[PPT] - Order of convergence of splitting schemes for both deterministic and PowerPoint Presentation

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Order of convergence of splitting schemes for both deterministic and stochastic nonlinear Schr¨

dinger

equations

Jie Liu National University of Singapore, Singapore Numerics for Stochastic Partial Differential Equations and their Applications, RICAM, Linz, Dec. 2016.

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Outline

The second order convergence of Strang-type splitting scheme for

nonlinear Schr¨

dinger equation.
Mass preserving splitting scheme for stochastic nonlinear Schr¨
dinger

equation with multiplicative noise ⋆ Explicit formula for the nonlinear step ⋆ first order strong convergence

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Strang-type splitting scheme for nonlinear Schr¨

dinger

equation

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

dinger equation

idu = ∆udt + V (x, |u|)udt in Rd. (1) Note three things:

If a ∈ R, idu

dt = 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) ≈ e

δt 2 BeδtAe δt 2 Bu(0).

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We will study the following scheme 1 un−1 → ˜ un−1

2 →

un−1

2 → un → ˜

un+1

2 →

un+1

2 → un+1 → · · · :

˜ un−1

2 = exp

−iδt

2 V (x, |un−1|)

un−1,

(2)

un−1

2 = exp {−iδt∆} ˜

un−1

2,

(3) un = exp

−iδt

2 V (x, |

un−1

2|)

un−1

2.

(4) It can be implemented as

un−1

2 → ˜

un+1

2 →

un+1

2 (skip un) because

˜ un+1

2 = exp

−iδt

2 V (x, |un|)

un = exp
−iδtV (x, |
un−1

2|)

un−1

2.

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

1Hardin & Tappert 1973, Taha & Ablowitz 1984.

2C. Lubich, On splitting methods for Schr¨
dinger-Poisson and cubic nonlinear Schr¨
dinger equations.
Math. Comp., 77 (2008) 2141–2153.

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Second order convergence

Theorem 1. Consider (1) and its numerical scheme (2)–(4) in R3. Assume V (x, |u|) = |u|2, and take any γ > 3/2 and any σ ∈ {0, 1, 2, 3, ...}. If u0γ = Mγ < ∞, then there are constants T and C1 depending on Mγ such that for any δt > 0, max

n=0,1,...,[T /δt] unγ ≤ C1.

(5) If u0σ+4 = Mσ+4 < ∞, then there are constants T and C2 depending

n Mσ+4 such that for any δt > 0,

max

n=0,1,...,[T /δt] u(tn) − unσ ≤ C2δt2.

(6) Assume in addition that there are constants ¯ T and ¯ Mσ+4 so that sup0≤t≤ ¯

T u(s)σ+4 =

¯ Mσ+4 < ∞, then there is a constant δt0 > 0 so that when δt ≤ δt0, the T in (6) can be taken as ¯ T.

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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, δt

2 )

On any interval [tn−1

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

φN(t) =      e{−iδt∆} lim

t↑tn−1

2 φN(t)

when t = tn−1

2,

e

{−i

(t−tn−1

2)V (x,|φN(tn−1 2)|)

}φN(tn−1

2)

when t ∈ [tn−1

2, tn+1 2).

Then φN(tn−1

2) =

un−1

2

and φN(tn) = un.

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Integral formulation of the scheme

Recall φN(t) =              e{−iδt∆} lim t↑tn−1 2 φN(t) when t = tn−1 2, e {−i

(t−tn−1

2)V (x,|φN(tn−1 2)|)

}

φN(tn−1 2) when t ∈ [tn−1 2, tn+1 2).

When t ∈ [tn−1

2, tn+1 2),

dφN(t) = −iV (x, |φN(t)|)φN(t)dt and therefore φN(t) = φN(tn−1

2) − i

t

tn−1

2

V (x, |φN(s)|)φN(s)ds. (7)

When t = tn+1

2,

φN(tn+1

2) = e{−iδt∆}

 φN(tn−1

2) − i

tn+1

2

tn−1

2

V (x, |φN(s)|)φN(s)ds   . (8)

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Integral formulation of the scheme

φN satisfies φN(t) = e−itn∆φN(0) − i t SN,n,t(s)

V (x, |φN(s)|)φN(s)
ds

when t ∈ [tn−1

2, tn+1 2), where φN(0) = u(0),

SN,n,t(s) = e−inδt∆I

[0,t

1 2](s) + n−1

j=1 I [tn−1

2−j,tn+1 2−j](s)e−i(jδt)∆ + I

[tn−1

2,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 S(t − s) (V (x, |u(s)|)u(s)) ds. Note that tn

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

tn

0 S(tn − s)ds on the partition

t

1 2

0 + n−1 j=1

tn+1

2−j

tn−1

2−j +

t

tn−1

2

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Error equation

Let r(t) = u(t) − φN(t). We have r(t) =(e−it∆ − e−itn∆)u0 − i t (S(t − s) − SN,n,t(s)) V (x, |φN(s)|)φN(s)ds − i t 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−1

2, tn+1 2) (but n is arbitrary). Note that only at t = tn, J1(tn)

vanishes instead of being of size O(δt).

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Error estimate

Step 1:

(First order error estimate over a short time interval) Since fgγ ≤ cfγgγ when γ > d/2 and e−it∆wσ = wσ, (e−it∆ − I)wα = t∆wα, one immediately get supt≤T φN(t)4 < C and sup

t≤T

r(t)2 ≤ Cδt for some T > 0. Here r(t) = u(t) − φN(t), rα = rHα

Step 2: (Error estimate over a short time interval) There is a constant

T 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 the standard discrete Gronwall inequality.

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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 prove the stability in H4: By Calculus inequality in the Sobolev space, |φN|2φN4 ≤ CφN2

2φN4.

(10) Hence φN(t)4 ≤φN(0)4 + C t (u(s)2 + φN(s) − u(s)2)2 φN(s)4ds.

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Stochastic Schr¨

dinger equation

Consider the stochastic Schr¨

dinger 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 are associated with {Ft : t ≥ 0}. Let {ek : k ∈ N} be an orthonormal basis

f L2(Rd, R). Then

ˆ W =

k

βk(t, ω)ek(x) and W = Φ ˆ W =

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.

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Applications of Stochastic NLS

It can be used to model the wave propagation in nonhomogeneous or

random media4. (Prof. Solna’s talk on Tuesday)

It has been introduced by Bang etc5 as a model for molecular

monolayers arranged in Scheibe aggregates with thermal fluctuations

f the phonons.
It has also been widely used in quantum trajectory theory6, which

determines the evolution of the state of a continuously measured quantum system (e.g. the continuous monitoring of an atom by the detection of its fluorescence light).

4V. Konotop, and L. V´

azquez, 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.

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⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 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)

2 The Stochastic Schr¨

dinger Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Linear Stochastic Differential Equations . . . . . . . . . . . . . . . . . . . . . 12 2.2.1 An Homogeneous Linear SDE in Hilbert Space . . . . . . . . 13 2.2.2 The Stochastic Evolution Operator. . . . . . . . . . . . . . . . . . . 14 2.2.3 The Square Norm of the Solution . . . . . . . . . . . . . . . . . . . . 17 2.3 The Linear Stochastic Schr¨

dinger Equation . . . . . . . . . . . . . . . . . . 19

2.3.1 A Key Restriction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3.2 A Change of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.4 The Physical Interpretation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.4.1 The POM of the Output and the Physical Probabilities . . 23 2.4.2 The A Posteriori States . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.4.3 Infinite Time Horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.4.4 The Conservative Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.5 The Stochastic Schr¨

dinger Equation . . . . . . . . . . . . . . . . . . . . . . . . 28

2.5.1 The Stochastic Differential of the A Posteriori State . . . . 28 2.5.2 Four Stochastic Schr¨

dinger Equations . . . . . . . . . . . . . . . 30

2.5.3 Existence and Uniqueness of the Solution . . . . . . . . . . . . . 33 2.5.4 The Stochastic Schr¨

dinger Equation as a Starting Point

38 2.6 The Linear Approach Versus the Nonlinear One . . . . . . . . . . . . . . . 40 2.7 Tricks to Simplify the Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.7.1 Time-Dependent Coefficients and Unitary Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

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Known results

A. de Bouard and A. Debussche, A stochastic nonlinear Schr¨
dinger 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

Schr¨

dinger 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 Schr¨

dinger equation, Applied

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

R. Marty, On a splitting scheme for the nonlinear Schr¨
dinger 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.

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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)|).

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Mass preserving splitting scheme

It is now clear that we can define the following splitting scheme initialized 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 vn2

L2 = ˜

vn2

L2 = vn−12

L2. We summarize this property

into the following Theorem: Theorem 3. The scheme (14)–(15) is unconditionally stable and is mass preserving, i.e, for any n vn2

L2 = v02 L2.

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First order strong convergence

Theorem 4. Consider (11) and the scheme (14)–(15) in Rd. Assume V (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

Ev0p

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 lim

K→∞ P

δt−1

max

n=0,1,...,[τL/δt] v(tn) − vnγ ≥ K

= 0.

(16) Moreover, for any ε > 0, there exists a random variable KL,ε with finite E|KL,ε|q for any q ≥ 1, such that max

n=0,1,...,[τL/δt] v(tn) − vnγ ≤ KL,εδt1−ε.

(17)

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Integral representation of the scheme

Fix T, δt and N = [T/δt]. Introduce a right continuous function ψN(x, t) with ψN(x, 0) = v(x, 0) = v0 and on any interval [tn−1, tn], ψN(t) =          e{−i[(t−tn−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

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Integral representation of the scheme

ψN(t) = e{−itn−1∆}ψN(0) − i t 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)

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Truncated stochastic Schrodinger equation

Because we do not have E|X|3 ≤ C(E|X|)3, the previous argument cannot be applied here. To prove error estimate, we need to study the following truncated equation (motivated the works of Profs. de Bouard & Debussche and Prof. Gy¨

ngy 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 wγ R

,

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] and θ(x) = 0 for x ≥ 2. γ is any integer > d/2.

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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 and consider (19) in Rd with ¯ VR(x, v) = θ (vγ/R) |v|2 − i

2FΦ.

Assume Φ ∈ L2(L2, Hγ+2). Take any T > 0 and any p ≥ 1, for any F0- measurable initial data v0 satisfying Ev0p

γ+2 = Mγ+2 < ∞, there is a

constant MR depending on R, such that E sup

t≤T

vR(t)p

γ+2 ≤ MR.

(20) Once again. the proof uses |vR|2vRγ+2 ≤ CvR2

γvRγ+2.

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Step 2: splitting scheme for the truncated stochastic NLS

Proposition 1. Take any integer γ > d/2 and consider the splitting scheme 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 data v0 satisfying Ev0p

γ+2 ≤ Mp,γ+2 < ∞, there is a constant CR which

depends on R, T, p, Mp,γ+2 and ΦL2(L2,Hγ+2) so that E max

n=0,1,...,[T/δt] vR(tn) − vn Rp γ ≤ CRδtp.

(21) Proof: Let r(t) = vR(t) − ψR

N(t). Then

r(t) =

7

k=1

Ik(t), (22)

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where I1(t) =

e{−i(t−tn−1)∆} − 1
e{−itn−1∆}v0,

(23) I2(t) = − i t SN,n,t(s)

VR(x, vR)vR(s) − VR(x, ψR

N)ψR N(s)

ds,

(24) I3(t) = − 1 2 t SN,n,t(s)r(s)FΦds, (25) I4(t) = − i t (S(t − s) − SN,n,t(s)) VR(x, vR)vR(s)ds, (26) I5(t) = − 1 2 t (S(t − s) − SN,n,t(s)) vR(s)FΦds, (27) I6(t) = − i t SN,n,t(s)r(s)dW(s), (28) I7(t) = − i t (S(t − s) − SN,n,t(s)) vR(s)dW(s). (29)

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Finally, E sup

0≤τ≤t

r(τ)p

γ ≤ C1

t

E sup

0≤τ≤s

r(τ)p

γ

ds + C2δtp.

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

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Step 3: from truncated equation to un-truncated equation

(Follows the idea of Gy¨

ngy and Nualart.) Next, for any σ ≥ 0, L > 0,

we define the stopping time τL = inf{t, v(t)σ ≥ L}. We then prove that for any ε ∈ (0, 1)

max

0≤n≤[τL/δt] vn − v(tn)σ ≥ ε

⊂
sup

0≤t≤τL

v(t)σ ≥ R − 1

∪
max

0≤n≤[τL/δt] vn R − 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 the splitting scheme for the un-truncated stochastic NLS.

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First order strong convergence

Theorem 5. Consider (11) and the scheme (14)–(15) in Rd. Assume V (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

Ev0p

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 lim

K→∞ P

δt−1

max

n=0,1,...,[τL/δt] v(tn) − vnγ ≥ K

= 0.