1 Introduction
The main aim of this paper is to develop a numerical method based on Carleman estimates to solve quasilinear elliptic PDEs with overdetermined boundary data. We consider this new method the second generation of Carlemanbased numerical methods while the first generation is called the convexification, which will be mentioned in detail later. Let be an open and bounded domain in , , with smooth boundary . Let and be two smooth functions defined on . Let be a function in the class Let be , symmetric, and positive definite, that is,
for some fixed . The following problem is of our interests.
Problem 1.1.
Assume that the overdetermined boundary value problem
(1.1) 
has a solution in . Compute the function .
Problem 1.1 is motivated by a class of nonlinear inverse problems in PDEs. One important goal of inverse problems is to reconstruct the internal structure of a domain from boundary measurements, which allow us to impose both Dirichlet and Neumann data of the unknown in (1.1). Recently, a unified framework to solve such inverse problems was developed by the research group of the first and second authors, which has two main steps. In the first step, by introducing a change of variables, one derives a PDE of the form (1.1) from the given inverse problem, in which and can be computed directly by the given boundary data. In the second step, one numerically solves (1.1) to find . The knowledge of directly yields that of the solution to the corresponding inverse problem under consideration. See [KhoaKlibanovLoc:SIAMImaging2020, LeNguyen:2020] and the references therein for some works in this framework. Moreover, this unified framework was successfully tested with experimental data in [VoKlibanovNguyen:IP2020, Khoaelal:IPSE2021, KlibanovLeNguyen:preprint2021]. Another motivation to study Problem 1.1 is to seek solutions to HamiltonJacobi equations under the circumstance that the Neumann data of the unknown can be computed by its Dirichlet data and the given form of the Hamiltonian, see e.g., [KlibanovNguyenTran:preprint2021, Assumption 1.1 and Remark 1.1]. Since inverse problems are out of the scope of this paper, we only focus on the applications in solving quasilinear elliptic PDEs and firstorder HamiltonJacobi equations.
A natural approach to solve (1.1) is based on optimization. That means one sets the computed solution to (1.1) as a minimizer of a mismatch functional, e.g.,
subject to the Cauchy boundary conditions and The methods based on optimization are widely used in the scientific community, especially in computational mathematics, physics and engineering. Although effective and popular, the optimizationbased approaches have some drawbacks:

In general, it is not clear that the obtained minimizer approximates the true solution to (1.1).

The mismatch functional is not convex, and it might have multiple minima and ravines (see an example in [ScalesSmithFischerLjcp1992] for illustration). To deliver reliable numerical solutions, one must know some good initial guesses of the true solutions.

The computation is expensive and time consuming.
Drawbacks # 1 and #2 can be treated by the convexification method, which is designed to globalize the optimization methods. The main idea of the convexification method is to employ some suitable Carleman weight functions to convexify the mismatch functionals. The convexity of weighted mismatch functionals is rigorously proved by Carleman estimates. Several versions of the convexification method have been developed in [KlibanovNik:ra2017, KhoaKlibanovLoc:SIAMImaging2020, Klibanov:sjma1997, Klibanov:nw1997, Klibanov:ip2015, KlibanovLeNguyen:preprint2021] since it was first introduced in [KlibanovIoussoupova:SMA1995]. Moreover, we recently discovered that the convexification method can be used to solve a large class of firstorder HamiltonJacobi equations [KlibanovNguyenTran:preprint2021].
In this paper, we introduce a new method to solve (1.1) based on linearization and Carleman estimates. Like the convexification method, our method delivers a reliable solution to (1.1) without requiring a good initial guess. This fact is rigorously proved. Unlike the convexification method which is time consuming, our new method quickly provides the desired solutions. Its converge rate is as for some
We find the numerical solution to (1.1) by repeatedly solving the linearization of (1.1) by a new “Carleman weighted” quasireversibility method. The classical quasireversibility method was first proposed in [LattesLions:e1969], and it has been studied intensively since then (see [Klibanov:anm2015] for a survey). By a Carleman weighted quasireversibility method, we mean that we let a suitable Carleman weight function involve in the cost functional suggested by the classical quasireversibility method. The presence of the Carleman weight function is the key for us to prove our convergence theorem. Our process to solve Problem 1.1 is as follows. We first choose any initial solution that might be far away from the true one. Denote this initial solution by the function . Linearizing (1.1) about , we obtain a linear PDE. We then solve this linear PDE by the Carleman weighted quasireversibility method to obtain an updated solution . Using the Carleman weighted quasireversibility method rather than the classical one in this step is the key to our success. By iteration, we repeat this step to construct a sequence . The convergence of this sequence to the true solution to (1.1) is proved by using Carleman estimates. We then apply this method to numerically solve some quasilinear elliptic equations. It is important to note that our approach works well for systems of quasilinear elliptic PDEs too.
Next, we use our method to solve some firstorder HamiltonJacobi equations. More precisely, to find viscosity solutions to the firstorder equation
we use the vanishing viscosity procedure and consider, for ,
with given Cauchy boundary data. Our new method is robust in the sense that it works for general nonlinearity that might not be convex in We refer the readers to [CrandallLions83, CrandallEvansLions84, Tran19] and the references therein for the theory of viscosity solutions. A weakness of our new approach in computing the viscosity solutions to HamiltonJacobi equations is that we need to require both Dirichlet and Neumann data of the unknown . See [KlibanovNguyenTran:preprint2021, Remark 1.1] for some circumstances that this requirement is fulfilled. There have been many important methods to solve HamiltonJacobi equations in the literature. For finite difference monotone and consistent schemes of firstorder equations and applications, see [BSnum, CLrate, OsFe, Sethian, Sou1] for details and recent developments. If is convex in and satisfies appropriate conditions, it is possible to construct some semiLagrangian approximations by the discretization of the Dynamical Programming Principle associated to the problem, see [FaFe1, FaFe2] and the references therein.
The paper is organized as follows. In Section 2, we recall a Carleman estimate and an example in which Problem 1.1 appears. In Section 3, we introduce the new iterative method based on linearization and Carleman estimates. In Section 4, we prove the convergence of our method. Some numerical results for quasilinear elliptic equations and firstorder HamiltonJacobi equations are presented in Section 5. Concluding remarks are given in Section 6.
2 Preliminaries
In this section, we recall a Carleman estimate, which plays a key role for the proof of the convergence theorem in this paper. We then present an inverse scattering problem in which Problem 1.1 appears.
2.1 A Carleman estimate
In this section, we present a simple form of Carleman estimates. Carleman estimates were first employed to prove the unique continuation principle, see e.g., [Carleman:1933, Protter:1960AMS], and they quickly became a powerful tool in many areas of PDEs afterwards. Let be a point in such that for all For each , define
(2.1) 
We have the following lemma.
Lemma 2.1 (Carleman estimate).
There exist positive constants depending only on , , , and such that for all function satisfying
(2.2) 
the following estimate holds true
(2.3) 
for all and . Here, depends only on the listed parameters.
Proof.
Lemma 2.1 is a direct consequence of [MinhLoc:tams2015, Lemma 5]. Let and such that . Here, for Extend to the whole such that for all . Using a change of variable and [MinhLoc:tams2015, Lemma 5], there exists a number depending on , and such that for all and , we have
(2.4) 
for some constant depending only on and . Since on and since , allowing to depend on and , we deduce the Carleman estimate (2.3) from (2.4). ∎
An alternative way to obtain (2.3) is to apply the Carleman estimate in [Lavrentiev:AMS1986, Chapter 4, §1, Lemma 3] for general parabolic operators. The arguments to obtain (2.3) using [Lavrentiev:AMS1986, Chapter 4, §1, Lemma 3] are similar to that in [LeNguyenNguyenPowell:JOSC2021, Section 3] with the Laplacian replaced by the operator .
Remark 2.1.
We specially draw the reader’s attention to different forms of Carleman estimates for all three main kinds of differential operators (elliptic, parabolic and hyperbolic) and their applications in inverse problems and computational mathematics [BeilinaKlibanovBook, BukhgeimKlibanov:smd1981, KlibanovLiBook]. It is worth mentioning that some Carleman estimates hold true for all functions satisfying and where is a part of , see e.g., [KlibanovNguyenTran:preprint2021, NguyenLiKlibanov:IPI2019], which can be used to solve quasilinear elliptic PDEs with the boundary data partly given.
2.2 An inverse scattering problem
As mentioned in Section 1, Problem 1.1
arises from nonlinear inverse problems. We present here an important example in the context of inverse scattering problems in the frequency domain. Let
be the spatially distributed dielectric constant of the medium and be an interval of wavenumbers with . Since the dielectric constant of the air is 1, we set for all For each let , , be the wave field generated by a point source at with wavenumber . The function is governed by the Helmholtz equation and the Sommerfeld radiation condition(2.5) 
The inverse scattering problem is formulated as the problem of computing the function , , from the measurements of
(2.6) 
for all , The knowledge of the function partly provides the internal structure of the domain . In other words, solving the inverse scattering problem allows us to examine a domain from external measurements, which has applications in security, sonar imaging, geographical exploration, medical imaging, nearfield optical microscopy, nanooptics, see, e.g., [ColtonKress:2013] and references therein for more details.
Remark 2.2.
In practice, measuring the wave field is expensive, and one does not measure both and . Rather, one measures the wave for all on a surface far away from We call the information of on the far field. Then, we employ a technique, called “data propagation” to compute the near field on . This data propagation is based on the angular spectrum representation, see, e.g., [NovotnyHecht:cup2012, Chapter 2] and [LiemKlibanovLocAlekFiddyHui:jcp2017, Section 4.2.1] for details in implementation. This technique allows us to compute the wave field in the region between the measurement surface and where . The knowledges of and follow.
There have been many important methods to solve inverse scattering problems in the literature. Each method has its own advantages and disadvantages. A common drawback of the widelyused method based on optimization to solve inverse scattering problems is the need of a good initial guess of the true solution . We recall from [LeNguyen:preprint2021] a method to solve the above inverse scattering problem in which such a need is relaxed. Denote by
where Then, satisfies
Let be the orthonormal basis of introduced in [Klibanov:jiip2017] and define
(2.7) 
We approximate
for a suitable cutoff number
. Then, the vector
“approximately” satisfies the system(2.8) 
where
for all and , see [LeNguyen:preprint2021, Section 6] for details. The Dirichlet and Neumann boundary conditions for can be computed by the knowledges of , , and (2.7). Solving the system of quasilinear elliptic equations (2.8) with the provided Dirichlet and Neumann data is basically a goal of Problem 1.1. Doing so is the key step to compute . See [LeNguyen:preprint2021] for convexification method to compute and the procedure to obtain from the knowledge of
We refer the reader to [KhoaKlibanovLoc:SIAMImaging2020, Khoaelal:IPSE2021, VoKlibanovNguyen:IP2020] for another setup of the inverse scattering problem and numerical results from experimental data. Since inverse problems are out of the scope of this paper, we only mention it here to explain the significance of Problem 1.1. In future works, we will use our solver for Problem 1.1 to solve inverse problems.
3 The iteration and linearization approach for Problem 1.1
Our approach to solve (1.1) is based on linearization and iteration. Assume that the solution to (1.1) is in the space for some where is the smallest integer that is greater than Define the set of admissible solutions
(3.1) 
Then, the assumption in Problem 1.1 implies that , and . We now construct a sequence that converges to the solution to (1.1). Take a function Assume by induction that we have the knowledge of for some . We find as follows. Assume that for some where
Plugging into (1.1), we have
(3.2) 
Heuristically, we assume at this moment that is small, that is, . This assumption plays the role in establishing a numerical scheme to find while it does not play a key role in the proof of the convergence theorem. By Taylor’s expansion, we approximate (3.2) as
(3.3) 
where
for all . Here, and are the partial derivative of with respect to its second variable and its gradient vector with respect to the third variable, respectively.
The next step is to compute a function satisfying (3.3). Since there is no guarantee for the existence of such a function , we only compute a “best fit” function by the Carlemanbased quasireversibility method described below. For each , , and , define the functional as
Here, the function is defined in (2.1) and is a regularization term.
For each , we minimize on . The unique minimizer is the desired function We then set
(3.4) 
The construction of the sequence above is summarized in Algorithm 1. We will prove that the sequence converges to in Section 4 as and . The presence of the Carleman weight is a key point for us to prove this convergence result.
4 The convergence analysis
In this section, we prove that the sequence generated by Algorithm 1 converges to the solution to (1.1). The following result is the main theorem in this paper.
Theorem 4.1.
Proof.
Fix . Let . Due to Step 3 in Algorithm 1, is the minimizer of . By the variational principle, we have
(4.2) 
for all Since , we can rewrite (4.2) as
(4.3) 
for all As is the solution to (1.1), we have
(4.4) 
for all It follows from (4.3) and (4.4) that for all
(4.5) 
Take . It follows from (4.5) that
(4.6) 
As , we can estimate
and  
These estimates, together with the inequality , imply
(4.7) 
Applying the Carleman estimate (2.3) for the function , we have
(4.8) 
Combining (4.7) and (4.8), we have
(4.9) 
Letting be sufficiently large, we can simplify (4.9) as
(4.10) 
Recall that . We get from (4.10) that
(4.11) 
Applying (4.11) for and denoting , we have
By induction, we have
(4.12) 
which implies (4.1). The proof is complete. ∎
Remark 4.1 (Removing the boundedness condition of in in Theorem 4.1).
In the case when , we need to assume that we know in advance that the true solution to (1.1) belongs to the ball of for some . This assumption does not weaken the result since can be arbitrarily large. Define the cutoff function as
and set Since , it is obvious that solves the problem
(4.13) 
Remark 4.2.
Remark 4.3.
As seen in the proof of Theorem 4.1, the efficiency of Algorithm 1 is guaranteed by Carleman estimate (2.3). Therefore, we call the proposed method described in Algorithm 1 the second generation of Carlemanbased numerical methods. The first generation of Carlemanbased numerical method was developed in [KlibanovIoussoupova:SMA1995], which is called the convexification. See [KlibanovNik:ra2017, KlibanovNguyenTran:preprint2021, LeNguyen:2020, LeNguyen:preprint2021] for some following up results. Like the convexification method, Algorithm 1 can be used to compute solutions to nonlinear PDEs without requesting an initial good guess. The advantage of our new method is the fast convergence rate, see Remark 4.2.
5 Numerical study
In this section, we present some numerical results obtained by Algorithm 1. For simplicity, we set and . On we arrange an grid points
where In our numerical scripts, .
In Step 1 of Algorithm 1, we choose a function . It is natural to find a function satisfying the equation obtained by removing from (1.1) the nonlinearity . We apply the Carlemanbased quasireversibility method to do so. That means, is the minimizer of
(5.1) 
subject to the boundary conditions in (5.7). To simplify the efforts in implementation, the norm in the regularization term is the norm rather than the norm. This change does not affect the performance of Algorithm 1. Algorithm 1 still provides satisfactory solutions to (1.1).
Remark 5.1.
We employ the Carlemanbased quasireversibility method to find for the consistency to Step 3 of Algorithm 1. Since can be chosen arbitrarily, one can use the quasireversibility method without the presence of the Carleman weight function . In our computation for all numerical examples below, , and The regularized parameter is . The threshold number
We minimize on by the least square MATLAB command “lsqlin”. The implementation for the quasireversibility method to minimize more general functionals than was described in [LeNguyen:2020, §5.3] and in [Nguyen:CAMWA2020, §5]. We do not repeat this process here. In Step 3 of Algorithm 1, given we minimize the functional on . Again, we refer the reader to [LeNguyen:2020, §5.3] and in [Nguyen:CAMWA2020, §5] for details in implementation. The scripts for other steps of Algorithm 1 can be written easily.
5.1 Quasilinear elliptic equations
The convexification method, first introduced in [KlibanovIoussoupova:SMA1995], was used to numerically solve quasilinear elliptic equations in [KlibanovNik:ra2017, KlibanovNguyenTran:preprint2021, LeNguyen:preprint2021]. Our approach here is of course different based on iteration and linearization. In particular, Step 3 in Algorithm 1 is very efficient as we only need to solve the linear equation (3.3) as opposed to solving directly a nonlinear quasilinear elliptic equation. In this subsection, we present two (2) numerical tests. In both tests, we choose the matrix to be
That means,
In test 1, we solve (1.1) when
(5.2) 
for all , and . The boundary conditions are given by
(5.3) 
for all The true solution of (1.1) is the function .
It is evident from Figure 1 that Algorithm 1 provides out of expectation solution for test 1. The relative error . One can see from Figure (c)c that Algorithm 1 converges at the third iteration.
In test 2, we solve (1.1) when
Comments
There are no comments yet.