∈J

We make a formal expansion:

1/2
W (t) =           (W (t), φ∗ )φj =
j                γj βj (t)φj
j∈J                      j∈J

EFEF, Marseille, 2007 – p. 24/2
Biorthogonal expansion

Let J ⊂ J be a ﬁnite subset. We investigate conditions under which we
can prove

J
WA (t) − WA (t) L2 (Ω,H) → 0, as J → J
J         J
WAh (t) − WA (t) L2 (Ω,H) → 0, as h → 0

together with error estimates, for the stochastic convolutions

t                                           t
1/2
WA (t) =             E(t − s) dW (s) =          γj               E(t − s)φj dβj (t)
0                            j∈J            0
t                                               t
J                                  J             1/2
WA (t)   =           E(t − s) dW (s) =           γj                  E(t − s)φj dβj (t)
0                             j∈J               0
t                                                     t
J                                                        1/2
WAh (t) =                Eh (t − s)Ph dW J (s) =         γj                    E(t − s)Ph φj dβj (t)
0                                 j∈J                 0

EFEF, Marseille, 2007 – p. 25/2
Error estimates (colored noise)

For continuous piecewise linear ﬁnite elements:

Theorem 4. Let −A be a self-adjoint nonpositive operator with compact inverse
β−1
generating an analytic semigroup on H . If T r(Q)    < ∞ and {φj }j∈J ⊂ D(A                   2      )
(β ≤ 2) is a frame for H , then

E( WA (t) − WAh (t) 2 ) ≤ C
J                                                       ˜ ˜
(A−1 φk , φl )(Qφk , φl )
k∈J \J l∈J \J
β−1            β−1
+ Ch  2β
(A    2    φk , A    2          ˜ ˜
φl )(Qφk , φl )
k,l∈J

To control the above sums more is needed!

EFEF, Marseille, 2007 – p. 26/2
Support, cancellation, smoothness

Suppose J ↔ (j, k), j = 0, 1, . . . and k = 0, 1, ..., K(j, d). Usually
K(j, d) ∼ 2dj .
Locality
˜
diam supp φj,k ∼ diam supp φj,k ∼ 2−j , j ≥ j0
Cancellation property
|(f, φj,k )| ≤ C2−j(s+d/2) f       W s,∞ (supp φj,k ) ,   s ≤ m, j ≥ j0 ,
˜
and
˜
|(f, φj,k )| ≤ C2−j(s+d/2) f                              s ≤ m, j ≥ j0 ,
W s,∞ (supp φj,k ) ,
˜

Smoothness (inverse inequality)
φj,k H s (Ω) ≤ C2sj φj,k L2 (Ω) , s ≤ γ,
and
˜
φj,k   H r (Ω)
˜
≤ C2rj φj,k   L2 (Ω) ,   r ≤ γ.
˜

EFEF, Marseille, 2007 – p. 27/2
Results in 1D (colored noise)

1
Let J ↔ {(j, k) : j = 0, 1, . . . , N } and (Qf )(x) :=           q(x, y)f (y) dy.
0

Haar Basis (no smoothness)
Theorem 5. Let    H := L2 [0, 1] and Au := −(au′ )′ + cu with
a ≥ a0 > 0, c ≥ 0 smooth functions. Let q ∈ W 1,∞ ([0, 1]2 ) and {φj,k } be the
Haar basis. Then, with 2−N := h,
J
E( WA (t) − WAh (t) 2 ) ≤ Ch2 .
Smooth wavelet basis
∈ W 3,∞ ([0, 1]2 ) and assume locality, cancelation for
Theorem 6. Let q
m ≥ 1, m ≥ 2 and smoothness with γ = γ = 1. Then, with 2−N := h,
˜                              ˜
J
E( WA (t) − WAh (t) 2 ) ≤ Ch4 | ln h|2 .

Good news: such wavelet basis exist!

EFEF, Marseille, 2007 – p. 28/2
Future work

Higher dimensions
Implementation
J
Weak convergence: |E[f (WA (t)) − f (WAh (t))]|

EFEF, Marseille, 2007 – p. 29/2

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