Exponential integrators for stochastic Schr\" odinger equations driven ...

EXPONENTIAL INTEGRATORS FOR STOCHASTIC SCHRÖDINGER EQUATIONS DRIVEN BY ITO NOISE

arXiv:1601.06623v1 [math.NA] 25 Jan 2016

RIKARD ANTON∗ AND DAVID COHEN† Abstract. We study an explicit exponential scheme for the time discretisation of stochastic Schrödinger equations driven by additive or multiplicative Ito noise. The numerical scheme is shown to converge with strong order 1 if the noise is additive and with strong order 1/2 for multiplicative noise. In addition, if the noise is additive, we show that the exact solutions of our problems satisfy trace formulas for the expected mass, energy, and momentum (i. e., linear drifts in these quantities). Furthermore, we inspect the behaviour of the numerical solutions with respect to these trace formulas. Several numerical simulations are presented and confirm our theoretical results. Key words. Stochastic partial differential equations; Stochastic Schrödinger equations; Numerical methods; Geometric numerical integration; Stochastic exponential integrators; Strong convergence; Trace formulas AMS subject classifications. 35Q55, 60H15, 65C20, 65C30, 65C50, 65J08

1. Introduction. We consider temporal discretisations of nonlinear stochastic Schrödinger equations driven by Ito noise idu = ∆u dt + F(x, u) dt + G(u) dW

in Rd × (0, ∞), in Rd ,

u(·, 0) = u0

(1.1)

√ where u = u(x,t), and i = −1. The product between G and dW is of Ito type, and further details on F and G and on the dimension d will be specified later. The stochastic process {W (t)}t≥0 is a square integrable complex-valued Q-Wiener process with respect to a normal filtration {Ft }t≥0 on a filtered probability space (Ω, F , P, {Ft }t≥0 ). The regularity of the covariance operator Q will be specified later in the text. The initial value u0 is an F0 -measurable complex-valued function, which will be further specified below. The Schrödinger equation is widely used within physics and takes several different forms depending on the situation. It is used in hydrodynamics, nonlinear optics and plasma physics to only mention a few areas. In certain physical situations it may be appropriate to incorporate some kind of randomness into the equation. One possibility is to add a driving random force to then obtain an equation of the form (1.1). See for example [12] and references therein for further details. Stochastic Schrödinger equations have received much attention from a more theoretical point of view during the last decades. Connected to the present article and without being exhaustive, we mention the works [13, 14, 18] on Ito problems and [12, 14, 16, 30, 18] for the Stratonovich setting. It is seldom possible to solve stochastic partial differential equations exactly, and efficient numerical schemes are therefore needed. For the time integration of the above stochastic Schrödinger equations, we will consider stochastic exponential integrators. These numerical methods are explicit and easy to implement, furthermore they offer good geometric properties. Exponential integrators are widely used and studied nowadays as witnessed by the recent review [23] for the time integration of deterministic problems. Applications of such schemes to the deterministic (nonlinear) Schrödinger equation can be found in, for example, ∗ Department of Mathematics and Mathematical Statistics, Umeå University, SE–901 87 Umeå, Sweden ([email protected]). † Department of Mathematics and Mathematical Statistics, Umeå University, SE–901 87 Umeå, Sweden ([email protected]). Department of Mathematics, University of Innsbruck, A–6020 Innsbruck, Austria ([email protected])

1

2

R. Anton and D. Cohen

[22, 4, 3, 7, 20, 8, 5] and references therein. These numerical methods were recently investigated for stochastic parabolic partial differential equations in, for example, [28, 24, 29] and for the stochastic wave equations in [9, 32, 10, 1]. We now review previous works on temporal discretisations of stochastic Schrödinger equations. In [17] a Crank-Nicolson scheme is studied for the equation with nonlinearity F(u). First order of convergence is obtained in the case of additive noise, and with multiplicative Ito noise the convergence rate is one half. Observe that this numerical scheme is implicit. A stochastic Schrödinger equation with Stratonovich noise is considered in [15], where, again, a Crank-Nicolson scheme is studied for the equation with nonlinearity F(x, u) = λ |u|2σ u, with λ = ±1 and σ > 0. The authors prove convergence to the exact solution and mass preservation of the scheme. Further, in [26] a mass-preserving splitting scheme for equation (1.1) with F(x, u) = V (x)u and G(u) = u is considered. The noise is of Stratonovich type and first order convergence is obtained. In [27], V (x) is replaced by |u|2 and first order convergence is again obtained. Still in the Stratonovich setting, [25] derives multi-symplectic schemes for stochastic Schrödinger equations. We finally mention [19, 2], in which thorough numerical simulations are presented for both additive noise and multiplicative Stratonovich noise. In the present work we show that • the exponential integrator applied to the linear stochastic Schrödinger equation with additive noise converges strongly with order 1 and satisfies exact trace formulas for the mass, the energy, and for the momentum; • the exponential integrator applied to the stochastic Schrödinger equation with a multiplicative potential and additive noise converges with strong order 1, but has a small error in the trace formulas for the mass and energy; • the exponential integrator applied to stochastic Schrödinger equations driven by multiplicative Ito noise strongly converges with order 1/2. We begin the exposition by introducing some notations and useful results that we will use in our proofs. After that we will follow a similar approach as in [17]. That is, we will begin by analysing the numerical method applied to the linear Schrödinger equation with additive noise in Section 3. Then we study stochastic Schrödinger equations with a multiplicative potential in Section 4 and finally we consider the stochastic Schrödinger equation with a multiplicative potential and multiplicative noise in Section 5. For each of the above problems, we analyse the speed of convergence of the exponential methods (in the strong sense) and for additive problems we show some trace formulas (such results could be interpreted as weak error estimates). Various numerical experiments accompany the presentation and illustrate the main properties of these exponential methods when applied to stochastic Schrödinger equations driven by Ito noise. 2. Notations and some useful results. Given two separable Hilbert spaces H1 and H2 with norms k·kH1 and k·kH2 respectively, we denote the space of bounded linear operators from H1 to H2 by L (H1 , H2 ). We denote by L2 (H1 , H2 ) the set of Hilbert-Schmidt operators from H1 to H2 with norm !1/2 ∞

kΦkL2 (H1 ,H2 ) :=

∑ kΦek k2H2

,

k=1

where {ek }∞ k=1 is any orthonormal basis of H1 . Furthermore, for Hilbert spaces H1 , H2 , and H3 we have (see the proof of Lemma 2.1 in [12]) that if S ∈ L2 (H1 , H2 ) and T ∈ L (H2 , H3 ), then T S ∈ L2 (H1 , H3 ) and kT SkL2(H1 ,H3 ) ≤ kT kL (H2 ,H3 ) kSkL2(H1 ,H2 ) .

(2.1)

3

Exponential integrators for stochastic Schrödinger equations

Also, we consider L2 (Rd ) the space of square integrable functions on Rd with inner product (u, v) = Re

Z

Rd

uv¯ dx.

(2.2)

For σ ∈ R, we further denote the fractional Sobolev space of order σ by H σ = H σ (Rd ) with norm k·kσ . To make the notations cleaner and more readable we will use the following shorter notations L2 = L2 (Rd ),

L (H1 ) = L (H1 , H1 ),

L2σ = L2 (L2 , H σ ).

We note that the operator −i∆, appearing in the stochastic Schrödinger equation, is the generator of a semigroup of isometries of bounded linear operators S(t) = e−it∆ . We will make use of the following result L EMMA 2.1. (See e.g. [26, Lemma 3.2]) Consider any σ ≥ 0 and S(t) = e−it∆ , t ≥ 0, then, for w ∈ H σ , one has kS(t)wkσ = kwkσ , and, for ∆w ∈ H σ and any t ≥ 0, k∆(S(t)w)kσ = k∆wkσ ,

k(S(t) − I)wkσ ≤ tk∆wkσ .

As a consequence of the above lemma and (2.1), we have, for Φ ∈ L2σ +2 and any t ≥ 0, k(S(t) − I)ΦkL2σ ≤ tkΦkL σ +2 . 2

We will also use the following result. L EMMA 2.2. For any σ ≥ 0 and t ≥ 0, we have kS(t) − IkL (H σ +1 ,H σ ) ≤ Ct 1/2 , for a constant C. The proof of this result is very similar to the proof of Lemma 2.1. Finally, throughout the paper, C (and C1 , C2 etc.) will denote a generic constant which may change from line to line. 3. The linear Schrödinger equation with additive noise. In this section, we study time discretisations of the linear stochastic Schrödinger equation idu − ∆u dt = dW

in Rd × (0, ∞), in Rd ,

u(0) = u0

(3.1)

where u0 is an F0 -measurable random variable and the noise {W (t)}t≥0 is a square integrable complex-valued Q-Wiener process with respect to the filtration. In this section, we have no restrictions on the dimension d. Global existence and uniqueness in H σ , σ ≥ 0, of the solution to equation (3.1) is guaranteed if u0 ∈ H σ a.s. and Q1/2 ∈ L2σ . The proof follows the same line as the proof of Theorem 7.4 in [11]. The mild solution of (3.1) reads u(t) = S(t)u0 − i where we recall the notation S(t) = e−it∆ .

Z t 0

S(t − r) dW (r),

4


We now consider the time integration of the above problem. This is done as follows. Let T > 0 be a fixed time horizon and N > 0 be an integer. We first divide the interval [0, T ] into subintervals 0 = t0 < t1 < . . . < tN−1 < tN = T of equal length k so that tn = nk. An exponential integrator with step size k is now derived by approximating the above stochastic integral, in the mild solution, at the left end point. We thus obtain a numerical approximation un of the exact solution u(tn ) of (3.1): n−1 Z t j+1

un = S(k)un−1 − iS(k)∆W n−1 = S(tn )u0 − i ∑

j=0 t j

S(tn − t j ) dW (r),

(3.2)

where ∆W n−1 = W n − W n−1 = W (tn ) − W (tn−1 ) denotes Wiener increments. We call this explicit numerical method an exponential integrator. 3.1. Error estimates. This subsection presents a result on the error of the exponential integrator (3.2) when applied to the linear stochastic Schrödinger equation (3.1). These error estimates are given in the next theorem. T HEOREM 3.1. Let σ ≥ 0, p ∈ N. Recall that un is the numerical approximation given by the exponential integrator (3.2) of the exact solution u(tn ) to the linear stochastic Schrödinger equation driven by a Q-Wiener process (3.1). Assume that u0 ∈ H σ a.s., and Q1/2 ∈ L2σ +2 . Then there exists a constant C such that i h 2p 1/2 2p k σ +2 . E max kun − u(tn )k2p σ ≤ Ck kQ L2

n=1,...,N

Proof. We have n−1 Z t j+1

un − u(tn ) = i ∑ =i

j=0 t j Z tn 0

(S(tn − r) − S(tn − t j )) dW (r)

(S(tn − r) − S (tn − [r/k]k)) dW (r),

where [r/k] denotes the integer part of r/k. The last equality is indeed correct since if t j ≤ r < t j+1 , then j ≤ r/k < j + 1 so that [r/k] = j and [r/k]k = jk = t j . Using Burkholder’s inequality [11, Lemma 7.2], and Lemma 2.1, we have Z t h i h i E max kun − u(tn)kσ2p ≤ E sup ki (S(t − r) − S (t − [r/k]k)) dW (r)k2p σ n=1,...,N

0

t∈[0,T ]

≤ CE = CE ≤ CE

Z

"

"

T

0

k(S([r/k]k − r) − I)Q1/2 k2L σ dr 2

N−1 Z t j+1

k(S(t j − r) − I)Q1/2 k2L σ 2

N−1 Z t j+1

|r − t j |2 kQ1/2 k2L σ +2 dr

∑

j=0 t j

∑

j=0 t j

≤ Ck2p kQ1/2 k2pσ +2 . L2

2

p

dr

! p#

! p#


5

3.2. A trace formula for the mass. Under appropriate boundary conditions, for example periodic boundary conditions or homogeneous Dirichlet boundary conditions, the mass (also called L2 -norm or density) M(u) :=

Z

|u|2 dx

of the deterministic linear Schrödinger equation i ∂∂ ut − ∆u = 0 is a conserved quantity. In the stochastic case, one immediately gets a trace formula for the mass of the exact solution as stated below. P ROPOSITION 3.2. Assume that the initial data is such that E[M(u0 )] is finite and that the covariance operator Q is trace-class. Then the exact solution of the linear Schrödinger equation with additive noise (3.1) satisfies the trace formula for the expected mass E[M(u(t))] = E[ku(t)k2L2 ] = E[M(u0 )] + tTr(Q)

for all time t.

Proof. Using Lemma 2.1 and the fact that the stochastic integrals are normally distributed with mean 0, we get Z t E[ku(t)k2L2 ] = E kS(t)u0k2L2 + S(t)u0, −i S(t − r) dW (r) 0 Zt Zt + −i S(t − r) dW (r), S(t)u0 + k S(t − r) dW (r)k2L2 0

= E[kS(t)u0k2L2 ] +

Zt 0

0

E[kS(t − r)Q1/2k2L 0 ] dr 2

= E[ku0 k2L2 ] + tTr(Q). R EMARK 3.3. We would like to point out that, as in the case of the linear stochastic wave equation treated in [9], an alternative proof of the above result can be obtained using Ito’s formula (see, for example [11, Theorem 4.17]). This remark is also valid for the other trace formulas given below. Our exponential integrator does indeed satisfy this trace formula for the mass as well, as seen in the next result. P ROPOSITION 3.4. With the same assumptions as in Proposition 3.2, the stochastic exponential integrator (3.2) satisfies the following trace formula for the mass E[M(un )] = E[M(un−1 )] + kTr(Q) = E[M(u0 )] + tn Tr(Q)

for all tn = nk.

Proof. Similarly to the proof of Proposition 3.2, we have E[M(un )] = E[M(un−1 )] + E[k

Z tn

tn−1

S(k) dW (r)k2L2 ]

= E[M(un−1 )] + kTr(Q). A recursion concludes the proof. We would like to examine the behaviour of the Euler-Maruyama scheme, the backward Euler-Maruyama scheme, and the midpoint rule with respect to the trace formula for the mass.

6


Since we will consider periodic domains and pseudospectral discretisations in the numerical experiments presented below, we will examine this case first. A pseudospectral spatial discretisation of (3.1) with a noise given by the representation W (x,t) =

∑ λn

1/2

βn (t)en (x),

n∈Z

where {βn (t)}n∈Z are i.i.d Brownian motions, λn are eigenvalues of Q, and {en (x)}n∈Z = { √12π einx }n∈Z is an orthonormal basis of L2 (0, 2π ), will lead to the following system of decoupled stochastic differential equations for the Fourier coefficients yk of the exact solution 1/2

idyk = −k2 yk dt + λk

dβk .

This motivates us to consider the scalar test problem (with a standard Brownian motion β and real numbers a, b) idy = ay dt + b dβ

(3.3)

and the quantity equivalent to the mass is thus the second moment E[|y|2 ]. We thus have the trace formula for the exact solution E[|y(t)|2 ] = E[|y(0)|2 ] + b2t

for all times t.

The following result states that the above classical numerical methods do not preserve the trace formula for this simple test problem, and in particular, are not suited when applied to pseudospectral discretisations of linear stochastic Schrödinger equations. P ROPOSITION 3.5. Consider a pseudospectral discretisation of the stochastic Schrödinger equation (3.1) with a trace-class noise and periodic boundary conditions yielding to equations of the form (3.3). We have the following results: 1. The Euler-Maruyama scheme yn+1 = yn − ikayn − ib∆W n produces a second moment that grows exponentially with time 1

E[|yn |2 ] ≥ e( 2 ka

2 )t n

E[|y0 |2 ] for tn = nk.

2. The backward Euler-Maruyama scheme yn+1 = yn − ikayn+1 − ib∆W n produces a second moment that grows at a slower rate than the exact solution E[|yn |2 ] ≤ E[|y0 |2 ] +

b2 a2 k

for all n ≥ 0,

hence lim E[|yn |2 ]/tn = 0. tn →∞

3. The midpoint rule

i

yn+1 + yn yn+1 − yn −a = b∆W n k 2

produces a second moment that underestimate the linear drift of the exact solution E[|yn |2 ] = E[|y0 |2 ] +

b2tn 2 2

1 + a 2k

for tn = nk.

7


Proof. The proof of this proposition is an easy adaptation of the results presented in [31]. We start by looking at the behaviour of the Euler-Maruyama scheme and compute, using properties of the Wiener increments, E[|yn+1 |2 ] = (1 + a2k2 )E[|yn |2 ] + b2k ≥ (1 + a2k2 )E[|yn |2 ] ≥ (1 + a2k2 )n+1 E[|y0 |2 ] 1

≥ e( 2 ka

2 )t n+1

E[|y0 |2 ].

For the backward Euler-Maruyama scheme we obtain, using a geometric series, E[|yn+1 |2 ] =

b2 k 1 b2 n 2 2 E[|y | ] + b k ≤ E[|yn |2 ] + ≤ E[|y0 |2 ] + 2 . 2 2 2 2 1+a k 1+a k a k

Finally, for the midpoint rule, we have

b2 k

E[|yn+1 |2 ] = E[|yn |2 ] +

1+

a2 k 2 2

= E[|y0 |2 ] +

b2tn+1 2 2

1 + a 2k

.

For homogeneous Dirichlet boundary conditions a similar result holds. P ROPOSITION 3.6. Consider the stochastic Schrödinger equation (3.1) with a traceclass noise and homogeneous Dirichlet boundary conditions. We have the following results: 1. The Euler-Maruyama scheme un+1 = un − ik∆un − i∆W n produces a numerical trace formula for the mass that grows exponentially with time E[M(un )] ≥ e( 2 kλ1 )tn E[M(u0 )] for tn = nk, 1

where λ1 denotes the smallest eigenvalue of the Laplacian. 2. The backward Euler-Maruyama scheme un+1 = un − ik∆un+1 − i∆W n produces a numerical trace formula for the mass that grows at a slower rate than the exact solution E[M(un )] ≤ E[M(u0 )] +

Tr(Q) , λ12 k

for all n ≥ 0,

where λ1 is the smallest eigenvalue of the Laplacian. Thus lim E[M(un )]/tn = 0. tn →∞

3. The midpoint rule [17]

i

un+1 − un un+1 + un −∆ = ∆W n k 2

underestimate the expected mass: E[M(un )] ≤ E[M(u0 )] +

tn 1 + k 4λ1 2

Tr(Q),

where λ1 is the smallest eigenvalue of the Laplacian. Proof.

for all n ≥ 0,

8


1. We start by looking at the behaviour of the Euler-Maruyama scheme and compute E[M(un )] = E[M(un−1 )] + k2 E[k∆un−1k2 ] + kTr(Q) ≥ E[M(un−1 )] + k2 λ12 E[kun−1 k2 ] + kTr(Q), where λ1 denotes the smallest eigenvalue of the Laplacian. Hence, one obtains E[M(un )] ≥ (1 + k2 λ12 )E[M(un−1 )] ≥ (1 + k2λ12 )n E[M(u0 )] ≥ e( 2 kλ1 )tn E[M(u0 )]. 1

2. For the backward Euler-Maruyama scheme we obtain un+1 + ik∆un+1 = un − i∆W n . Taking the norm and expectation we have E[kun+1 k2 ] + k2 E[k∆un+1k2 ] = E[kun k2 ] + E[k∆W n k2 ]. Using that the eigenvalues are positive and increasing, and using a similar argument as in the proof of Proposition 3.5, we get 1 E[kun k2 ] + kTr(Q) 2 2 1 + k λ1 Tr(Q) , ≤ E[ku0 k2 ] + kλ12

E[kun+1 k2 ] ≤

where λ1 is the smallest eigenvalue of the Laplacian. 3. Let (λ j , e j ) be the eigenpairs of the Laplace operator and let (αl , fl ) be the eigenpairs of the covariance operator Q. Writing u(t) = ∑∞j=1 c j (t)e j we have that the expected mass is given by ∞

E[M(u(t))] =

∑ E[|c j (t)|2 ].

j=1

The midpoint rule thus becomes ikλ j n+1 ∑ 1 + 2 cj ej = j=1 ∞

∞ ∞ ikλ j n ∑ 1 − 2 c j e j − i ∑ ∑ αl ∆βln ( fl , e j )e j . j=1 j=1 l=1 ∞

Thus, for every j ≥ 1 we have

∞ ikλ j n+1 ikλ j n 1+ cj = 1 − c j − i ∑ αl ∆βln ( fl , e j ). 2 2 l=1

Multiplying both sides with the conjugate and taking the expectation, we have ! ! ∞ k2 λ j2 k2 λ j2 n+1 2 E[|c j | ] = 1 + E[|cnj |2 ] + ∑ |αl |2 E[|∆βln |2 ]|( fl , e j )|2 1+ 4 4 l=1 ! 2 2 ∞ k λj E[|cnj |2 ] + k ∑ |αl |2 |( fl , e j )|2 . = 1+ 4 l=1

9


Using that {λ j }∞j=1 is a positive and increasing sequence, we get 2 n 2 E[|cn+1 j | ] = E[|c j | ] +

= E[|c0j |2 ] + ≤ E[|c0j |2 ] +

∞

k 1+

k2 λ

j

4

∑ |αl |2 |( fl , e j )|2

l=1 ∞

tn+1 1+

k2 λ

j

4

∑ |αl |2 |( fl , e j )|2

l=1 ∞

tn+1 1+

k2 λ

1

4

∑ |αl |2|( fl , e j )|2 .

l=1

Now using Parseval’s identity, we finally obtain

E[M(un+1 )] =

∞

∞

0 2 2 ∑ E[|cn+1 j | ] ≤ ∑ E[|c j | ] + j=0 ∞

j=0

= = =

∑ E[|c0j |2 ] +

tn+1 1+

k2 λ 4

tn+1

1

∞

∞

l=1 ∞

j=0

∑ |αl |2 ∑ |( fl , e j )|2 ∑

|αl |2 k fi k2 k 2 λ1 1 + 4 l=1 j=0 ∞ tn+1 ∞ |αl |2 E[|c0j |2 ] + k 2 λ1 1 + 4 l=1 j=0 tn+1 Tr(Q). E[M(u0 )] + 2 1 + k 4λ1

∑

∑

3.3. A trace formula for the energy. Again, under appropriate boundary conditions, for example periodic boundary conditions or homogeneous Dirichlet boundary conditions, it is well known that the energy

H(u(t)) :=

1 2

Z

|∇u|2 dx

of the deterministic linear Schrödinger equation i ∂∂ ut − ∆u = 0 remains constant along the exact solution, see for example [6]. When additive noise is introduced into the problem, we get a linear drift in the expected value of the total energy as stated in the following result. P ROPOSITION 3.7. Assume that the initial data is such that E[H(u0 )] is finite, and that k∇Q1/2 kL 0 is bounded. Then the exact solution of the linear Schrödinger equation with 2 additive noise (3.1) satisfies the trace formula for the expected energy t E H(u(t)) = E H(u0 ) + Tr(∇Q∇) 2

for all time t.

10


Proof. Using Lemma 2.1 and the fact that the Ito integral is normally distributed with mean 0, we have Zt

E[k∇u(t)k2L2 ] = E[k∇(S(t)u0 ) − i ∇S(t − r) dW (r)k2L2 ] 0 Zt 2 = E k∇(S(t)u0)kL2 + k ∇S(t − r) dW (r)k2L2 0 Zt Z t − ∇(S(t)u0 ), i ∇S(t − r)dW (r) − i ∇S(t − r)dW (r), ∇(S(t)u0 ) = E[k∇u0 k2L2 ] +

Zt 0

0

0

E[k∇(S(t − r)Q1/2 )k2L 0 ] dr 2

= E[k∇u0 k2L2 ] + tTr(∇Q∇). We now show that our exponential integrator satisfies the very same energy trace formula as the exact solution to the linear stochastic Schrödinger equation (3.1). P ROPOSITION 3.8. With the same assumptions as in Proposition 3.7, the exponential integrator (3.2) satisfies the following trace formula k E H(un ) = E H(un−1) + Tr(∇Q∇) 2 tn = E H(u0 ) + Tr(∇Q∇) for all tn = nk. 2 Proof. Similarly to the proof of the previous proposition, we have 1 E H(un ) = E[k∇un k2L2 ] 2 Z tn 1 = E[k∇(S(k)un−1) − i ∇S(k) dW (r)k2L2 ] 2 tn−1 Z 1 tn 1 E[k∇(S(k)Q1/2 )k2L 0 ] dr = E[k∇un−1k2L2 ] + 2 2 2 tn−1 k = E H(un−1 ) + Tr(∇Q∇). 2 3.4. A formula for the momentum. In the deterministic case, one has an additional conserved quantity, the momentum Z

p(u) := i (u∇u¯ − u∇u) ¯ dx. The next result investigate the behaviour of this quantity in the stochastic case. P ROPOSITION 3.9. Assume that E[p(u0 )] < ∞ and Q1/2 ∈ L21 . Then the exact solution of the linear Schrödinger equation with additive noise (3.1) exhibits the following formula for the expected momentum D E E[p(u(t))] = E[p(u0 )] − 2tIm Q1/2 , ∇Q1/2 0 for all time t. L2

11


Here, we denote by h·, ·i the usual L2 inner product hu, vi :=

Z

Rd

uv¯ dx,

for u, v ∈ L2 . And further denote the Hilbert-Schmidt inner product from L2 to L2 with the above inner product by h·, ·iL 0 . 2 Proof. The momentum can be written as p(u(t)) = i hu(t), ∇u(t)i − i h∇u(t), u(t)i . Using the mild solution, we have Z t Z t hu(t), ∇u(t)i = S(t)u0 − i S(t − r) dW (r), ∇S(t)u0 − i ∇S(t − r) dW (r) 0 0 Z t = hS(t)u0, ∇S(t)u0 i + S(t)u0 , −i S(t − r)∇dW (r) 0 Zt + −i S(t − r)dW (r), S(t)∇u0 0 Zt Z t + −i S(t − r)dW (r), −i S(t − r)∇dW (r) . 0

0

Taking the expectation, the terms containing one integral is zero. Using also the Ito isometry we obtain Z t Z t S(t − r)dW (r), S(t − r)∇dW (r) E[hu(t), ∇u(t)i] = E[hS(t)u0 , ∇S(t)u0 i] + E 0 0 Z t D E 1/2 1/2 = E[hu0 , ∇u0 i] + E S(t − r)Q , S(t − r)∇Q dr L20 0 Z t D E Q1/2 , ∇Q1/2 0 dr = E[hu0 , ∇u0 i] + E L2 0 E D 1/2 1/2 = E[hu0 , ∇u0 i] + t Q , ∇Q . 0 L2

Similarly, we have E D E[h∇u(t), u(t)i] = E[h∇u0 , u0 i] + t ∇Q1/2 , Q1/2

L20

.

For the expected momentum we finally obtain E[p(u(t))] = E[i hu(t), ∇u(t)i − i h∇u(t), u(t)i] D E E D 1/2 1/2 1/2 1/2 = iE[hu0 , ∇u0 i − h∇u0 , u0 i] + it Q , ∇Q − ∇Q , Q L20 L20 D E

= E[p(u0 )] + it Q1/2 , ∇Q1/2 0 − Q1/2 , ∇Q1/2 L 0 2 L E2 D 1/2 1/2 = E[p(u0 )] − 2tIm Q , ∇Q . 0 L2

12


As we see in the proposition below, the exponential method satisfies the same trace formula for the momentum. P ROPOSITION 3.10. With the same assumptions as in Proposition 3.9, the exponential integrator (3.2) exhibits the following formula for the expected momentum. E D E[p(un )] = E[p(u0 )] − 2tn Im Q1/2 , ∇Q1/2 0 for all tn = nk. L2

Proof. Writing un = S(k)un−1 − i

Z tn

S(k)dW (r),

tn−1

we can use the same techniques as in the proof for the exact solution. We obtain E D E[p(un )] = E[p(un−1 )] − 2kIm Q1/2 , ∇Q1/2 0 L E 2 D 1/2 1/2 = E[p(u0 )] − 2tnIm Q , ∇Q . 0 L2

3.5. Numerical experiments for the linear stochastic Schrödinger equation. This subsection illustrates the above properties (error estimates and trace formulas) of the stochastic exponential integrator (3.2) when applied to the linear stochastic Schrödinger equation (3.1). Let us first consider the error in the time integration of the linear stochastic Schrödinger equation. For this, we consider problem (3.1) on the interval [0, 2π ] with periodic boundary conditions. The initial value is taken to be u0 = 0 and the eigenvalues of the covariance operator Q are given by λn = 1/(1 + n8) for n ∈ Z. Such a regular noise is needed for the midpoint rule to be convergent, see [17, Proposition 3.1] and the discussion below. The spatial discretisation is done by a pseudospectral method (with M = 28 Fourier modes) using the following representation of the noise W (x,t) =

∑ λn

1/2

βn (t)en (x),

n∈Z

where {βn (t)}n∈Z are i.i.d Brownian motion and {en (x)}n∈Z = { √12π einx }n∈Z is an orthonor-

mal basis of L2 (0, 2π ). The resulting system of stochastic differential equations is then integrated in time by the stochastic exponential integrator (3.2) (SEXP), the Crank-Nicolson scheme from [17, Section 3], which reduces to the stochastic implicit midpoint rule (MP), and the classical backward Euler-Maruyama methods (BEM). Note that the classical explicit Euler-Maruyama would require an unreasonable small time step and is therefore omitted in our computational experiments. Observe also that MP and BEM are semi-implicit methods whereas SEXP is explicit. The rates of mean-square convergence (measured in the L2 -norm at the end of the interval of integration [0, 0.5]) of these integrators are presented in Figure 3.1. The expected rate of convergence O(k2 ) of the stochastic exponential integrator, as stated in Theorem 3.1, can be confirmed. Here, the exact solution is approximated by the stochastic midpoint rule with a very small time step kexact = 2−10 and Ms = 750000 samples are used for the approximation of the expected values. Note that here and in all our numerical experiments, enough samples are taken in order for the Monte-Carlo errors to be negligible.

13

Exponential integrators for stochastic Schrödinger equations 10

Ms = 750000

0

10 -2

10

Error

-4

10 -6

10 -8

Slope 1 Slope 2 SEXP MP BEM

10 -10

10 -12 -3 10

10

-2

10

-1

10

0

k

F IGURE 3.1. Linear stochastic Schrödinger equation: Mean-square errors for the stochastic exponential integrator (SEXP), the midpoint rule (MP), and backward Euler-Maruyama (BEM). The dotted, resp. dash-dotted, lines have slopes 1 and 2.

We now turn to the trace formulas. Figure 3.2 illustrates the energy trace formula from Proposition 3.7 and compare the results obtained with our stochastic exponential method, with the backward Euler-Maruyama method, and with the midpoint rule. M = 27 Fourier modes are used for the spatial discretisation; Ms = 10000 samples are used for the approximatation of the expected values; the time interval is [0, 2500], and all schemes use a step size k = 0.1. The other parameters are the same as in the above numerical experiments. The numerical linear drift in the expected value of the energy of the stochastic exponential integrator (3.2), as stated in Proposition 3.8, is observed in this figure. One also observes an excellent behaviour of the midpoint rule in reproducing the linear growth of the expected energy correctly. This is in contrast with the wrong behaviour of the backward Euler-Maruyama method. Note that this fact was already observed for the trace formula for the linear stochastic wave equation [9]. Using the same parameters as in the previous numerical experiments, a similar behaviour of the numerical methods is observed for the mass trace formula given in Proposition 3.2. These results are not displayed. However, when considering a less regular noise, for example when the eigenvalues of Q are given by λn = 1/(1 + n2), which is of trace-class, one observes that the midpoint rule does not perform as well as the stochastic exponential method, see Figure 3.3. This further illustrates the theoretical results obtained in Proposition 3.5. 4. Linear stochastic Schrödinger equations with a multiplicative potential. We now consider the stochastic Schrödinger equation idu = ∆u dt + V (x)u dt + dW u(·, 0) = u0 ,

in Rd × (0, ∞), in Rd ,

(4.1)

where V (x) is a real-valued potential. The initial data u0 is an F0 -measurable complexvalued random variable, and the Q-Wiener process {W (t)}t≥0 is again complex-valued and square integrable with respect to the filtration. Further assumptions on d, u0 , V (x) and Q will be made when the results of this section are presented. In this section we aim to prove convergence rate of our exponential integrator, trace formulas for the mass and energy, and

14

R. Anton and D. Cohen Ms = 10000, k = 0.10, M = 128 12 Exact SEXP MP BEM

10

8

Energy

6

4

2

0 0

500

1000

1500

2000

2500

Time

F IGURE 3.2. Linear stochastic Schrödinger equation: Energy trace formulas for the numerical solutions given by the stochastic exponential method (SEXP), the midpoint rule (MP) and the backward Euler-Maruyama scheme (BEM). Time interval and time step: [0,2500], resp. k = 0.1. Ms = 10000, k = 0.10, M = 128 0.07 Exact SEXP MP BEM

0.06

0.05

0.04

Mass 0.03

0.02

0.01

0 0

2

4

6

8

10

Time

F IGURE 3.3. Linear stochastic Schrödinger equation: Mass trace formulas for the numerical solutions given by the stochastic exponential method (SEXP), the midpoint rule (MP) and the backward Euler-Maruyama scheme (BEM). Time interval and time step: [0,10], resp. k = 0.1.

finally present numerical experiments that confirm these theoretical findings. We use the same semigroup approach as in Section 3 and write the mild equation of (4.1) as u(t) = S(t)u0 − i

Z t 0

S(t − r)V (x)u(r) dr − i

Z t 0

S(t − r) dW (r),

(4.2)

where we recall that S(t) = e−it∆ . We have the following result regarding existence, uniqueness and boundedness of solutions of (4.1). 1/2 ∈ L σ . Assume P ROPOSITION 4.1. Assume that E[ku0 k2p σ ] < ∞ for some p ∈ N, and Q 2 also that


15

(i) σ > d2 and V (x) ∈ H σ , or (ii) σ = 0 and d = 1, 2, 3 and V (x) ∈ H 2 , or (iii) σ = 1 and d = 2, 3 and V (x) ∈ H 3 .

Then (4.1) has a unique solution on [0, T ], for T ∈ (0, ∞), which is given by (4.2) and satisfies E[ sup ku(t)kσ2p] ≤ C. 0≤t≤T

Proof. We first observe that with the different assumptions (i)-(iii), we have (i) H σ forms an algebra. In particular we have, for example, kV (x)u(t)kσ ≤ CkV (x)kσ ku(t)kσ . (ii) In this case we have H 2 ⊂ L∞ , so whenever V (x) ∈ H 2 we have kV (x)u(t)kL2 ≤ kV (x)kL∞ ku(t)kL2 ≤ Cku(t)kL2 . (iii) We still have H 2 ⊂ L∞ and, if V (x) ∈ H 3 , then kV (x)u(r)k1 ≤ Cku(r)k1 . This can be seen by the following calculation: kV (x)u(r)k21 = kV (x)u(r)k2L2 + k∇(V (x)u(r))k2L2

= kV (x)u(r)k2L2 + k∇V (x) · u(r) + V(x)∇u(r)k2L2

≤ kV (x)u(r)k2L2 + 2(k∇V (x) · u(r)k2L2 + kV (x)∇u(r)k2L2 )

≤ 3kV (x)k2L∞ (ku(r)k2L2 + k∇u(r)k2L2 ) + 2k∇V(x)k2L∞ ku(r)k2L2 ≤ Cku(r)k21 ,

since k∇V (x)kL∞ < ∞ if V (x) ∈ H 3 . Existence and uniqueness now follow from a standard fixed point argument (see [11, Theorem 7.4]) using what was mentioned above about the different cases (i)-(iii). To prove boundedness, we use the mild equation (4.2).

E

Z t 2p S(t − r)V (x)u(r) drk + E sup k sup ku(t)kσ2p ≤ C E sup kS(t)u0k2p σ σ

0≤t≤T

+E

0≤t≤T

sup k

0≤t≤T

Z t 0

0

0≤t≤T

S(t − r) dW (r)k2p σ

.

We now proceed by estimating the terms on the right-hand side. First we have, by Lemma 2.1,

E

sup kS(t)u0 kσ2p = E[ku0 kσ2p] ≤ C.

0≤t≤T

For the second term we need to treat each case (i)-(iii) separately.

16


(i) Using Lemma 2.1, the fact that H σ is an algebra, and Hölder’s inequality, we have " Z # Z t

E

sup k

0≤t≤T

0

≤E

kS(t − r)V (x)u(r)kσ dr

sup

"

0

0≤t≤T

Z

sup 0≤t≤T

t 0

≤ CkV (x)k2p σ E 



0≤t≤T

≤ CT 2p−1E Z T 0

kV (x)kσ ku(r)kσ dr

"

 ≤ CE  sup 

≤C

2p

t

≤E S(t − r)V (x)u(r) drk2p σ

Z

sup

Z

t

0

sup

t

0

0≤t≤T

ku(r)kσ dr

ku(r)k2p σ dr

Z t

0≤t≤T 0

2p #

1 Z

t

2p

ku(r)k2p σ dr

E sup ku(s)kσ2p dr.

2p # 2p

1 2p−1 dr 0

0≤s≤r

2p In the last step we have used that ku(r)k2p σ ≤ sup ku(s)kσ . 0≤s≤r

(ii) If instead σ = 0 and d = 1, 2, 3 and thus H σ = L2 is not an algebra, then we have, using H 2 ⊂ L∞ , " " Z Z # # 2p

t

E

sup

0≤t≤T

0

≤ kV (x)k2p L∞ E

kS(t − r)V (x)u(r)kL2 dr

≤ CE

"

sup 0≤t≤T

2p

t

sup

0≤t≤T

Z

0

ku(r)kL2 dr

t 0

ku(r)kL2 dr

2p #

.

The same estimate follows from similar calculations as in (i). (iii) We now assume σ = 1, d = 2, 3 and V (x) ∈ H 3 . The proof follows the same lines as in (ii), noting that kV (x)u(r)k1 ≤ Cku(r)k1 , as seen above. For the third and last term, using Burkholder’s inequality, we obtain Z T p Z t kQ1/2 k2L σ dt E sup k S(t − r) dW (r)kσ2p ≤ CE 0≤t≤T

0

0

≤ CT Altogether we arrive at Z 2p E sup ku(t)kσ ≤ C1 + C2 0≤t≤T

0

T

p

2

E[kQ1/2 k2p ] ≤ C. L2σ

E[ sup ku(s)k2p σ ] dt, 0≤s≤t

and an application of Gronwall’s lemma completes the proof.

2p−1 2p

2p    

17


4.1. Error analysis of the stochastic exponential method. Our exponential scheme for the time integration of (4.1) now reads un = S(k)un−1 − ikS(k)V (x)un−1 − iS(k)∆W n−1 n−1 Z t j+1

= S(tn )u0 − i ∑

j=0 t j

n−1 Z t j+1

S(tn − t j )V (x)u dr − i ∑ j

j=0 t j

S(tn − t j ) dW (r),

(4.3)

where, as before, k denotes the step size, tn = nk for n = 0, . . . , N, and ∆W n−1 = W (tn ) − W (tn−1 ). We first show that these numerical approximations are bounded. 1/2 ∈ L σ . As in P ROPOSITION 4.2. Assume that E[ku0 k2p σ ] < ∞ for some p ∈ N, and Q 2 Proposition 4.1, we also assume (i) σ > d2 , and V (x) ∈ H σ , or (ii) σ = 0 and d = 1, 2, 3, and V (x) ∈ H 2 , or (iii) σ = 1 and d = 2, 3, and V (x) ∈ H 3 . Then the numerial solution given by (4.3), with step size k, satisfies 2p

E[kun kσ ] ≤ C

for 0 ≤ tn = nk ≤ T.

Proof. We prove the case (i), the other cases are treated as in the proof of Proposition 4.1. We have " # n−1

E[kun kσ2p] ≤ C E[kS(tn )u0 kσ2p] + E kik ∑ S(tn − t j )V (x)u j k2p σ "

j=0

n−1 Z t j+1

+ E ki ∑

j=0 t j

S(tn − t j ) dW (r)k2p σ

#!

.

The first term is estimated as in Proposition 4.1. For the second term, using Hölder’s inequality and the fact that V (x) ∈ H σ , we have n−1

n−1

j=0

j=0

kk ∑ S(tn − t j )V (x)u j kσ ≤ k ∑ kV (x)u j kσ n−1

≤ Ck ∑ ku j kσ j=0

≤ Ck

n−1

∑

1 2p

j=0

!1

2p

ku j k2p σ

.

For the third term we use the Burkholder inequality and obtain " # Z Z n−1

E k∑

t j+1

j=0 t j

S(tn − t j ) dW (r)kσ2p = E k ≤ CE ≤ C,

tn

S(tn − [r/k]k) dW (r)k2p σ

0

Z

where, as before, [r/k] denotes the integer part of r/k.

0

T

kQ1/2 k2L σ dt 2

p

18


Altogether we arrive at n−1

E[kun kσ ] ≤ C1 + C2 k ∑ E[ku j kσ ] 2p

2p

j=0

and an application of Gronwall’s lemma concludes the proof. We are now ready to prove the main result of this section on the convergence of our numerical method. T HEOREM 4.3. Consider the time discretisation of the stochastic Schrödinger equation (4.1) given by the exponential integrator (4.3). Let d = 1, 2, 3, σ ≥ 0, and assume that σ +2 , and Q1/2 ∈ L σ +2 . Then there exists a conE[ku0 k2p σ +2 ] < ∞ for some p ∈ N, V (x) ∈ H 2 stant C such that ≤ Ck2p . E max kun − u(tn )k2p σ n=1,...,N

R EMARK 4.4. Using similar assumptions as in Hypothesis 4.1 and 4.2 in [17], we could replace the term V (x)u in (4.1) by a general Lipschitz function F(u). That is, if we assume σ > d2 , and F : H σ → H σ is a C2 function that is bounded as well as its derivatives up to order 2, that (F(u(t)) ∈ L2p (Ω; L∞ (0, T ; H σ +2 )), that the solution (u(t))t∈ [0,T ] is in L2p (Ω; L∞ (0, T ; H σ +2 )) and L4p (Ω, L∞ (0, T, H σ +1)), and u0 ∈ L2p (Ω; H σ +2 ), then the convergence order stated in Theorem 4.3 remains unchanged. Proof. As before, there are three cases to consider: (i) σ > d2 , (ii) σ = 0 and d = 1, 2, 3, (iii) σ = 1 and d = 2, 3. We will prove the case (i) where σ > d2 , so that H σ forms an algebra. In the cases (ii) and (iii), H σ does not form an algebra, but these two cases can be treated in a similar way as in the proof of Proposition 4.1. We consider the error n un − u(tn ) = ErrVn + ErrW ,

where we have defined n−1 Z t j+1

ErrVn := i ∑

j=0 t j

(S(tn − r)V (x)u(r) − S(tn − t j )V (x)u j ) dr,

and n ErrW

n−1 Z t j+1

:= i ∑

j=0 t j

(S(tn − r) − S(tn − t j )) dW (r).

This then gives us E[ max ku n=1,...,N

n

2p − u(tn)kσ ]

n 2p n 2p ≤ C E[ max kErrV kσ ] + E[ max kErrW kσ ] . n=1,...,N

n=1,...N

The second term on the right-hand side is the same term as in the proof of Theorem 3.1 and we thus get n 2p E[ max kErrW kσ ] ≤ Ck2p . n=1,...,N


19

In order to estimate ErrVn , we first write it as n−1 Z t j+1

ErrVn = i ∑

j=0 t j

S(tn − r)V (x)(u(r) − u(t j )) dr

n−1 Z t j+1

(S(tn − r) − S(tn − t j ))V (x)u(t j ) dr

n−1 Z t j+1

S(tn − t j )V (x)(u(t j ) − u j ) dr

+i ∑

j=0 t j

+i ∑

j=0 t j n =: I1 + I2n + I3n .

The estimate for I1n is more complicated than the estimates for I2n and I3n , so we save it for last. Using Lemma 2.1 and that H σ +2 is an algebra, we have

kI2n kσ

≤

n−1 Z t j+1

k(S(tn − r) − S(tn − t j ))V (x)u(t j )kσ dr

=

n−1 Z t j+1

kS(tn − t j )(S(t j − r) − I)V (x)u(t j )kσ dr

n−1 Z t j+1

|t j − r|k∆(V (x)u(t j ))kσ dr

∑

j=0 t j

∑

j=0 t j

≤

∑

j=0 t j

≤ Ck2

n−1

∑ kV (x)u(t j )kσ +2

j=0

n−1

≤ Ck2 kV (x)kσ +2 ∑ ku(t j )kσ +2 . j=0

Now, using Hölder’s inequality we have

n−1

∑ ku(t j )kσ +2 ≤ ∑

j=0

j=0

≤ Ck−1

!1

2p

n−1

ku(t j )k2p σ +2

k

1−2p 2p

sup ku(t)kσ2p+2

0≤t≤T

1

2p

.

Thus 2p 2p 2p E[ max kI2n k2p σ ] ≤ Ck E[ sup ku(t)kσ +2 ] ≤ Ck , n=1,...,N

by Proposition 4.1.

0≤t≤T

20


Again using the fact that H σ is an algebra, and then Hölder’s inequality, we have kI3n kσ ≤

n−1 Z t j+1

∑

j=0 t j

kV (x)(u(t j ) − u j )kσ dr

≤ CkV (x)kσ

n−1 Z t j+1

∑

j=0 t j

ku(t j ) − u j kσ dr

n−1

≤ Ck ∑ ku(t j ) − u j kσ j=0

∑

≤ Ck ≤ Ck

j=0 1 2p

!1

2p

n−1

ku(t j ) − u j k2p σ

∑

1−2p 2p

!1

2p

n−1 j=0

k

ku(t j ) − u j k2p σ

.

Thus E[ max kI3n k2p σ ] ≤ Ck n=1,...,N

N−1

max ku(tl ) − ul kσ2p ]. ∑ E[l=0,..., j

j=0

In order to estimate I1n we use the mild formulation of the exact solution on the interval [t j , r] u(r) = S(r − t j )u(t j ) − i

Z r tj

S(r − ρ )V(x)u(ρ ) dρ − i

Z r tj

S(r − ρ ) dW (ρ ),

and therefore u(r) − u(t j ) = (S(r − t j ) − I)u(t j ) − i

Z r tj

S(r − ρ )V (x)u(ρ ) dρ − i

Z r tj

S(r − ρ ) dW (ρ ).

We have I1n

n−1 Z t j+1

S(tn − r)V (x)(u(r) − u(t j )) dr

n−1 Z t j+1

S(tn − r)V (x)(S(r − t j ) − I)u(t j ) dr

=i∑

j=0 t j

=i∑

j=0 t j

n−1 Z t j+1

+∑

j=0 t j

n−1 Z t j+1

+∑

S(tn − r)V (x) S(tn − r)V (x)

j=0 t j n =: J1 + J2n + J3n .

Z

r

tj

Z

r

tj

S(r − ρ )V (x)u(ρ ) dρ S(r − ρ ) dW (ρ ) dr

dr

Using Lemma 2.1, the fact that H σ is an algebra, Hölder’s inequality, and Proposition 4.1, we


21

obtain

kJ1n kσ ≤

n−1 Z t j+1

∑

j=0 t j

kS(tn − r)V (x)(S(r − t j ) − I)u(t j )kσ dr

≤ CkV (x)kσ ≤ Ck2 ≤ Ck

n−1 Z t j+1

∑

j=0 t j

|r − t j |ku(t j )kσ +2 dr

n−1

∑ ku(t j )kσ +2

j=0

sup

0≤t≤T

ku(t)k2p σ +2

1

2p

≤ Ck, so that 2p

E[ max kJ1n kσ ] ≤ Ck2p . n=1,...,N

Next, we estimate J2n . Using Lemma 2.1, the fact that H σ is an algebra, and Hölder’s inequality, we get

kJ2n kσ

≤

n−1 Z t j+1

∑

j=0 t j

kS(tn − r)V (x)

≤ CkV (x)kσ

n−1 Z t j+1

n−1 Z t j+1

≤C ∑

j=0 t j

n−1 Z t j+1

∑

j=0 t j

|r − t j |

tj

2p−1 2p

r

tj

S(r − ρ )V (x)u(ρ ) dρ kσ dr

kV (x)kσ ku(ρ )kσ dρ dr

Z

r

tj

ku(ρ )kσ2p dρ !1

1

2p

dr

2p

≤C ∑

|r − t j |

≤ Ck

ku(t)kσ2p

j=0 t j

Z r

Z

sup

t j ≤ρ ≤r

!1

ku(ρ )kσ2p

dr

2p

sup 0≤t≤tn

,

and thus, by Proposition 4.1, E[ max kJ2n kσ2p ] ≤ Ck2p E[ sup ku(t)kσ2p] ≤ Ck2p . n=1,...,N

0≤t≤T

Finally, using Fubini’s theorem (when changing the order of integration, we go from t j ≤ r ≤ t j+1 and t j ≤ ρ ≤ r to t j ≤ ρ ≤ t j+1 and ρ ≤ r ≤ t j+1 ), the Burkholder inequality,

22


Lemma 2.1, and the fact that H σ is an algebra, we have " # Z r n−1 Z t j+1 2p n 2p S(tn − r)V (x) S(r − ρ ) dW (ρ ) drkσ E[ max kJ3 kσ ] =E max k ∑ n=1,...,N j=0 t j

n=1,...,N

=E =E ≤E

" "

"

tj

n−1 Z t j+1 Z t j+1

max k ∑ max k

n=1,...,N

sup k

0≤t≤T

Z tn Z [ ρ +1]k k ρ

0

Z tZ 0

ρ [ k +1]k

ρ

" Z Z t ≤ C sup E k ≤ C sup E 0≤t≤T

" Z

= C max E n=1,...,N

≤ C max E n=1,...,N

"

"

"

t 0

k

∑

#

ρ

[ k +1]k

S(t − r)V (x)S(r − ρ ) dr dW (ρ )kσ2p

Z [ ρ +1]k k ρ

n−1 Z t j+1

S(t − r)V (x)S(r − ρ )Q

Z t j+1

j=0 t j

∑

k

n−1 Z t j+1

Z

∑

j=0 t j

N−1 Z t j+1 j=0 t j

S(tn − r)V (x)S(r − ρ ) dr dW (ρ )kσ2p

2p S(t − r)V (x)S(r − ρ ) dr dW (ρ )kσ

ρ

0

0≤t≤T

≤ CE

ρ

n=1,...,N j=0 t j

S(tn − r)V (x)S(r − ρ ) dr dW (ρ )kσ2p

ρ

|t j+1 − ρ |2 dρ

drk2L σ 2

S(t − r)V (x)S(r − ρ )Q

t j+1

ρ

1/2

kV (x)kσ kQ

! p#

1/2

kL2σ dr

1/2

2

#

# dρ

#

! p#

drk2L σ 2 dρ

dρ

!p#

! p#

≤ Ck2p ,

where we recall that [ ρk + 1] denotes the integer part of ρk + 1, where k is the step size of the scheme. To summarize, we have obtained the following bound 2p E[ max kun − u(tn )k2p σ ] ≤ C1 k + C2 k n=1,...,N

N−1

max kul − u(tl )k2p σ ], ∑ E[l=0,..., j

j=0

and an application of Gronwall’s lemma completes the proof. 4.2. A trace formula for the mass. With appropriate boundary conditions, such as when the domain is the d-dimensional torus Td , the expected mass Z |u|2 dx E[M(u)] = E Td

e g), such as the still exhibits linear growth. On a smooth compact Riemannian manifold (M, e ⊂ L∞ (M) e if d = 1, 2, 3 (see [21, Theorem 2.7]). In d-dimensional torus, we still have H 2 (M) particular, Propositions 4.1 and 4.2 and Theorem 4.3 above remain valid when Rd is replaced by Td . Therefore, in this subsection, we assume that the spatial domain is the d-dimensional torus Td . P ROPOSITION 4.5. Let d = 1, 2, 3 and assume that the initial data is such that E[M(u0 )] is finite, that V (x) ∈ H 2 , and that Q is trace-class. Then the exact solution of (4.1), given by


23

equation (4.2), satisfies the trace formula E[M(u(t))] = E[M(u0 )] + tTr(Q), for t ≥ 0. Proof. This follows from the Ito formula [11, Theorem 4.17]. We have Z t Zt M ′ (u(s)), ϕ (s) ds M ′ (u(s)), Φ(s) dW (s) + M(u(t)) = M(u0 ) + +

Z t 1 0

2

0

0

′′

Tr[M (u(s))(Φ(s)Q

1/2

)(Φ(s)Q

1/2 ∗

) ] ds,

where ϕ (s) = −i∆u(s) − iV (x)u(s) and Φ(s) = I. The expected value of the first integral on the right-hand side is zero. For the second integral, we have M ′ (u(s)), ϕ (s) = (u(s), ϕ (s)) + (ϕ (s), u(s)) = (u(s), −i∆u(s)) + (u(s), −iV (x)u(s)) + (−i∆u(s), u(s)) + (−iV(x)u(s), u(s)) = 0.

Thus, we arrive at Z

1 t E[Tr(M ′′ (u(s))(Q1/2 )(Q1/2 )∗ )] ds 2 0 = E[M(u0 )] + tTr(Q).

E[M(u(t))] = E[M(u0 )] +

We now investigate how the expected mass of our exponential integrator behaves. As seen below we do not get an exact trace formula as for the linear problem considered in Section 3, but instead we get a trace formula with an error of size O(k) on compact intervals. P ROPOSITION 4.6. Let d = 1, 2, 3 and assume that the initial data is such that E[M(u0 )] is finite, that V (x) ∈ H 2 , and that Q is trace-class. Then the stochastic exponential integrator un given by (4.3) satisfies the following almost trace formula for the expected mass E[M(un )] = E[M(u0 )] + tn Tr(Q) + O(k) for 0 ≤ tn = nk ≤ T. Proof. First note that, by assumption, we have E[ku0 k2L2 ] < ∞. The numerical method reads un = S(k)un−1 − iS(k)V (x)un−1 k − i

Z tn

S(k) dW (r)

tn−1

and a straight-forward calculation gives kun k2L2 = kS(k)un−1k2L2 + (S(k)un−1, −iS(k)V (x)un−1 k) Z tn n−1 S(k) dW (r) + (−iS(k)V (x)un−1 k, S(k)un−1 ) + S(k)u , −i tn−1 Z tn n−1 2 n−1 + kiS(k)V(x)u kkL2 + −iS(k)V (x)u k, −i S(k) dW (r) tn−1 Zt Zt n n n−1 n−1 S(k) dW (r), −iS(k)V (x)u k + −i S(k) dW (r), S(k)u + −i + ki

Z tn

tn−1

tn−1

S(k) dW (r)k2L2 .

tn−1

24


Next we observe that (S(k)un−1 , −iS(k)V (x)un−1 k) + (−iS(k)V (x)un−1 k, S(k)un−1 ) = 0 and the expected value of the terms containing one stochastic integral is zero. We thus have, using Lemma 2.1, Proposition 4.2 (case (ii)), and Ito’s isometry, Zt n S(k) dW (r)k2L2 E[kun k2L2 ] = E[kun−1k2L2 ] + k2 E[kV (x)un−1 k2L2 ] + E k = E[M(un−1 )] + Ck2 +

Z tn

tn−1

= E[M(u

n−1

2

tn−1

kQ1/2 k2L 0 dr 2

)] + Ck + kTr(Q).

An iteration completes the proof. 4.3. A trace formula for the energy. As for the mass, with suitable boundary conditions, the expected energy will still grow linearly. We again consider the domain to be the d-dimensional torus and the energy is given by H(u) =

1 2

Z

Td

|∇u|2 dx −

1 2

Z

Td

V (x)|u|2 dx.

P ROPOSITION 4.7. Let d = 1, 2, 3 and assume that the initial data is such that E[H(u0 )] is finite, and that Q1/2 ∈ L21 . If d = 1, we assume V (x) ∈ H 2 and if d = 2, 3 we assume V (x) ∈ H 3 . Then the exact solution of (4.1), given by equation (4.2), satisfies the trace formula t E[H(u(t))] = E[H(u0 )] + (Tr(∇Q∇) − Tr(Q1/2V (x)Q1/2 )). 2 Proof. Similar to the trace formula for the mass, we use the Ito formula H(u(t)) = H(u0 ) + +

Z t 1 0

2

Z t

Z H ′ (u(s)), Φ(s) dW (s) +

H ′ (u(s)), ϕ (s) ds

t

0

0

′′

Tr[H (u(s))(Φ(s)Q

1/2

)(Φ(s)Q

1/2 ∗

) ] ds,

where ϕ (s) = −i∆u(s) − iV (x)u(s) and Φ(s) = I. The expected value of the first integral is seen to be zero, and a similar calculation as in the proof of Proposition 4.5 shows that the second integral is zero as well. To calculate the third integral, we note that Tr(H ′′ (u(s))(Q1/2 )(Q1/2 )∗ ) =

∑ H ′′ (u(s))(Q1/2 en , Q1/2en )

n≥1

=

∑ (∇Q1/2 en, ∇Q1/2 en )L2 − ∑ (V (x)Q1/2en , Q1/2 en)L2

n≥1

= Tr(∇Q∇) − Tr(Q

n≥1

1/2

V (x)Q

1/2

).

The energy for the numerical approximation satisfies an almost trace formula: we get a small error of size O(k), as seen in the next proposition. P ROPOSITION 4.8. Let d = 1, 2, 3 and assume that E[ku0 k22 ] ≤ C. If d = 1 then assume V (x) ∈ H 2 . If d = 2, 3 then assume V (x) ∈ H 3 . Also assume that Q1/2 ∈ L22 . Then the stochastic exponential integrator un given by (3.2) satisfies the following almost trace formula for the expected energy E[H(un )] = E[H(u0 )] +

tn (Tr(∇Q∇) − Tr(Q1/2V (x)Q1/2 )) + O(k) for 0 ≤ tn = nk ≤ T. 2


25

Proof. First we note that with these assumptions we have finite initial energy, since Z E[H(u0 )] = E[k∇u0 k2L2 ] − E V (x)|u0 |2 dx ≤

Td 2 E[ku0 k1 ] + kV(x)k2L∞ E[ku0 k2L2 ]

≤ C.

We add and subtract the energy for the exact solution E[H(un )] = E[H(u(tn ))] + E[H(un) − H(u(tn ))] tn = E[H(u0 )] + (Tr(∇Q∇) − Tr(Q1/2V (x)Q1/2 )) + E[H(un ) − H(u(tn))]. 2 Thus we need to show that |E[H(un ) − H(u(tn ))]| ≤ Ck. We have (omitting the constant 1/2 for ease of presentation) |E[H(un ) − H(u(tn))]| ≤ |E[k∇un k2L2 − k∇u(tn)k2L2 ]| Z Z + E V (x)|u(tn )|2 dx − V (x)|un |2 dx Td Td =: I1n + I2n .

We begin by estimating I1n . We have, using that the inner product defined by (2.2) is symmetric, integration by parts and the boundary conditions, Cauchy-Schwarz and Hölder’s inequalities, I1n = |E[(∇(un + u(tn )), ∇(un − u(tn)))]|

= |E[(∆(un + u(tn)), un − u(tn ))]| ≤ E[k∆(un + u(tn ))kL2 kun − u(tn)kL2 ] 1/2 1/2 ≤ E[k∆(un + u(tn ))k2L2 ] E[kun − u(tn)k2L2 ] 1/2 1/2 . ≤ E[kun + u(tn )k22 ] E[kun − u(tn)k2L2 ]

Using Propositions 4.1, 4.2, and Theorem 4.3 we have I1n ≤ Ck. Similarly, we have I2n

Z 2 n 2 = E V (x)(|u(tn )| − |u | ) dx Td

≤ kV (x)kL∞ |E[ku(tn )k2L2 − kunk2L2 ]|

≤ CE[ku(tn ) + unkL2 ku(tn ) − unkL2 ] 1/2 1/2 ≤ C E[ku(tn ) + unk2L2 ] E[ku(tn ) − unk2L2 ]

≤ Ck.

26

R. Anton and D. Cohen 10

Ms = 750000

0

10 -1

10 -2

10

-3

Error 10 -4

10

-5

Slope 1 Slope 2 SEXP CN SEM

10 -6

10 -7 -3 10

10

-2

10

-1

10

0

k

F IGURE 4.1. Stochastic Schrödinger equation with potential: Mean-square errors for the stochastic exponential integrator (SEXP), the Crank-Nicolson scheme (CN), and the semi-implicit Euler-Maruyama (SEM). The dotted, resp. dash-dotted, lines have slopes 1 and 2.

4.4. Numerical experiments for the stochastic Schrödinger equations with a multiplicative potential. This subsection illustrates some of the above properties (error estimates and trace formula for the mass) of the stochastic exponential integrator (4.3) when applied to the linear stochastic Schrödinger equation with a multiplicative potential on the interval [0, 2π ] with periodic boundary conditions (4.1). In these numerical experiments, we con2 . We compare sider a potential V (x) = 1+sin1 2 (x) , [4], and initial values with u0 (x) = 2−cos(x) the stochastic exponential integrator (SEXP) with the Crank-Nicolson scheme (CN) and the semi-implicit Euler-Maruyama scheme, explicit in V (x)u, (SEM). Figure 4.1 illustrates the convergence errors of the above numerical methods for a noise with covariance operator having the eigenvalues λn = 1/(1 + n6) for n ∈ Z. The spatial discretisation is done by a pseudospectral method with M = 28 Fourier modes. The rates of mean-square convergence (measured in the L2 -norm at the end of the interval of integration [0, 0.5]) of these numerical methods are presented in this figure. The expected rate of convergence O(k2 ) of the stochastic exponential integrator, as stated in Theorem 4.3, can be confirmed. Here, the exact solution is approximated by the stochastic exponential method with a very small time step kexact = 2−9 and Ms = 750000 samples are used for the approximation of the expected values. We now consider eigenvalues of Q given by λn = 1/(1 + n2) and examine the numerical trace formulas for the mass. Figure 4.2 is produced using M = 27 Fourier modes for the spatial discretisation; Ms = 10000 samples for the approximatation of the expected values; the time interval is [0, 5]; and a step size k = 0.1. As stated by Proposition 4.6, in this figure, one can observe the small error in the preservation of the trace formula for the mass of the numerical solution given by the exponential integrator. 5. Stochastic Schrödinger equations driven by multiplicative noise. We continue our exposition of the exponential integrator applied to the stochastic Schrödinger equation by considering the equation driven by a multiplicative Ito noise idu = ∆u dt + V (x)u dt + u dW u(·, 0) = u0

in Rd × (0, ∞), in Rd .

(5.1)

27

Exponential integrators for stochastic Schrödinger equations Ms = 10000, k = 0.10, M = 128 0.04 Exact SEXP CN SEM

0.035

0.03

0.025

Mass

0.02

0.015

0.01

0.005

0 0

1

2

3

4

5

Time

F IGURE 4.2. Stochastic Schrödinger equation with potential: Mass trace formulas for the numerical solutions given by the stochastic exponential method (SEXP), the Crank-Nicolson scheme (CN) and the semi-implicit EulerMaruyama scheme (SEM). Time interval and time step: [0,5], resp. k = 0.1.

As before, V (x) is a real-valued potential, u0 is an F0 -measurable complex-valued random variable, and the Q-Wiener process is complex-valued and square integrable. Further assumptions on d, u0 , V (x), and Q will be made in the results of this section. Again, the equation can be written in its mild form

u(t) = S(t)u0 − i

Zt 0

S(t − r)V (x)u(r) dr − i

Z t 0

S(t − r)u(r) dW (r),

where S(t) = e−it∆ . P ROPOSITION 5.1. Let σ > d2 , and assume that E[ku0 k2p σ ] < ∞ for some p ∈ N, V (x) ∈ and Q1/2 ∈ L2σ . Then there exists a unique solution u(t) on [0, T ], for some T > 0, to problem 5.1 which satisfies

Hσ ,

E[ sup ku(t)kσ2p] ≤ C. 0≤t≤T

R EMARK 5.2. The result of Proposition 5.1 holds with σ = 0 if we assume d = 1, 2, 3, 2 1/2 ∈ L 2 . E[ku0 k2p 2 0 ] < ∞ for some p ∈ N, V (x) ∈ H , and Q Proof. The existence and uniqueness of a solution again follows by a standard fixed point argument as used in [11, Theorem 7.4]. From the mild equation and the proof of Proposition

28


4.1, we have E

sup 0≤t≤T

ku(t)kσ2p

≤ C E[ sup kS(t)u0k2p σ ] +E

+E

0≤t≤T

sup k

0≤t≤T

sup k

0≤t≤T Z T

≤ C1 + C2 + C3 E

Z t

0

0

S(t − r)V (x)u(r) drkσ2p

Z t

S(t − r)u(r) dW (r)kσ2p

0

E[ sup ku(s)kσ2p] dt 0≤s≤t Z t

sup k

0≤t≤T

0

S(t − r)u(r) dW (r)kσ2p

.

We need to bound the stochastic integral. Using the Burkholder inequality, that H σ forms an algebra, and Hölder’s inequality, we have p Z T Z t 2p 1/2 2 E sup k S(t − r)u(r) dW (r)kσ ≤ CE ku(t)Q kL σ dt 0≤t≤T

0

2

0

≤ CE ≤ CE ≤C

Z Z

Z T 0

0

T

0 T

ku(t)k2σ kQ1/2k2L σ 2

ku(t)kσ2p dt

dt

p

E[ sup ku(s)k2p σ ] dt. 0≤s≤t

The result follows from Gronwall’s lemma. We will need the following lemma regarding regularity of the exact solution. L EMMA 5.3. Let σ > d2 and assume E[ku0 kσ2p+1 ] < ∞ for some p ∈ N, V (x) ∈ H σ +1 , and Q1/2 ∈ L2σ +1 . Then, for 0 ≤ s ≤ t < ∞, the exact solution of (5.1) satisfies the following regularity estimate E[ku(t) − u(s)kσ2p] ≤ C|t − s| p . R EMARK 5.4. The result of Lemma 5.3 holds with σ = 0 if we assume d = 1, 2, 3, 2 1/2 ∈ L 2 . E[ku0 k2p 2 2 ] < ∞ for some p ∈ N, V (x) ∈ H , and Q Proof. Using the mild equation, we have u(t) − u(s) = (S(t − s) − I)u(s) − i −i

Z t s

Zt s

S(t − r)V (x)u(r) dr

S(t − r)u(r) dW (r).

We proceed to estimate the three terms on the right hand side. The first one, due to Lemma 2.2, reads k(S(t − s) − I)u(s)kσ ≤ kS(t − s) − IkL (H σ +1 ,H σ ) ku(s)kσ +1 ≤ C|t − s|1/2ku(s)kσ +1 ,

29


so that, by Proposition 5.1, p E[k(S(t − s) − I)u(s)k2p σ ] ≤ C|t − s| .

To estimate the second term, we use that H σ forms an algebra and Hölder’s inequality k

Z t s

S(t − r)V (x)u(r) drkσ ≤

Z t s

≤C

kV (x)u(r)kσ dr

Z t s

kV (x)kσ ku(r)kσ dr

≤ C|t − s|

sup ku(r)kσ2p

s≤r≤t

1

2p

.

Taking expectation of the 2p-th power of the above expression and using Proposition 5.1, we get E[k

Z t s

2p S(t − r)V (x)u(r) drk2p σ ] ≤ C|t − s| .

Finally, we have Zt Z t p 2p 1/2 2 E k S(t − r)u(r) dW (r)kσ ≤ CE ku(r)Q kL σ dr s

2

s

≤ C|t − s| p E[ sup

s≤r≤t

ku(r)k2p σ ]

≤ C|t − s| p . 5.1. Error analysis for multiplicative noise. As before, we let N > 0 be an integer and divide the interval [0, T ] into subintervals 0 = t0 < t1 < . . . < tN−1 < tN = T of equal length k so that tn = nk. We discretise the mild equation and our exponential integrator now reads un = S(k)un−1 − ikS(k)V (x)un−1 − iS(k)un−1∆W n−1 n−1 Z t j+1

= S(tn )u0 − i ∑

j=0 t j

n−1 Z t j+1

S(tn − t j )V (x)u j dr − i ∑

j=0 t j

S(tn − t j )u j dW (r),

(5.2)

where ∆W n−1 = W (tn ) − W (tn−1 ). As seen in the theorem below, because of the multiplicative noise, the order of convergence is reduced to one half. T HEOREM 5.5. Consider the time integration of the stochastic Schrödinger equation (5.1) given by the exponential integrator (5.2). Let σ > d2 and assume that E[ku0 k2p σ +1 ] < ∞ σ +1 σ +1 1/2 for some p ∈ N. Assume also that V (x) ∈ H and Q ∈ L2 . Then there exists a constant C such that p E[ max kun − u(tn )k2p σ ] ≤ Ck . n=1,...N

R EMARK 5.6. Following remarks 5.2 and 5.4, the result of Theorem 5.5 holds with σ = 0 2 1/2 ∈ L 2 . if d = 1, 2, 3, E[ku0 k2p 2 2 ] < ∞ for some p ∈ N, V (x) ∈ H , and Q

30


R EMARK 5.7. We could replace the nonlinear term and the u in front of the noise in equation (5.1) with more general bounded Lipschitz functions F : H σ → H σ and Φ : H σ → L2σ . The above result holds with similar assumptions as in Hypothesis 5.1 and 5.2 in [17], that is if we assume that the solution (u(t))t∈[0,T ] is in L2p (Ω; L∞ (0, T ; H σ +1 ), (F(u(t)))t∈[0,T ] is in L2p (Ω; L∞ (0, T ; H σ +1 )), (Φ(u(t)))t∈[0,T ] is in L2p (Ω; L∞ (0, T ; L2σ +1 )), and u0 ∈ L2p (Ω; H σ +1 ). Proof. Using the mild formulation, we have n−1 Z t j+1

un − u(tn) = i ∑

j=0 t j

(S(tn − r)V (x)u(r) − S(tn − t j )V (x)u j ) dr

n−1 Z t j+1

+i ∑

(S(tn − r)u(r) − S(tn − t j )u j ) dW (r)

j=0 t j n =: ErrVn + ErrW .

We add and subtract suitable terms to get n−1 Z t j+1

ErrVn = i ∑

j=0 t j

S(tn − r)V (x)(u(r) − u(t j )) dr

n−1 Z t j+1

(S(tn − r) − S(tn − t j ))V (x)u(t j ) dr

n−1 Z t j+1

S(tn − t j )V (x)(u(t j ) − u j ) dr

+i ∑

j=0 t j

+i ∑

j=0 t j n =: I1 + I2n + I3n ,

and n−1 Z t j+1

n ErrW =i∑

j=0 t j

S(tn − r)(u(r) − u(t j )) dW (r)

n−1 Z t j+1

(S(tn − r) − S(tn − t j ))u(t j ) dW (r)

n−1 Z t j+1

S(tn − t j )(u(t j ) − u j ) dW (r)

+i ∑

j=0 t j

+i ∑

j=0 t j n =: J1 + J2n + J3n .

We now proceed to estimate each of these terms. For I1n , using again that [r/k] denotes the integer part of r/k and then Hölder’s inequality, we have kI1n kσ ≤

n−1 Z t j+1

∑

j=0 t j

kV (x)(u(r) − u(t j ))kσ dr

≤ CkV (x)kσ ≤C

Z tn

≤ CT

0

n−1 Z t j+1

∑

j=0 t j

ku(r) − u(t j )kσ dr

ku(r) − u([r/k]k)kσ dr

2p−1 2p

Z

0

tn

ku(r) − u([r/k]k)k2p σ dr

1

2p

,

31


so that, using Lemma 5.3, E[ max kI1n k2p σ ] ≤ CE n=1,...,N

Z

T 0

ku(r) − u([r/k]k)kσ2p dr

≤C

N−1 Z t j+1

E[ku(r) − u(t j )kσ ] dr

≤C

N−1 Z t j+1

|r − t j | p dr

∑

j=0 t j

∑

j=0 t j p

2p

≤ Ck .

By Lemma 2.2 and Hölder’s inequality, we have kI2n kσ

≤

n−1 Z t j+1

∑

j=0 t j

kS(tn − t j )(S(t j − r) − I)V (x)u(t j )kσ dr

n−1 Z t j+1

≤C ∑

|t j − r|1/2kV (x)kσ +1 ku(t j )kσ +1 dr

j=0 t j

∑

≤ Ck3/2 ≤ Ck

j=0

p+1 2p

!1

2p

n−1

ku(t j )k2p σ +1

n−1

∑ ku(t j )kσ2p+1

j=0

!1

n−1

∑1

2p 2p−1

j=0

! 2p−1 2p

2p

,

so that, by Proposition 5.1, p+1 E[ max kI2n k2p σ ] ≤ Ck n=1,...,N

N−1

∑ E[ku(t j )k2p σ +1 ]

j=0

p

≤ Ck . Again using Hölder’s inequality, we have for I3n kI3n kσ ≤

n−1 Z t j+1

∑

j=0 t j

kV (x)(u(t j ) − u j )kσ dr

n−1

≤ Ck ∑ kV (x)kσ ku(t j ) − u j kσ j=0

≤ Ck

1 2p

n−1

∑

j=0

ku(t j ) − u j kσ2p

!1

2p

,

and thus E[ max kI3n k2p σ ] ≤ Ck n=1,...,N

N−1

max ku(tl ) − ul kσ2p ]. ∑ E[l=0,..., j

j=0

When estimating the stochastic integrals we will use the Burkholder inequality and that [r/k]

32


is the integer part of r/k. Using also Hölder’s inequality, and Lemma 5.3, we have for J1n Z t E[ max kJ1n kσ2p ] ≤ E sup k S(t − [r/k]k)(u(r) − u([r/k]k)) dW (r)k2p σ n=1,...,N

0

0≤t≤T

≤ CE

Z

T

0

k(u(t) − u([t/k]k))Q1/2k2L σ 2

E ≤ CkQ1/2 k2p L2σ

" Z

T 0

ku(t) − u([t/k]k)kσ2p dt

≤C

N−1 Z t j+1

E[ku(t) − u(t j )k2p σ ] dt

≤C

N−1 Z t j+1

|t − t j | p dt

∑

j=0 t j

∑

j=0 t j p

dt

p

1p

T

p−1 p

!p#

≤ Ck . Similarly for E[ max

n=1,...,N

J2n ,

we have

kJ2n k2p σ ]≤

E

≤ CE

sup k

0≤t≤T

Z

0

≤ CT p−1 ≤C

T

Z t 0

k(S([t/k]k − t) − I)u([t/k]k)Q1/2k2L σ 2

Z T

dt

p

i h dt E k(S([t/k]k − t) − I)u([t/k]k)Q1/2k2p σ L 2

0 N−1 Z t j+1

∑

S(t − [r/k]k)(S([r/k]k − r) − I)u([r/k]k) dW(r)kσ2p

E[kS(t j − t) − I)u(t j )Q1/2 k2p dt Lσ 2

j=0 t j

≤ CkQ1/2 k2pσ +1 L2

N−1 Z t j+1

∑

j=0 t j

|t j − t| pE[ku(t j )k2p σ +1 ] dt

≤ Ck p . For J3n , we have E[ max

n=1,...,N

kJ3n kσ2p]

≤ CE ≤ CE ≤ CT

sup k

0≤t≤T

Z

p−1

Z t 0

S(t − [r/k]k)(u([r/k]k) − u

T 0

k(u([t/k]k) − u

kQ1/2 k2p L2σ

[t/k]k

N−1 Z t j+1

∑

j=0 t j

)Q1/2 k2L σ 2

[r/k]k

dt

) dW (r)k2p σ

p

E[ku(t j ) − u j kσ2p ] dt

N−1

≤ Ck

max ku(tl ) − ul k2p σ ]. ∑ E[l=0,..., j

j=0

Putting all these estimates together yields E[ max kun − u(tn )kσ2p ] ≤ C1 k p + C2 k n=1,...,N

and Gronwall’s lemma completes the proof.

N−1

max kul − u(tl )kσ2p ], ∑ E[l=0,..., j

j=0

33

Exponential integrators for stochastic Schrödinger equations 10

10

Error

Ms = 750000

0

-1

10 -2

10

-3

Slope 1 SEXP CN BEM

10 -4 -3 10

10

-2

10

-1

10

0

k

F IGURE 5.1. Stochastic Schrödinger equation with multiplicative noise: Mean-square errors for the stochastic exponential integrator (SEXP), the Crank-Nicolson scheme (CN), and the semi-implicit Euler-Maruyama (SEM). The dotted line has slope 1.

5.2. Numerical experiments for Schrödinger equations with a multiplicative noise. This subsection illustrates the convergence properties of the stochastic exponential integrator (5.2) when applied to the stochastic partial differential equation (5.1) on the interval [0, 2π ] with periodic boundary conditions. In these numerical experiments, we set V (x) = 0 and 2 u0 (x) = e−5(x−π ) . We compare the stochastic exponential integrator (SEXP) with the CrankNicolson scheme (CN) and the semi-implicit Euler-Maruyama scheme (SEM). Figure 5.1 illustrates the convergence errors of the above numerical methods for a noise with covariance operator having the eigenvalues λn = 1/(1 + |n|5.1) for n ∈ Z. The spatial discretisation is done by a pseudospectral method with M = 28 Fourier modes. The rates of mean-square convergence (measured in the L2 -norm at the end of the interval of integration [0, 0.5]) of these numerical methods are presented in this figure. The expected rate of convergence O(k1 ) of the stochastic exponential integrator, as stated in Theorem 5.5, can be confirmed. Here, the exact solution is approximated by the stochastic exponential method with a very small time step kexact = 2−10 and Ms = 750000 samples are used for the approximation of the expected values. REFERENCES [1] R. A NTON , D. C OHEN , S. L ARSSON , AND X. WANG, Full discretisation of semi-linear stochastic wave equations driven by multiplicative noise, Submitted for publication, (2015). [2] M. BARTON -S MITH , A. D EBUSSCHE , AND L. D I M ENZA, Numerical study of two-dimensional stochastic NLS equations, Numer. Methods Partial Differential Equations, 21 (2005), pp. 810–842. [3] H. B ERLAND , A. L. I SLAS , AND C. M. S CHOBER, Conservation of phase space properties using exponential integrators on the cubic Schrödinger equation, J. Comput. Phys., 225 (2007), pp. 284–299. [4] H. B ERLAND , B. O WREN , AND B. S KAFLESTAD, Solving the nonlinear SchrÃ˝udinger equation using exponential integrators, Modeling, Identification and Control, 27 (2006), pp. 201–218. [5] B. C ANO AND A. G ONZÁLEZ -PACHÓN, Exponential time integration of solitary waves of cubic Schrödinger equation, Appl. Numer. Math., 91 (2015), pp. 26–45. [6] T. C AZENAVE AND A. H ARAUX, An Introduction to Semilinear Evolution Equations, vol. 13 of Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press, Oxford University Press, New York, 1998. Translated from the 1990 French original by Yvan Martel and revised by the authors.

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