Convergence of the time-discretized monotonic schemes

M2AN, Vol. 41, N^o1, 2007, pp. 77–93 www.edpsciences.org/m2an

DOI: 10.1051/m2an:2007008

CONVERGENCE OF THE TIME-DISCRETIZED MONOTONIC SCHEMES

Julien Salomon

Abstract. Many numerical simulations in (bilinear) quantum control use the monotonically conver- gent Krotov algorithms (introduced by Tannoret al. [Time Dependent Quantum Molecular Dynamics (1992) 347–360]), Zhu and Rabitz [J. Chem. Phys. (1998) 385–391] or their unified form described in Maday and Turinici [J. Chem. Phys. (2003) 8191–8196]. In Madayet al. [Num. Math. (2006) 323–338], a time discretization which preserves the property of monotonicity has been presented. This paper introduces a proof of the convergence of these schemes and some results regarding their rate of convergence.

Mathematics Subject Classification. 49J20, 68W40.

Received: January 2, 2006. Revised: November 3, 2006.

Introduction

Quantum control has recently been subjected to significant developments through encouraging experimental results [7, 15]. At computational level [4], the introduction of monotonic Krotov algorithms (introduced by Tannor et al. in [18]), Zhu and Rabitz [20] or the unified form proposed by Maday and Turinici in [10] has enabled us to design efficient methods being implemented to obtain laser fields that control the molecular dynamics [13]. From a mathematical point of view, it can be proved that these procedures monotonically increase the values of a given criterium. Yet, few results are available about the convergence of the control sequences obtained by these schemes. In [14], a first necessary condition for convergence has been obtained in case of large penalization of the L²-norm of the control. A functional analysis of the schemes has been displayed in [6] in the abstract framework of semi-group theory. To complete these works, a first analysis of a time-discretized version of the monotonic algorithms is presented here.

Let us briefly introduce the model and the corresponding optimal control framework used in this paper.

Consider a quantum system described by its wavefunction ψ =ψ(x, t). The relevant spatial coordinates, de- noted by x, belong to a general space Ω ⊂R^3p, where p is the number of particles considered (the symbolx will often be omitted in the following developments in order to make it simple). Absorbing boundary condi- tions [17] can be used to treat numerically the case of unbounded domains. We assume homogeneous Dirichlet

Keywords and phrases. Quantum control, monotonic schemes, optimal control, Lojasiewicz inequality.

∗ The author acknowledges financial support from the Action Concert´ee Incitative “Simulation Mol´eculaire” of the MENRT (France).

1 Universit´e Pierre et Marie Curie, Paris 6, Laboratoire Jacques-Louis Lions, 175 rue du Chevaleret 75013 Paris, France.

2 Universit´e Paris-Dauphine, Paris 9, CEREMADE, Place du Mar´echal Lattre de Tassigny, 75775 Paris Cedex 16, France.

Julien.Salomon@dauphine.fr

c EDP Sciences, SMAI 2007

Article published by EDP Sciences and available at http://www.edpsciences.org/m2anor http://dx.doi.org/10.1051/m2an:2007008

boundary conditions. The dynamics of this system is characterized by itsinternal Hamiltonian:

H =H₀+V.

In this equationH₀is the kinetic part:

H₀=−1 2

p n=1

1 m_n∆_rn,

where m_n is the mass of the particle n and ∆_rn the Laplace operator with respect to its coordinates. The operatorV =V(x) is the potential part. In case the domain is bounded andV is smooth, the spectrum ofH is discrete, seee.g. [2]. In corresponding numerical simulations, spectral decompositions can be used to discretize the wavefunction.

A way to control such a system is to light it with a laser pulse modelled by a scalar electric field ε(t). The contribution of this laser field is taken into account by introducing a perturbative term in the Hamiltonian which then reads H−µ ε. Like V, the dipolar momentum µ is a function of x. In many models, the value µ(x) = +∞ is only reached for|x|= +∞. Neglecting the interactions with far-off particles, we suppose that the operatorµis bounded onL²(Ω). The evolution ofψis then governed by the Schr¨odinger equation (we are making use of atomic units,i.e. = 1):

i_∂t^∂ψ=Hψ−µ ε ψ

ψ(t= 0) =ψ_init, (1)

whereψ_initis the initial condition forψ, subjected to the constraint: ψinit_L²(Ω)= 1. In numerical simulations, the ground state,i.e. a unitary eigenvector ofH associated to the lowest eigenvalue, is generally taken as the initial state. The Hamiltonian being real, the evolution through (1) preserves theL²-norm ofψso that

ψ(t)_L²_(Ω)= 1. (2) Remark 0.1. The analysis presented here after will be carried out in a general setting and prevails for all space discretizations preserving property (2).

In order to design relevant fields of control, the optimal control framework usually introduces a cost functional to assess their quality. A general example of such functional is given by:

J(ε) =ψ(T)|O|ψ(T) −α _T

0

ε²(t)dt, (3)

whereαandT are two positive real numbers andOis a positive symmetric operator, also calledobservable, that encodes the target: the larger the value ψ(T)|O|ψ(T)is, the better the control objectives are met (here and in what follows, we notef|A|g=

Ωf(x)A g(x)dx, wheref andgare complex valued functions andAa given operator). In this paper, we assume that O is bounded on L²(Ω). This assumption is true in many physical models. However, the analysis presented here after is relevant ifO is only bounded on sub-domains containing the system. The Euler-Lagrange equations corresponding toJ can be written by means of the introduction of a Lagrange multiplierχ, calledadjoint statein what follows. The system obtained by this way is [20]:

⎧⎨

⎩

i_∂t^∂ψ= (H−µ ε)ψ ψ(t= 0) =ψ_init

(4)

⎧⎨

⎩

i_∂t^∂χ= (H−µ ε)χ χ(t=T) =Oψ(T)

(5) α ε(t) =−Imχ(t)|µ|ψ(t).

An efficient tool to solve these equations is provided by the monotonic schemes. A general formulation of these algorithms is given by the following system (see [10]), which is defined recursively:

⎧⎨

⎩

i_∂t^∂ψ^k = (H−µ ε^k)ψ^k ψ^k(t= 0) =ψ_init ε^k(t) = (1−δ)ε^k−1(t)− δ

αImχ^k−1(t)|µ|ψ^k(t)

⎧⎨

⎩

i_∂t^∂χ^k = (H−µε^k)χ^k χ^k(t=T) =O ψ^k(T)

ε^k(t) = (1−η)ε^k(t)−η

αImχ^k(t)|µ|ψ^k(t), (6)

whereδ, η∈[0,2], with (δ, η)= (0,0). A significant property of these schemes is that the cost functional values monotonically increase during the iterations since:

J(ε^k+1)−J(ε^k) =ψ^k+1(T)−ψ^k(T)|O|ψ^k+1(T)−ψ^k(T) +

2 δ−1

_T

0

α(ε^k+1(t)−ε˜^k(t))²dt+ 2

η −1 _T

0

α(˜ε^k(t)−ε^k(t))²dt≥0.

Implicit and explicit relevant time discretizations of these schemes have been designed and tested in [11, 12].

Their main property is to maintain the monotonic character at the discrete level. This paper aims at proving the convergence of these algorithms.

The paper is structured as follows: the time discretizations of the Schr¨odinger equation (1) and the cost functional defined in (3) will be presented in Section 1. In Section 2, we will study the variations in the discrete cost functional with respect toε. In Section 3, we will recall the main features of the implicit time discretized monotonic schemes. Section 4 is dedicated to proving the convergence of these schemes. Finally, we will give some estimates of the rate of convergence in Section 5. Sufficient condition of convergence of the explicit scheme is provided in the appendix.

1. Discrete optimal control setting

For any given integer N, let us introduce the discretization parameter ∆T defined by N∆T =T and ε_j,

˜

ε_j,ψ_j,χ_j that stand respectively for approximations ofε(j∆T), ˜ε(j∆T),ψ(j∆T),χ(j∆T). We denote in the followingε= (ε_j)_{0≤j≤N−1}, ˜ε= (˜ε_j)_{0≤j≤N−1},ψ= (ψ_j)_0≤j≤N andχ= (χ_j)_0≤j≤N.

To solve (1) numerically with enough accuracy, a propagation method based on the Strang’s second-order split-operator technique [16] is generally used (seee.g. [1, 19, 20]). This method also simplifies algebraic manip- ulations as it will appear later in this study (see Eq. (21)).

In this part, for any prescribed sequencesε, ˜ε, the approximations of the stateψand the adjoint stateχare thus defined by the semi-discretized propagation equations:

⎧⎨

⎩

ψ^ε_j+1 = e^H0∆2iTe^{V−iµεj∆T}e^H0∆2iTψ_j^ε ψ₀^ε =ψ_init,

(7)

⎧⎨

⎩

χ^ε_j^˜ = e^−H0∆2iTe^{−V+µ˜i εj∆T}e^−H0∆2iTχ^ε_j+1^˜ χ^ε_N^˜ =Oψ_N^ε.

(8)

Note that actuallyχ^ε˜also depends onε. We omit this dependence for purpose of simplicity. We refer to [12] for more details about full discretization. This discretization of (4) and (5) preserves theL²-norm of the propagated vectors, which means:

∀j= 0...N −1, ψ_j^ε_L²(Ω)=ψinit_L²(Ω)= 1, (9) χ^ε_j^˜_L²(Ω)=χ^ε_N^˜_L²(Ω)≤ O∗, (10) where O∗ denotes the norm of the operatorO overL²(Ω). We consider the following time discretization of the cost functional defined in (3):

J_∆T(ε) =ψ^ε_N|O|ψ_N^ε −α∆T

N−1 j=0

|ε_j|².

Let us also introduce norms onR^N corresponding to the time discretization:

ε₁= ∆T

N−1

j=0

|ε_j|, ε₂=

∆T

N−1

j=0

|ε_j|²¹₂ .

Note that the inner product associated with the norm.2 is defined by:

ε.ε= ∆T

N−1

j=0

ε_j ε_j.

2. Variations in J

In this section, we investigate some variational properties ofJ_∆T.

2.1. Gradient of J

Given a fieldε, the dependence ofψ^ε_N with respect to the field can be explicitly stated with the formula:

ψ_N^ε =

N−1

j=0

e^H0∆2iTe^{−i(V−µεj)∆T}e^H0∆2iT ψ_init.

The computation of the gradient ofJ_∆T is simplified by the introduction of the adjoint stateχ^ε. Indeed, note first that the partial derivative ofψ_N^ε|O|ψ^ε_Nwith respect toε_j0 reads:

∂ψ^ε_N

∂ε_j0 = ∆T

N−1

j=j0+1

e^H0∆2iTe^{V−iµεj∆T}e^H0∆2iT

× e^H0∆2iTe

V−µεj0

i ∆Tiµe^H0∆2iT

×

j0−1 j=0

e^H0∆2iTe^{V−iµεj∆T}e^H0∆2iT ψ_init.

Consequently, the following equations hold true:

∂ψ^ε_N|O|ψ^ε_N

∂ε_j0 = 2 Re ∂ψ^ε_N

∂ε_j0|O|ψ_N^ε

=−2∆TImχ^ε_j0|µ|ψ˘^ε_j0, where:

χ^ε_j= e^H0∆2iTχ^ε_j, ψ˘_j^ε= e^H0∆2iTψ_j^ε.

The gradient of J_∆T is then given by:

∇J_∆T(ε).δε=−2∆T

N−1 j=0

(Imχ^ε_j|µ|ψ˘_j^ε+αε_j)δε_j. (11)

The previous computation allows us to define setC of the critical points ofJ_∆T: C=

ε/ ∀j= 0...N −1, Imχ^ε_j|µ|ψ˘^ε_j+α ε_j= 0

. (12)

2.2. About the Hessian of J

For some reasons that will be developed in Section 4.1, it can be useful to find conditions to make sure that the Hessian matrixH_J∆T(ε) ofJ_∆T is invertible.

Denoting by (∇J_∆T(ε))the-th component of∇J_∆T(ε), the following equation prevails:

∂

∂ε_m(∇J_∆T(ε))=−2

Im ∂

∂ε_mχ^ε|µ|ψ˘^ε

+ Im

χ^ε|µ| ∂

∂ε_mψ˘^ε

+α δ_,m ,

where δ_,m is the Kronecker’s symbol. Repeating the analysis carried out in the previous section, we find out that H_J∆T(ε) reads:

H_J∆T(ε) =−2(S(ε) +α Id), whereIdis the identity matrix andS(ε) = (s_,m)_{= 0...N−1}

m= 0...N−1 has the following property (see (9) and (10)):

|s_,m| ≤2∆Tµ²_∗O∗, (13) whereµ_∗is the norm of the operatorµonL²(Ω). This estimate enables us to claim the following result.

Lemma 2.1. Let be ε∈R^N. Suppose thatα >2µ²_∗O_∗T. ThenH_J∆T(ε) is invertible.

Proof. Let us denote byh_,m the coefficients of H_J∆T(ε). According to (13), we obtain, for any= 0...N−1:

|h_,| ≥2(α−2∆Tµ²_∗O∗)

>4µ²_∗O∗(T−∆T)

≥

N−1 m=0,m=

|hm,|.

Thus,H_J∆T(ε) is a dominant diagonal matrix and the result comes as a consequence.

2.3. Variations in ψ

and χ

The variations in the stateψ^εand adjoint stateχ^εwith respect toεcan also be estimated by means of direct computations. Given two discrete fieldsε andε, the differenceψ_j+1^ε −ψ^ε_j+1 reads:

ψ_j+1^ε −ψ_j+1^ε = e^H0∆2iT e

V−µε

i j∆T −e^{V−iµεj∆T}

e^H0∆2iTψ^ε_j+ e^H0∆2iTe

V−µε

i j∆Te^H0∆2iT

ψ_j^ε−ψ^ε_j . The mean value inequality then yields:

ψ_j+1^ε −ψ^ε_j+1L²(Ω)≤∆Tµ∗|ε_j−ε_j|+ψ^ε_j−ψ^ε_jL²(Ω).

Hence:

ψ^ε_j−ψ^ε_jL²_(Ω)≤∆Tµ_∗ j−1 l=0

|ε_l−ε_l| ≤ µ_∗ε−ε1.

According to a similar argument:

χ^ε_j−χ^ε_jL²_(Ω)≤∆Tµ∗|ε_j−ε_j|+χ^ε_j+1 −χ^ε_j+1L²_(Ω), and consequently:

χ^ε_j−χ^ε_jL²_(Ω) ≤∆Tµ∗ N−1

l=j

|ε_l−ε_l|+O∗ψ^ε_N−ψ_N^ε_L²_(Ω)

≤ µ∗ε−ε1+O∗ψ_N^ε −ψ_N^ε_L²(Ω)

≤ µ∗(1 +O∗)ε−ε1.

Remark 2.2. For algorithmic reasons, the adjoint state may be propagated with a field ˜εthat does not coincide with ε(see (6)), the field bringing about the final conditionχ^ε_N^˜ =O ψ^ε_N. However, the previous estimate still holds insofar as we have:

χ^ε_j^˜−χ^ε_j^˜_L²(Ω) ≤ µ∗˜ε−ε˜ 1+O∗ψ_N^ε −ψ_N^ε_L²(Ω)

≤ µ∗(ε˜−ε˜1+O∗ε−ε1). (14)

3. Implicit discrete monotonic schemes

This section aims at recalling the definition of the implicit discrete monotonic schemes and presenting some of their properties.

3.1. Definition of the schemes

Consider two real numbersδ, η∈[0,2[ and the operatorµ^∗(h) defined by:

µ^∗(h) = e^−iµh∆T −Id

−ih∆T · (15)

This function is an approximation ofµinasmuch as:

µ^∗(h)−µ_∗≤∆Tµ²_∗|h|, (16) which can be obtained by the mean value inequality, since^dµ_dh^∗(h)∗≤∆Tµ²_∗.

Given initial control fields ε⁰, ˜ε⁰ and their associated state ψ⁰ and adjoint state χ⁰, suppose that for some k≥1,ψ^k−1,χ^k−1,ε^k−1, ˜ε^k−1are known. The computation of ψ^k,χ^k,ε^k, ˜ε^k is achieved as follows:

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

ψ_j+1^k = e^H0∆2iTe^{V−µεki j∆T}e^H0∆2iTψ^k_j

ε^k_j = (1−δ)˜ε^k−1_j −_α^δ Imχ^k−1_j |µ^∗(˜ε^k−1_j −ε^k_j)|ψ˘^k_j ψ^k₀ =ψ_init,

(17)

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

χ^k_j = e^−H0∆2iTe^{−V+µ˜i εkj∆T}e^−H0∆2iTχ^k_j+1

˜

ε^k_j = (1−η)ε^k_j −^η_αImχ˘^k_j|µ^∗(ε^k_j −ε˜^k_j)|ψ^k_j+1 χ^k_N =O ψ_N^k,

(18)

where:

˘

χ^ε_j = e^−H0∆2iTχ^ε_j,ψ^ε_j = e^−H0∆2iTψ_j^ε. Subsequently, the initial value ε⁰of the monotonic schemes is considered fixed.

3.2. Properties of sequence (ε

)

A first property of (ε^k)_k∈Nand (˜ε^k)_k∈Ndefined in (17) and (18) is that these sequences are bounded. Indeed, the following result can be proved by induction (see [12]).

Lemma 3.1. Given an initial fieldε⁰, let us defined M by:

M = max

ε⁰∞,max

1, δ 2−δ, η

2−η

O∗µ∗

α .

The sequences (ε^k)_k∈N and(˜ε^k)_k∈N match the following conditions:

∀k∈N, ∀j= 0...N−1, |ε^k_j| ≤M, |ε˜^k_j| ≤M. (19) 3.2.2. Monotonic convergence

A computation of the variation in the cost functional values leads to:

J_∆T(ε^k+1)−J_∆T(ε^k) =ψ_N^k+1−ψ_N^k|O|ψ_N^k+1−ψ_N^k +_N−1

j=0 2 Re

˘

χ^k_j+1|e^{µ(εkji−˜εkj)∆T}−Id|ψ^k_j+1

−α∆T((˜ε^k_j)²−(ε^k_j)²)

+_N−1

j=0 2 Re

χ^k_j|e^−µ(εk

j+1−˜εkj)

i ∆T−Id|ψ˘^k+1_j

−α∆T((ε^k+1_j )²−(˜ε^k_j)²).

(20)

Note that this equality holds for any sequence of controls (ε^k)_k∈N and (˜ε^k)_k∈N. In the case of the implicit schemes (17)–(18), this expression reads:

J_∆T(ε^k+1)−J_∆T(ε^k) =

ψ_N^k+1−ψ_N^k|O|ψ_N^k+1−ψ^k_N

+α∆T

N−1

j=0

2

δ−1 (˜ε^k_j −ε^k+1_j )²+ 2

η −1 (ε^k_j −ε˜^k_j)²

=

ψ_N^k+1−ψ_N^k|O|ψ_N^k+1−ψ^k_N

+α 2

δ −1 ε˜^k−ε^k+12₂+α 2

η −1 ε^k−ε˜^k2₂. (21) This result, combined with the bounded character of (ε^k)_k∈N leads to the following theorem.

Theorem 3.2. The implicit schemes (17)–(18)ensure the monotonic convergence of the cost functional J_∆T insofar as:

J_∆T(ε^k)≤J_∆T(ε^k+1). (22)

Furthermore there exists l_ε0 such that:

k→+∞lim J_∆T(ε^k) =l_ε0.

Proof. Inequality (22) is a trivial consequence of (21), sinceOis a positive operator. Moreover, we know that:

∀k∈N, J_∆T(ε^k)≤ O∗,

hence the existence ofl_ε0.

We keep the notationl_ε0 in the sequel.

3.2.3. Limit points of (ε^k)_k∈N

Having reached these preliminary results, we are now in a position to describe the limit points set C_ε0 of sequence (ε^k)_k∈N.

Lemma 3.3. Keeping the previous notations:

C_ε0 ⊂C, whereC is the set of critical points, defined by (12).

Proof. Consider a convergent subsequence (ε^kn)_n∈N of (ε^k)_k∈N and its limit ε^∞. By means of continuity and sinceε^kn−ε˜^kn−12→0 (see (21)), we find the following limits:

˜

ε^kn−1 →ε^∞, µ^∗(ε^k_j−ε˜^k−1_j )→µ,

ψ^kn →ψ^ε∞, χ^kn−1 →χ^ε∞. Whenntends to +∞, Equation (17) becomes:

∀j = 0...N−1, ε^∞_j =−1

αImχ^ε_j^∞|µ|ψ˘_j^ε∞,

which is the desired conclusion.

Thanks to (19), a standard argument of compactness shows that:

d(ε^k, C_ε0)→0, (23)

where d(ε^k, C_ε0) is the distance associated to .2between ε^k and set C_ε0. Furthermore, inequality (22) then implies:

J_∆T(C_ε0) =l_ε0. (24)

Finally, note that sinceC is compact (see (12)),C_ε0 is also compact.

3.3. Estimates of the gradient

This section introduces an estimate of the norm of the gradient at the points of the monotonic schemes. Let us recall that according to (11), we know that:

∇J_∆T(ε^k)1= 2∆T

N−1

j=0

|Imχ^ε_j^k|µ|ψ˘^k_j+α ε^k_j|.

Lemma 3.4. There existsλ≥0 such that:

∇J_∆T(ε^k)1≤λ(ε^k−ε˜^k−12+ε˜^k−1−ε^k−12). (25) Proof. If we focus on one coefficient ofJ_∆T(ε^k), we find:

Imχ^ε_j^k|µ|ψ˘_j^k+α ε^k_j = Imχ^k−1_j |µ^∗(˜ε^k−1_j −ε^k_j)|ψ˘_j^k+α ε^k_j + Imχ^k−1_j |µ−µ^∗(˜ε^k−1_j −ε^k_j)|ψ˘_j^k

+ Imχ^ε_j^k−χ^k−1_j |µ|ψ˘^k_j. (26) Let us estimate each term of this decomposition. Definition (17) of the algorithm implies that:

Imχ^k−1_j |µ^∗(˜ε^k−1_j −ε^k_j)|ψ˘^k_j+α ε^k_j = α

δ(˜ε^k−1_j −ε^k_j). (27) Then, thanks to (16), we obtain:

|Imχ^k_j|µ−µ^∗(˜ε^k−1_j −ε^k_j)|ψ˘_j^k| ≤∆Tµ²_∗O∗|ε^k_j −ε˜^k−1_j |. (28) Lastly, inequality (14) (see Rem. 2.2) yields:

|Imχ^ε_j^k−χ^k−1_j |µ|ψ˘_j^k| ≤µ²_∗(ε^k−ε˜^k−11+O∗ε^k−ε^k−11). (29) Combining (27)–(29) with (26), we come to equation (25) where

λ= 2√ T

α

δ + (T+ ∆T)µ²_∗(1 +O_∗)

.

4. Convergence of the implicit schemes

The above mentioned results enable us to prove the convergence of the schemes. First we will present a general inequality due to Lojasiewicz [8, 9] (see also [3]), and use it to prove that (ε^k)_k∈N defined by (17)–(18) is Cauchy.

4.1. Lojasiewicz inequality

The Lojasiewicz inequality enables us to estimate the variation in an analytic function by its gradient. This result is detailed in the next theorem.

Theorem 4.1(Lojasiewicz inequality). LetΓ :Rⁿ→Rbe an analytic function in a neighborhood of a pointa in Rⁿ. Then there exists σ >0 and0< θ≤¹₂ such that:

∀x∈Rⁿ, x∈B(a, σ) ∇Γ(x) ≥ |Γ(x)−Γ(a)|^1−θ,

whereB(a, σ)denotes the ball centered ina with a radius equal toσ,. is a norm onRⁿ.

The real numberθis called a Lojasiewicz exponent ofa. A more precise result can be obtained if the Hessian matrix of Γ, denoted byH_Γ(a), is invertible (see,e.g. [5]).

Lemma 4.2. Suppose that H_Γ(a)is invertible, then there exists σ >0and κ >0 such that:

∀x∈Rⁿ, x∈B(a, σ) ∇Γ(x) ≥κ|Γ(x)−Γ(a)|¹². In order to apply these results to our problem, let us define the shifted cost functional:

J_∆T(ε) =l_ε0−J_∆T(ε).

It is easy to check that J_∆T is analytic. Now, considerain C_ε0. By means of (24), we find thatJ_∆T(a) = 0.

Consequently, Theorem 4.1 ensures that there exist 0< θ_a≤1/2 and σ_a>0 such that:

∀ε∈R^N, ε−a1< σ_a ∇J∆T(ε)1≥ |J_∆T(ε)|^1−θa.

The compactness of C_ε0 enables us to extract from the set{B(a,^σ₂^a), a∈C_ε0}a family F_ε0 ={B(a_i,^σ₂^ai)}_i∈I, whereI is a finite set of indexes such that:

C_ε0 ⊂F_ε0.

Let us denote by θ ∈ ]0,1/2], the lower bound of {θ_ai}i∈I and by σ, the lower bound of {^σ₂^ai}i∈I. Given ε such that d(ε, C_ε0)< σ, there exists i_ε such that ε∈ B(a_iε, σ_aiε) and we come to the following version of Theorem 4.1:

Lemma 4.3. Keeping the above notations:

∀ε∈R^N, d(ε, C_ε0)< σ, ∇J_∆T(ε)1≥ |J_∆T(ε)|^1−θ.

In caseH_J∆T is invertible onC_ε0, a similar analysis can be led to obtain the corresponding version of Lemma 4.2.

Lemma 4.4. Suppose that H_J∆T(ε)is invertible for all ε∈C_ε0,

∃σ>0, ∃κ>0, ∀ε∈R^N, d(ε, C_ε0)< σ, ∇J_∆T(ε)1≥κ|J_∆T(ε)|¹².

Summarizing the above mentioned results, we have obtained that there exist σ >0,κ >0 andθ∈]0,¹₂] such that:

d(ε, C_ε0)<σ, ∇J_∆T(ε)1≥κ|J_∆T(ε)|^1−θ, (30) withθ= 1/2 whenH_J∆T(ε) is invertible.

4.2. Cauchy property of the monotonic sequences

Theorem 4.5. Sequence(ε^k)_k∈N defined by(17)–(18) converges towards a critical point ofJ_∆T. Proof. First suppose that:

∀k∈N, J_∆T(ε^k)= 0.

According to (23), there existsk₀such that (30) holds for allε^k withk≥k₀, with ˜θ= 1/2 ifH_J∆T is invertible onC_ε0. Consider an integerk≥k₀. Thanks to the results above:

(J_∆T(ε^k))^θ−(J_∆T(ε^k+1))^θ≥ θ (J_∆T(ε^k+1))^1−θ

J_∆T(ε^k+1)−J_∆T(ε^k)

≥ κθ

∇J_∆T(ε^k+1)1

2

δ−1 ε^k+1−ε˜^k2₂+ 2

η −1 ε˜^k−ε^k2₂

≥ κθ a _(δ,η)

∇J_∆T(ε^k+1)1

ε^k+1−ε˜^k2+ε˜^k−ε^k2

₂

≥ κθ a _(δ,η) λ

ε^k+1−ε˜^k2+˜ε^k−ε^k2

(31)

≥ κθ a _(δ,η)

λ ε^k+1−ε^k2 (32)

wherea_(δ,η)=_max(δ,η)¹ −¹₂. Givenq∈N, inequality (32) leads to:

(J_∆T(ε^k))^θ−(J_∆T(ε^k+q))^θ≥ κθ a _(δ,η) λ

k+q−1

l=k

ε^l+1−ε^l2

≥ κθ a _(δ,η)

λ ε^k+q−ε^k2. Since

(J_∆T(ε^k))^θ

k∈N is a Cauchy sequence, we conclude that the sequence (ε^k)_k∈Nis also Cauchy.

In case there exists k₁ such thatJ_∆T(ε^k1) = 0, the monotonicity of the algorithm implies thatJ_∆T(ε^k1) = J_∆T(ε^k1+1) =J_∆T(ε^k1+2) =... and according to (21) the sequence (ε^k)_k∈N is constant fork≥k₁.

Lastly, the limit that has been reached belongs toCaccording to Lemma 3.3.

Remark 4.6. A similar proof can be developed whenδ= 0, η = 0 andδ= 0, η= 0. In caseδ= 0, η= 0, the convergence is trivial.

5. Rate of convergence

In this section, we will present both theoretical and numerical results regarding the rate of convergence of the discrete monotonic schemes.

5.1. Theoretical results

The rate of convergence can be evaluated by making a second use of Lojasiewicz inequality. The result is summarized in the next theorem.

Theorem 5.1. Let us denote byε^∞ the limit of (ε^k)_k∈Ndefined by (17)–(18)and byθandκthe real numbers appearing in (30), whereC_ε0 ={ε^∞}.

If θ < ¹₂, then there existsc >0 such that:

ε^k−ε^∞2≤ck⁻

θ˜

1−2˜θ. (33)

If θ= ¹₂, then there existc andτ such that:

ε^k−ε^∞2≤ce^−τk. (34) Proof. As in the proof of Theorem 4.5, let bek₀≥0 such that

∀l≥k₀ ∇J∆T(ε^l)1≥κ|J_∆T(ε^l)|^1−θ. (35) Let us fixk≥k₀ and introduce ∆^k defined by:

∆^k = ∞ l=k

ε^l+1−ε˜^l2+ε˜^l−ε^l2.

Sticking to a general way of reasoning, we may assume that ∆^k>0 for anyk≥k₀. By summing (31) between kand +∞, we obtain:

(J_∆T(ε^k))^θ≥κθ a˜ _(δ,η) λ ∆^k. This estimate, combined with (35), yields:

∇J_∆T(ε^k)1≥κκθ a _(δ,η) λ ∆^k

¹⁻_˜^θ˜

θ . From Lemma 3.4, we get:

λ(∆^k−1−∆^k)≥κκθ a _(δ,η) λ ∆^k

¹⁻_˜^θ˜

θ , which can be rewritten as follows:

∆^k−1−∆^k

(∆^k)^β ≥ν, (36)

withν >0 andβ= ¹⁻_θ˜^θ˜.

Suppose thatθ=¹₂,i.e.,β= 1. Equation (36) reads:

(1 +ν)^k0∆^k0 1

1 +ν

k

≥∆^k,

and (34) is proved withc= (1 +ν)^k0∆^k0 andτ = ln(1 +ν).

Now let us consider the case in which θ < ¹₂. Letr∈]0,1[, and first suppose that:

(∆^k)^β ≥r(∆^k−1)^β.

Since 1−β <0, the functions→s^1−β is convex and consequently:

(∆^k)^1−β−(∆^k−1)^1−β ≥(β−1)∆^k−1−∆^k (∆^k−1)^β

≥(β−1)r∆^k−1−∆^k (∆^k)^β

≥(β−1)rν.

In the other case:

(∆^k)^β ≤r(∆^k−1)^β, and consequently:

(∆^k)^1−β−(∆^k−1)^1−β ≥(∆^k)^1−β−(r^β1∆^k)^1−β= (1−r^1−ββ )(∆^k)^1−β≥(1−r^1−ββ )(∆^k0)^1−β. Thus, in any case, there existsν ≥0 such that:

(∆^k)^1−β−(∆^k−1)^1−β≥ν. (37)

Consider nowk> k. Inequality (37) implies that for small enough ac, we get:

∆^k ≤

ν(k−k) + (∆^k)^2−1θ˜ ₋ ^θ˜

1−2˜θ ≤ck⁻

θ˜ 1−2˜θ,

and (33) follows.

Remark 5.2. Thanks to Lemma 2.1, we have thus obtained that if:

α >2µ²_∗O∗T, (38)

the convergence of the schemes is at least linear.

5.2. Numerical results

In order to test the performance of the algorithm we have chosen a case already treated in the literature. The system under consideration is theO−H bond that vibrates in a Morse type potential. We refer to [20] for the numerical details concerning this system. The objective is to localize the wavefunction at a given locationx; this is expressed through the requirement thatψ(T)|O|ψ(T)is maximized, where the observableO is defined byO(x) = √^γ0πe^{−γ20(x−x)2}. In our test,β= 1,µ_∗= 0.68151,O_∗= 12.375 andT = 131000. Numerical tests carried out on this example show that though (38) is not true at all, the convergence of the algorithm is linear, as it appears in Figure 1. Therefore some further analysis has to be done to improve Lemma 2.1 and the results of Section 5.1.

Appendix: convergence of the explicit discrete monotonic scheme

In [12], an explicit scheme has also been presented. The proof of convergence of this scheme, although analogous to that of the implicit scheme, is slightly more elaborate. We briefly introduce here a sufficient condition for convergence.