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A Schrödinger-type algorithm for solving the Schrödinger equations via Phragmén–Lindelöf inequalities
Journal of Inequalities and Applications volume 2019, Article number: 144 (2019)
Abstract
In this article, we consider the numerical method for solving the Schrödinger equations via Phragmén–Lindelöf inequalities under the order induced by a symmetric cone with the function involved being monotone. Based on the Phragmén–Lindelöf inequalities, the underlying system of inequalities is reformulated as a system of smooth equations, and a Schrödinger-type method is proposed to solve it iteratively so that a solution of the system of the Schrödinger equations is found. By means of the Schrödinger type inequalities, the algorithm is proved to be well defined and to be globally convergent under weak assumptions and locally quadratically convergent under suitable assumptions. Preliminary numerical results indicate that the algorithm is effective.
1 Introduction
In this paper, we consider the following Schrödinger equation (see [18, 19]):
where \(n\geq 3\), \(0<\alpha <n\), \(2-\frac{\alpha }{n}< p\leq 2^{\ast } _{\alpha }=\frac{2n-\alpha }{n-2}\) and
- (A1):
-
\(V\in C(\mathbb{R}^{n},\mathbb{R^{+}})\) is coercive, that is,
$$ \lim_{|x|\to +\infty }V(x)=+\infty . $$
In 2016, Qiao et al. [19] first considered the bound and ground states for the nonlinear Schrödinger equations under the condition
- (A2):
-
\(\inf_{\mathbb{R}^{n}} V>0\), and there exists a positive constant r satisfying
$$ \operatorname{meas}\bigl\{ x\in \mathbb{R}^{n}, \vert x-y \vert \leq r, V(x)\leq M\bigr\} \to 0 $$as \(|y|\to \infty \), where \(M>0\) and meas stands for the Lebesgue measure.
The nonlinear system (1.1) has been proved to possess wide application fields in many real world problems such as anomalous diffusion [2, 4, 15], disease models [6, 9, 21], ecological models [26], synchronization of chaotic systems [1, 27], etc.
Put
is the Nevanlinna norm (see [8]), problem (1.1) is equivalent to the following Schrödinger problem defined by
where \(x\in \Re ^{n}\).
The Phragmén–Lindelöf inequalities (see [23]) have the main objective to solve the so-called Schrödinger Phragmén–Lindelöf subproblem model to get the trial step \(\tau _{k}\)
In 2015, a modified Phragmén–Lindelöf inequality was proved by Wan [23]:
where p, c, and γ are three positive numbers.
Recently, another adaptive Schrödinger Phragmén–Lindelöf inequality has been defined by Qiao et al. [17]:
where \(\mathcal{B}_{k}\) is defined by
\(y_{k}=u(x_{k+1})-u(x_{k})\) and \(s_{k}=x_{k+1}-x_{k}\). This Schrödinger Phragmén–Lindelöf method can possess the global convergence without the nondegeneracy (see [1, 7, 11, 26] etc.), which shows that this paper made a further progress in theory. And there exist many applications about the Schrödinger Phragmén–Lindelöf inequalities (see [3, 25, 27, 28] etc.) for nonsmooth optimizations and other problems.
We further consider the Schrödinger Phragmén–Lindelöf model for the nonlinear system \(u(x)\) at \(x_{k}\) (see [17])
where \(\nabla u(x_{k})\) is the Jacobian matrix of \(u(x)\) at \(x_{k}\).
It is well known that the above model (1.5) can be written as the following extension (see [20, 23, 24]):
If we set the Schrödinger Phragmén–Lindelöf matrix \(\nabla u(x_{k})\), then we can use the Schrödinger Phragmén–Lindelöf matrix \(\mathcal{B}_{k}\) instead of it. Thus, our Schrödinger Phragmén–Lindelöf model can be defined as follows:
where \(\mathcal{B}_{k}=\mathcal{H}_{k}^{-1}\) and \(\mathcal{H}_{k}\) is generated by
where \(\rho _{k}=\frac{1}{s_{k}^{\mathcal{T}}y_{k}}\), \(\mathcal{V} _{k}=I-\rho _{k}y_{k}s_{k}^{\mathcal{T}}\) (see [23] etc.).
Let \(\tau _{k}^{p}\) be the solution of (1.7). Define
and predict reduction by
Based on definitions of \(A\tau _{k}(\tau _{k}^{p})\) and \(P\tau _{k}(\tau _{k}^{p})\), we design their ratio by
Therefore, the Schrödinger-type algorithm for solving (1.1) is stated as follows.
Algorithm
- Initial::
-
Let \(\mathfrak{B}_{0}=\mathfrak{H}_{0}^{-1}\in \Re ^{n}\times \Re ^{n}\) be a symmetric and positive definite matrix. \(x_{0}\in \Re ^{n}\) and \(\varrho =0\). ρ, c, and ϵ are three positive constants. Let \(l:=0\);
- Step 1::
-
Stop if \(\|\chi (x_{l})\|<\epsilon \) holds;
- Step 2::
-
Solve (1.7) with \(\triangle =\triangle _{l}\) to obtain \(\varsigma _{l}^{\varrho }\);
- Step 3::
-
Compute \(A\varsigma _{l}(\varsigma _{l}^{\varrho })\), \(\mathcal{P}\varsigma _{l}(\varsigma _{l}^{\varrho })\), and the ratio \(r_{l}^{\varrho }\). If \(r_{l}^{\varrho }<\rho \), let \(\varrho =\varrho +1\), go to Step 2. If \(r_{l}^{\varrho }\geq \rho \), go to the next step;
- Step 4::
-
Set \(x_{l+1}=x_{l}+\varsigma _{l}^{\varrho }\), \(y_{l}=\chi (x_{l+1})-\chi (x_{l})\), update \(\mathfrak{B}_{l+1}= \mathfrak{H}_{l+1}^{-1}\) by (1.8) if \(y_{l}^{\mathfrak{T}} \varsigma _{l}^{p}>0\), otherwise set \(\mathfrak{B}_{l+1}=\mathfrak{B} _{l}\);
- Step 5::
-
Let \(l:=l+1\) and \(\varrho =0\). Go to Step 1.
In this paper, we further focus on convergence results of the above algorithm under the following assumptions.
Assumptions
- (A):
-
Define the set Ω by
$$ \varOmega =\bigl\{ x|\varphi (x)\leq \varphi (x_{0})\bigr\} . $$(1.12)It is easy to see that Ω is bounded.
- (B):
-
The nonlinear system \(\chi (x)\) is twice continuously differentiable in \(\varOmega _{1}\), which is an open convex set containing Ω.
- (C):
-
The following Phragmén–Lindelöf relation
$$ \bigl\Vert \bigl[\nabla \chi (x_{l})- \mathfrak{B}_{l}\bigr]\chi (x_{l}) \bigr\Vert =O\bigl( \bigl\Vert \varsigma _{l}^{p} \bigr\Vert \bigr) $$(1.13)holds.
- (D):
-
The sequence matrices \(\{\mathfrak{B}_{l}\}\) are uniformly bounded in \(\varOmega _{1}\).
It follows from Assumption (B) that (see [10, 22])
where \(M_{L}\) is a positive real number.
2 Convergence results
We first have the following new Phragmén–Lindelöf inequalities.
Lemma 2.1
Let \(\tau _{k}^{p}\) be the solution of (1.1). Then
holds.
Proof
Define
It follows from (1.7) that
Consider \(V(x)\) is a minimizer for both \(S_{\alpha , p}\). By the continuity of \(\mathcal{J}\), we know that
where \(0\leq t<\gamma \).
So
It follows from \(t\geq \gamma \) that
Here, we use that \(\langle J'(u_{0}), tV\rangle =0\) and \(V(x)\) is a solution of (1.1). By the definition of \(\tau _{k}^{p}\) [14, 16] and it being the solution of (1.7), we get
for any \(\alpha \in [0,1]\).
Therefore
□
Lemma 2.2
Let Assumptions (A), (B), (C), and (D) hold. Then
where \(\tau _{k}\) is the solution of (1.7).
Proof
It follows from (1.9) and (1.10) that
□
Theorem 2.1
Let Assumptions (A), (B), (C), and (D) hold. Then Algorithm either finitely stops or generates an infinite sequence \(\{x_{k}\}\) satisfying
where \(\{x_{k}\}\) is defined as in Algorithm.
Proof
We know that \(t^{-}(u)\) is a continuous function of u. Consequently, the manifold \(\varLambda ^{-}\) disconnects \(D^{1,2}( {\Re ^{n}})\) in exactly two connected components \(\mathcal{U}_{1}\) and \(\mathcal{U}_{2}\), where
So \(D^{1,2}({\Re ^{n}})=\varLambda ^{-}\cup \mathcal{U}_{1} \cup \mathcal{U}_{2}\). In particular, \(u_{0}\in \varLambda ^{+} \subset \mathcal{U}_{1}\). Since
we have
uniformly for \(t\in \mathbb{R}\).
On the other hand,
where \(t\geq \tilde{t}\).
So that we can fix a positive number \(t_{0}\) such that
which yields that
Combining this and the fact \(u_{0}\in \mathcal{U}_{1}\), we know that
for some \(0< t_{1}< t_{0}\).
So
And there exists a minimizing sequence \(\{u_{n}\}\subset \varLambda ^{-}\) satisfying
where \(w\in \varLambda ^{-}\).
So that
which implies \(\|u_{n}\|\) has an upper bound.
It follows from \(\{u_{n}\}\subset \varLambda ^{-}\) that
Thus, \(\|u_{n}\|_{D}\) has a uniform positive lower bound.
Similarly,
By Lemma 2.2 and
we obtain that
which leads to a contradiction.
In the case \(h>0\). Applying Lemma 2.1 to \(u_{1}\) and \(|u_{1}|\), we know that there exists \(t^{-}(|u_{1}|)\) such that
Moreover,
So
which implies that \(u_{1}\geq 0\). According to the maximum principle, we get \(u_{1}>0\).
It is easy to see that \(\|u_{n}\|\) is bounded, which yields that
and
as \(n\to \infty \).
So
and
Notice that
which yields that
It follows from (2.4) that
and
which also leads to a contradiction.
Suppose that
holds. Using Assumption (C) we get (2.2). It follows from (2.5) that the subsequence \(\{k_{j}\}\) satisfies
Set
So we assume that
holds, where \(k\in K\).
It follows from the definition of Algorithm and Lemma 2.1 that
Lemma 2.2 gives us that the sequence \(\{\nu (x_{k})\}\) is convergent, which yields that
Then \(p_{k} \rightarrow +\infty \) when \(k\rightarrow +\infty \) and \(k\in K\). It follows that
is unacceptable.
If we put \(x_{k+1}'=x_{k}+\tau _{k}'\), then we have
By applying Lemma 2.1 and the definition of \(\triangle _{k}\), we know that
By applying Lemma 2.2, we know that
So
By applying \(p_{k}\rightarrow +\infty \) as \(k\rightarrow +\infty \), we know that
which also gives a contradiction to (2.8). □
3 Numerical results
This section reports some numerical results of Algorithm.
3.1 Problems
Define
Problem 1
The Schrödinger differential function (see [12])
where \(l=1,2,3,\ldots ,n\).
Initial guess:
Problem 2
Logarithmic function
where \(l=1,2,3,\ldots ,n\).
Initial points:
Problem 3
Schrödinger differential function (see [5, pp. 471–472])
where \(l=1,2,3,\ldots ,n\).
Initial points:
Problem 4
Trigexp function (see [5, p. 473])
where \(l=1,2,3,\ldots ,n\).
Initial guess:
Problem 5
Let \(u(x)\) be the gradient of
Then
where \(l=1,2,3,\ldots ,n\).
Initial points:
Parameters: \(c=0.2\), \(\epsilon =10^{-2}\), \(\rho =0.03\), \(p=3\), \(m=6\) \(\mathcal{H}_{0}\) is the unit matrix.
The method for (1.3) and (1.7): the Dogleg method [13, 25].
Code experiments: run on a PC with Intel Pentium(R) Xeon(R) E5507 CPU 2.27 GHz, 6.00 GB of RAM, and Windows 7 operating system.
Code software: MATLAB r2017a.
Stop rules: the program stops if \(\|u(x)\|\leq 1e-4\) holds.
Other cases: we will stop the program if the iteration number is larger than ten hundred.
3.2 Results and discussion
The column meaning in the following tables:
Dim: the dimension. NI: the number of iterations.
NG: the norm function number. Time: the CPU-time in seconds.
Numerical results of Table 1 show the performance of these two algorithms about NI, NG, and Time. It is not difficult to see that both of these algorithm can successfully solve all these ten nonlinear problems.
It is easy to see that the NI and the NG of Algorithm have won since their performance profile plot is on top right. And the Time of Algorithm YL has superiority over Algorithm. Both of these two algorithms have good robustness.
4 Conclusions
In this paper, we considered the numerical method for solving the Schrödinger equations via Phragmén–Lindelöf inequalities under the order induced by a symmetric cone with the function involved being monotone. Based on the Phragmén–Lindelöf inequalities, the underlying system of inequalities was reformulated as a system of smooth equations, and a Schrödinger-type method was proposed to solve it iteratively so that a solution of the system of the Schrödinger equations was found. By means of the Schrödinger type inequalities, the algorithm was proved to be well defined and to be globally convergent under weak assumptions and locally quadratically convergent under suitable assumptions. Preliminary numerical results indicate that the algorithm was effective.
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Zhao, L. A Schrödinger-type algorithm for solving the Schrödinger equations via Phragmén–Lindelöf inequalities. J Inequal Appl 2019, 144 (2019). https://doi.org/10.1186/s13660-019-2098-3
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DOI: https://doi.org/10.1186/s13660-019-2098-3