1

Order of convergence of splitting schemes for bothdeterministic and stochastic nonlinear Schrodinger

equations

Jie Liu

National University of Singapore, Singapore

Numerics for Stochastic Partial Differential Equations and theirApplications, RICAM, Linz, Dec. 2016.

2

Outline

• The second order convergence of Strang-type splitting scheme fornonlinear Schrodinger equation.

• Mass preserving splitting scheme for stochastic nonlinear Schrodingerequation with multiplicative noise

? Explicit formula for the nonlinear step

? first order strong convergence

3

Strang-type splitting scheme for nonlinear Schrodingerequation

Let i =√−1 and V be a real-valued function. Consider the following

nonlinear Schrodinger equation

idu = ∆udt+ V (x, |u|)udt in Rd. (1)

Note three things:

• If a ∈ R, idudt = au ⇒ u(t) = e−iatu(0). Hence |u(t)| = |u(0)|.

• The exact solution of idu = V (x, |u|)udt, u(0) = u0 satisfies |u(t)| =|u(0)| and therefore u(x, t) = exp {−itV (x, |u0(x)|)}u0(x).

• Recall the Strang splitting for du = (Au+Bu)dt:

u(δt) = eδt(A+B)u(0) ≈ eδt2 BeδtAeδt2 Bu(0).

4

We will study the following scheme 1

un−1 → un−12 → ◦

un−12 → un → un+1

2 → ◦un+1

2 → un+1 → · · · :

un−12 = exp

{−iδt

2V (x, |un−1|)

}un−1, (2)

◦un−

12 = exp {−iδt∆} un−1

2, (3)

un = exp

{−iδt

2V (x, | ◦un−1

2|)}◦un−

12. (4)

It can be implemented as◦un−

12 → un+1

2 → ◦un+1

2 (skip un) because

un+12 = exp

{−iδt

2V (x, |un|)

}un = exp

{−iδtV (x, | ◦un−1

2|)}◦un−

12.

The second order convergence in L2 norm has been proved by Lubich2.

1Hardin & Tappert 1973, Taha & Ablowitz 1984.2C. Lubich, On splitting methods for Schrodinger-Poisson and cubic nonlinear Schrodinger equations.

Math. Comp., 77 (2008) 2141–2153.

5

Second order convergence

Theorem 1. Consider (1) and its numerical scheme (2)–(4) inR3. Assume V (x, |u|) = |u|2, and take any γ > 3/2 and anyσ ∈ {0, 1, 2, 3, ...}. If ‖u0‖γ = Mγ < ∞, then there are constants Tand C1 depending on Mγ such that for any δt > 0,

maxn=0,1,...,[T/δt]

‖un‖γ ≤ C1. (5)

If ‖u0‖σ+4 = Mσ+4 < ∞, then there are constants T and C2 dependingon Mσ+4 such that for any δt > 0,

maxn=0,1,...,[T/δt]

‖u(tn)− un‖σ ≤ C2δt2. (6)

Assume in addition that there are constants T and Mσ+4 so thatsup0≤t≤T ‖u(s)‖σ+4 = Mσ+4 < ∞, then there is a constant δt0 > 0so that when δt ≤ δt0, the T in (6) can be taken as T .

6

Integral formulation of the scheme

Fix T , δt, and N = [T/δt]. Define a right continuous function φN(x, t)with

• φN(x, 0) = u0(x).

• φN(t) = e{−i[tV (x,|φN(0)|)]}φN(0) when t ∈ [0, δt2 )

• On any interval [tn−12, tn+1

2) (n = 1, 2, ...),

φN(t) =

e{−iδt∆} lim

t↑tn−12φN(t) when t = tn−

12,

e{−i

[(t−tn−

12)V (x,|φN(tn−

12)|)

]}φN(tn−

12) when t ∈ [tn−

12, tn+1

2).

ThenφN(tn−

12) =

◦un−

12 and φN(tn) = un.

7

Integral formulation of the scheme

Recall φN (t) =

e{−iδt∆} lim

t↑tn−12φN (t) when t = t

n−12 ,

e

{−i[

(t−tn−12)V (x,|φN (t

n−12)|)

]}φN (t

n−12) when t ∈ [t

n−12 , t

n+12).

• When t ∈ [tn−12, tn+1

2), dφN(t) = −iV (x, |φN(t)|)φN(t)dt andtherefore

φN(t) = φN(tn−12)− i

∫ t

tn−12

V (x, |φN(s)|)φN(s)ds. (7)

• When t = tn+12,

φN(tn+12) = e{−iδt∆}

φN(tn−12)− i

∫ tn+12

tn−12

V (x, |φN(s)|)φN(s)ds

.(8)

8

Integral formulation of the scheme

φN satisfies

φN(t) = e−itn∆φN(0)− i

∫ t

0

SN,n,t(s)(V (x, |φN(s)|)φN(s)

)ds

when t ∈ [tn−12, tn+1

2), where φN(0) = u(0),SN,n,t(s) = e−inδt∆I

[0,t12 ]

(s) +∑n−1j=1 I[tn−

12−j,tn+1

2−j](s)e−i(jδt)∆ + I

[tn−12 ,t]

(s).

Let S(t− s) = e−i(t−s)∆. The exact solution u(t) of NLS satisfies

u(t) = e−it∆u(0)− i∫ t

0

S(t− s) (V (x, |u(s)|)u(s)) ds.

Note that∫ tn

0SN,n,tn(s)ds is the midpoint rule approximation of∫ tn

0S(tn − s)ds on the partition

∫ t120

+∑n−1j=1

∫ tn+12−j

tn−12−j

+∫ ttn−

12

9

Error equation

Let r(t) = u(t)− φN(t). We have

r(t) =(e−it∆ − e−itn∆)u0

− i∫ t

0

(S(t− s)− SN,n,t(s))V (x, |φN(s)|)φN(s)ds

− i∫ t

0

S(t− s) [V (x, |u(s)|)u(s)− V (x, |φN(s)|)φN(s)] ds

=:J1(t) + J2(t) + J3(t) (9)

for t ∈ [tn−12, tn+1

2) (but n is arbitrary). Note that only at t = tn, J1(tn)vanishes instead of being of size O(δt).

10

Error estimate

• Step 1: (First order error estimate over a short time interval)Since ‖fg‖γ ≤ c‖f‖γ‖g‖γ when γ > d/2 and ‖e−it∆w‖σ = ‖w‖σ,‖(e−it∆ − I)w‖α = t‖∆w‖α, one immediately get supt≤T ‖φN(t)‖4 <C and

supt≤T‖r(t)‖2 ≤ Cδt

for some T > 0. Here r(t) = u(t)− φN(t), ‖r‖α = ‖r‖Hα

• Step 2: (Error estimate over a short time interval) There is a constantT which depends on the initial data so that

‖r(tn)‖0 ≤ C3δt

n−1∑j=1

‖r(tj)‖0 + C5δt2

for all n ≤ [T/δt]. So, second order convergence follows from thestandard discrete Gronwall inequality.

11

Step 3: (Error estimate up to the blowing up time of the exact solution)Note that to prove the 2nd order convergence in L2, one has to provethe stability in H4:

By Calculus inequality in the Sobolev space,

‖|φN |2φN‖4 ≤ C‖φN‖22‖φN‖4. (10)

Hence

‖φN(t)‖4 ≤‖φN(0)‖4

+ C

∫ t

0

(‖u(s)‖2 + ‖φN(s)− u(s)‖2)2 ‖φN(s)‖4ds.

12

Stochastic Schrodinger equation

Consider the stochastic Schrodinger equation with multiplicative noise:

idv = ∆vdt+ V (x, |v|)vdt+ v ◦ dW in Rd, (11)

where v is a complex valued function, i =√−1, W is a real valued

Wiener process, ◦ means Stratonovich product, V is also real valued. Let{βk : k ∈ N} be a sequence of independent Brownian motions that areassociated with {Ft : t ≥ 0}. Let {ek : k ∈ N} be an orthonormal basisof L2(Rd,R). Then

W =∑k

βk(t, ω)ek(x)

andW = ΦW =

∑k

βk(t, ω)(Φek(x)).

Φ is a Hilbert-Schmidt operator from L2 to Hα 3.3To present our scheme, we need α = 0. To prove stability and convergence, we will need larger α.

13

Applications of Stochastic NLS

•• It can be used to model the wave propagation in nonhomogeneous orrandom media4. (Prof. Solna’s talk on Tuesday)

• It has been introduced by Bang etc5 as a model for molecularmonolayers arranged in Scheibe aggregates with thermal fluctuationsof the phonons.

• It has also been widely used in quantum trajectory theory6, whichdetermines the evolution of the state of a continuously measuredquantum system (e.g. the continuous monitoring of an atom by thedetection of its fluorescence light).

4V. Konotop, and L. Vazquez, Nonlinear random waves, World Scientific, NJ, 1994.5O. Bang, P.L. Christiansen, F. If, K. Ø. Rasmussen and Y. B. Gaididei, Temperature effects in a

nonlinear model of monolayer Scheibe aggregates, Phys. Rev. E, 49 (1994) 4627–4636.6A. Barchielli and M. Gregoratti, Quantum trajectories and measurements in continuous time: the

diffusive case, Lecture Notes in Physics 782, Springer, Berlin, 2009.

14

20 2 The Stochastic Schrodinger Equation

‖ψ(t)‖2 = ‖ψ0‖2 exp

{∑j

[∫ t

0m j (s)dW j (s) − 1

2

∫ t

0m j (s)2ds

]}. (2.24)

Moreover, ∀p ≥ 1,

sup0≤t≤T

EQ[‖ψ(t)‖2p

] ≤ EQ

[sup

0≤t≤T‖ψ(t)‖2p

]< +∞. (2.25)

Proof. Being an Ito process, ψ is continuous and this holds for its square norm.By Assumption 2.10 and the definitions (2.22), (2.23), equation (2.17) reduces to

‖ψ(t)‖2 = ‖ψ0‖2 +∑

j

∫ t

0m j (s) ‖ψ(s)‖2 dW j (s). (2.26)

By Proposition 2.9, the positive continuous process ‖ψ(t)‖2 is a square-integrablemartingale. By taking m as given, (2.26) is a Doleans equation whose solution isunique and given by (2.24) (cf. Proposition A.41 and (Eqs. (A.23), (A.24), (A.25),(A.26)).

By inequality (2.19), we have

∑j

m j (t)2 ≤ 4

∥∥∥∑j R j (t)∗ R j (t)

∥∥∥ ,

∫ T

0

∑j

m j (t)2dt ≤ 4 sup

t∈[0,T ]

∥∥∥∑j R j (t)∗ R j (t)

∥∥∥ T . (2.27)

Then, the last statement follows from Proposition A.42.

In the following, we shall call linear stochastic Schrodinger equation the originalSDE (2.2) for an H-valued process ψ under all Assumptions 2.1, 2.2, 2.3 and 2.10,i.e.

⎧⎪⎪⎨⎪⎪⎩

dψ(t) =⎛⎝−iH (t) − 1

2

d∑j=1

R j (t)∗ R j (t)

⎞⎠ψ(t) dt +

d∑j=1

R j (t)ψ(t) dW j (t) ,

ψ(0) = ψ0 , ψ0 ∈ H.

(2.28)

Of course, the solution is the continuous, adapted stochastic process ψ(t) =A0

t ψ0, where the stochastic evolution operator Ast and its adjoint As

t∗ still satisfy the

SDEs (2.7) and (2.11) with K (t) = −iH (t)− 1

2

d∑j=1

R j (t)∗ R j (t) and H (t) = H (t)∗.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Quantum Open Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Approaches to Continuous Measurements . . . . . . . . . . . . . . . . . . . . 21.3 Classical SDEs in Continuous Measurement Theory . . . . . . . . . . . 31.4 The Plan of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Part I General Theory

2 The Stochastic Schrodinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Linear Stochastic Differential Equations . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 An Homogeneous Linear SDE in Hilbert Space . . . . . . . . 132.2.2 The Stochastic Evolution Operator . . . . . . . . . . . . . . . . . . . 142.2.3 The Square Norm of the Solution . . . . . . . . . . . . . . . . . . . . 17

2.3 The Linear Stochastic Schrodinger Equation . . . . . . . . . . . . . . . . . . 192.3.1 A Key Restriction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3.2 A Change of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4 The Physical Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.4.1 The POM of the Output and the Physical Probabilities . . 232.4.2 The A Posteriori States . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.4.3 Infinite Time Horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.4.4 The Conservative Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.5 The Stochastic Schrodinger Equation . . . . . . . . . . . . . . . . . . . . . . . . 282.5.1 The Stochastic Differential of the A Posteriori State . . . . 282.5.2 Four Stochastic Schrodinger Equations . . . . . . . . . . . . . . . 302.5.3 Existence and Uniqueness of the Solution . . . . . . . . . . . . . 332.5.4 The Stochastic Schrodinger Equation as a Starting Point 38

2.6 The Linear Approach Versus the Nonlinear One . . . . . . . . . . . . . . . 402.7 Tricks to Simplify the Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.7.1 Time-Dependent Coefficients and UnitaryTransformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

ix

15

Known results

• A. de Bouard and A. Debussche, A stochastic nonlinear Schrodinger equation with

multiplicative noise, Commun. Math. Phys. 205 (1999) 161-181.

• A. de Bouard and A. Debussche, A semi-discrete scheme for the stochastic nonlinear

Schrodinger equation, Numerische Math. 96 (2004) 733–770.

• A. de Bouard and A. Debussche, Weak and strong order of convergence of a

semi discrete scheme for the stochastic nonlinear Schrodinger equation, Applied

Mathematics and Optimization, 54 (2006) 369–399.

• R. Marty, On a splitting scheme for the nonlinear Schrodinger equation in a random

medium. Commun. Math. Sci., 4 (2006) 679–705.

• Z. Brzezniak and A. Millet, On the splitting method for Schrodinger-like evolution

equations. Stochastic Analysis and Related Topics, 2012 57–90.

16

Exact formula for the nonlinear part

Let us for the moment drop the ∆v term in stochastic NLS:

idv = V (x, |v|)vdt+ v ◦ dW (t). (12)

For the above equation, we have the following observation:

Theorem 2. The following v(x, t) is a solution of (12)

v(x, t) = v(x, 0) exp {−i[tV (x, |v(x, 0)|) +W (x, t)]} . (13)

In particular, |v(x, t)| = |v(x, 0)| and V (x, |v(x, t)|) = V (x, |v(x, 0)|).

17

Mass preserving splitting scheme

It is now clear that we can define the following splitting schemeinitialized with v0 = v(x, 0):

vn = exp{−i[δtV (x, |vn−1|) +

(W (x, tn)−W (x, tn−1)

)]}vn−1, (14)

vn = exp {−iδt∆} vn. (15)

It is clear ‖vn‖2L2 = ‖vn‖2

L2 = ‖vn−1‖2L2. We summarize this property

into the following Theorem:

Theorem 3. The scheme (14)–(15) is unconditionally stable and is masspreserving, i.e, for any n

‖vn‖2L2 = ‖v0‖2L2.

18

First order strong convergence

Theorem 4. Consider (11) and the scheme (14)–(15) in Rd. AssumeV (x, |v|) = |v|2. Take any integer γ > d/2 and assume Φ ∈L2(L2, Hγ+2). Assume the F0-measurable initial data v0 satisfies(E‖v0‖p

Hγ+2

)1/p< Mγ+2 < ∞ uniformly for p ≥ 1. Then, for any

L > 0 with associated stopping time τL = inf{t, ‖v(t)‖γ ≥ L}, we have

limK→∞

P(δt−1 max

n=0,1,...,[τL/δt]‖v(tn)− vn‖γ ≥ K

)= 0. (16)

Moreover, for any ε > 0, there exists a random variable KL,ε with finiteE|KL,ε|q for any q ≥ 1, such that

maxn=0,1,...,[τL/δt]

‖v(tn)− vn‖γ ≤ KL,εδt1−ε. (17)

19

Integral representation of the scheme

Fix T , δt and N = [T/δt]. Introduce a right continuous functionψN(x, t) with ψN(x, 0) = v(x, 0) = v0 and on any interval [tn−1, tn],

ψN(t) =

e{−i[(t−t

n−1)V (x,|ψN(tn−1)|)+(W (t)−W (tn−1))]}

×ψN(tn−1) t ∈ [tn−1, tn),

e{−iδt∆} limt↑tn ψN(t) t = tn.

We have ψN(tn) = vn

20

Integral representation of the scheme

ψN(t) = e{−itn−1∆}ψN(0)− i

∫ t

0

SN,n,t(s)×(V (x, |ψN(s)|)ψN(s)ds+ ψN(s)dW (s)

),

where

SN,n,t(s) =

n−1∑j=1

I[tn−1−j,tn−j](s)e−i(jδt)∆ + I[tn−1,t](s). (18)

21

Truncated stochastic Schrodinger equation

Because we do not have E|X|3 ≤ C(E|X|)3, the previous argumentcannot be applied here. To prove error estimate, we need to study thefollowing truncated equation (motivated the works of Profs. de Bouard& Debussche and Prof. Gyongy and his co-workers)

idvR = ∆vRdt+ VR(x, vR)vRdt+ vRdW in Rd, (19)

with initial condition vR(x, 0) = v0(x). Here

θR(w) = θ

(‖w‖γR

),

VR(x,w) = θR(w)V (x, |w|),

VR(x,w) = VR(x,w)− i

2FΦ.

θ ∈ C∞(R) is a cut-off function satisfying θ(x) = 1 for x ∈ [0, 1] andθ(x) = 0 for x ≥ 2. γ is any integer > d/2.

22

Step 1: Stability of the truncated stochastic NLS

One can prove that even though we only truncate in the Hγ norm in vR,the solution automatically becomes Hγ+2-regular:

Lemma 1. (de Bouard and Debussche) Take any integer γ > d/2 andconsider (19) in Rd with VR(x, v) = θ (‖v‖γ/R) |v|2 − i

2FΦ. AssumeΦ ∈ L2(L2, Hγ+2). Take any T > 0 and any p ≥ 1, for any F0-measurable initial data v0 satisfying E‖v0‖pγ+2 = Mγ+2 < ∞, there is aconstant MR depending on R, such that

E supt≤T‖vR(t)‖pγ+2 ≤MR. (20)

Once again. the proof uses ‖|vR|2vR‖γ+2 ≤ C‖vR‖2γ‖vR‖γ+2.

23

Step 2: splitting scheme for the truncated stochasticNLS

Proposition 1. Take any integer γ > d/2 and consider the splittingscheme in Rd with VR(x, v) = θR(v)|v|2. Assume Φ ∈ L2(L2, Hγ+2).For any R, T > 0 and any p ≥ 1, for any F0-measurable initial datav0 satisfying E‖v0‖pγ+2 ≤ Mp,γ+2 < ∞, there is a constant CR whichdepends on R, T, p,Mp,γ+2 and ‖Φ‖L2(L2,Hγ+2) so that

E maxn=0,1,...,[T/δt]

‖vR(tn)− vnR‖pγ ≤ CRδtp. (21)

Proof: Let r(t) = vR(t)− ψRN(t). Then

r(t) =

7∑k=1

Ik(t), (22)

24

where

I1(t) =(e{−i(t−t

n−1)∆} − 1)e{−it

n−1∆}v0, (23)

I2(t) =− i∫ t

0

SN,n,t(s)(VR(x, vR)vR(s)− VR(x, ψRN)ψRN(s)

)ds, (24)

I3(t) =− 1

2

∫ t

0

SN,n,t(s)r(s)FΦds, (25)

I4(t) =− i∫ t

0

(S(t− s)− SN,n,t(s))VR(x, vR)vR(s)ds, (26)

I5(t) =− 1

2

∫ t

0

(S(t− s)− SN,n,t(s)) vR(s)FΦds, (27)

I6(t) =− i∫ t

0

SN,n,t(s)r(s)dW (s), (28)

I7(t) =− i∫ t

0

(S(t− s)− SN,n,t(s)) vR(s)dW (s). (29)

25

Finally,

E sup0≤τ≤t

‖r(τ)‖pγ ≤ C1

∫ t

0

(E sup

0≤τ≤s‖r(τ)‖pγ

)ds+ C2δt

p.

The C1 term comes from the estimates of I2, I3, I6 and the C2 termcomes from the estimates of I1, I4, I5, I7.

26

Step 3: from truncated equation to un-truncatedequation

(Follows the idea of Gyongy and Nualart.) Next, for any σ ≥ 0, L > 0,we define the stopping time τL = inf{t, ‖v(t)‖σ ≥ L}. We then provethat for any ε ∈ (0, 1){

max0≤n≤[τL/δt]

‖vn − v(tn)‖σ ≥ ε}

⊂

{sup

0≤t≤τL‖v(t)‖σ ≥ R− 1

}∪{

max0≤n≤[τL/δt]

‖vnR − vR(tn)‖σ ≥ ε}(30)

for any R > 0. This follows from A ⊂ B ∪ {B ∩A}.

(30) allows us to prove the convergence in probability before τL for thesplitting scheme for the un-truncated stochastic NLS.

27

First order strong convergence

Theorem 5. Consider (11) and the scheme (14)–(15) in Rd. AssumeV (x, |v|) = |v|2. Take any integer γ > d/2 and assume Φ ∈L2(L2, Hγ+2). Assume the F0-measurable initial data v0 satisfies(E‖v0‖p

Hγ+2

)1/p< Mγ+2 < ∞ uniformly for p ≥ 1. Then, for any

L > 0 with associated stopping time τL = inf{t, ‖v(t)‖γ ≥ L}, we have

limK→∞

P(δt−1 max

n=0,1,...,[τL/δt]‖v(tn)− vn‖γ ≥ K

)= 0. (31)

Moreover, for any ε > 0, there exists a random variable KL,ε with finiteE|KL,ε|q for any q ≥ 1, such that

maxn=0,1,...,[τL/δt]

‖v(tn)− vn‖γ ≤ KL,εδt1−ε. (32)