A modified quasi reversibility method for inverse source problem of poisson equation

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A modified quasi-reversibility method for inverse source problem of Poisson equation

Jin Wen, Li-Ming Huang & Zhuan-Xia Liu

To cite this article: Jin Wen, Li-Ming Huang & Zhuan-Xia Liu (2021): A modified quasi-reversibility method for inverse source problem of Poisson equation, Inverse Problems in Science and

Engineering, DOI: 10.1080/17415977.2021.1902516

To link to this article: https://doi.org/10.1080/17415977.2021.1902516

Published online: 22 Mar 2021.

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A modified quasi-reversibility method for inverse source

problem of Poisson equation

Jin Wen , Li-Ming Huang and Zhuan-Xia Liu

Department of Mathematics, Northwest Normal University, Lanzhou, 730070 Gansu, People’s Republic of China

ABSTRACT

In this article, we consider an inverse source problem for Poisson

equation in a strip domain That is to determine source term in the

Poisson equation from a noisy boundary data This is an ill-posed

problem in the sense of Hadamard, i.e., small changes in the data can

cause arbitrarily large changes in the results Before we give the main

results about our proposed problem, we display some useful lemmas

at first Then we propose a modified quasi-reversibility regularization

method to deal with the inverse source problem and obtain a

conver-gence rate by using an a priori regularization parameter choice rule.

Numerical examples are provided to show the effectiveness of the

proposed method.

ARTICLE HISTORY

Received 9 November 2020 Accepted 24 February 2021

KEYWORDS

Inverse source problem; Poisson equation; a modified quasi-reversibility

regularization method; a priori choice; convergence analysis

2010 MATHEMATICS SUBJECT

CLASSIFICATIONS

35R30; 65N20

1 Introduction

Inverse source problem is of great importance in many branches of engineering and sci-ence; such as heat source determination [1,2], heat conduction problem [3 5], Stephan design problem [6] and pollutant detection To our best knowledge, there are also a vari-ety of researches on inverse source problems in the Poisson equation adopted numerical methods; for examples, logarithmic potential method [7], the projective method [8], the Green’s function method [9], the dual reciprocity boundary element method [10,11] and the method of fundamental solution (MFS) [12–14]

Quasi-reversibility method is originally introduced by Lattes and Lions [15], and later studied by Melnikova and Filinkov [16] The idea consists in replacing the final bound-ary value problem with an approximate solution of the final boundbound-ary value problem

In the initial method of the quasi-reversibility, the author [17] replaced the heat opera-tor∂/∂t − ∂2/∂x2by a perturbed operator P ε = (∂/∂t) + A − εAA∗, perturbing the final

condition we get an approximate solution from the final boundary value problem with a small parameterε The authors [18] take f (A) = A − A2using logarithmic convexity to obtain well-posed solution as above Lattes and Lions [15] The final value problem in [19] is considered about perturbing the final conditions to obtain an approximate non-local prob-lem after operator perturbation In [20], the quasi-reversibility method is to approach the ill-posed second order Cauchy problem depending on a (small) regularization parameter,

CONTACT Jin Wen wenj@nwnu.edu.cn; wenjin0421@163.com

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based on the fundamental solution for a second order elliptic operator Furthermore they propose the mixed quasi-reversibility method, and give some nice results The ill-posed problem of the wave equation in [21] is replaced with a boundary value problem for

a fourth order equation by using the method of quasi-reversibility They consider the

wave equation Lu := 1

c (x)2∂ tt u − u = f , constructing Tikhonov functional firstly J ε (u) =

1

2Lu − f 2+ ε2u2, it is equivalent with the abstract Euler equation J ε(u ε )(v) = 0, for all

v ∈ H2

0(Q T ), then through a minimizer u ε of calculation for above equation, they obtain perturb term to approximate the solution of ill-posed problem with a small parameterε.

In [22], from the original quasi-reversibility method, the mixed quasi-reversibility method with variable parameterλ is extended in a system of two second-order equations

involv-ing two functions u and λ, the aim is to find an approximation (u ε,λ) of (u, λ) as a

solution of the weak formulation and(δ, ε) denotes α for small ε > 0 and δ > 0 The

method of quasi-reversibility proposed by [23] is a particular case of Tikhonov

regular-ization and A = or + k2, which provides corresponding error estimate with a priori choice forε as a function of δ In [24], the article addsμ2f xx (x) to the left-hand side of the

equationu(x, y) − k2u(x, y) − f (x), the quasi-reversibility regularization solution and

a priori convergence estimate are obtained There are some important references about inverse source problem by using the quasi-reversibility method recently, such as the inverse source problems for parabolic equations [25,26], and hyperbolic equations [27–29]

In this article, we consider the following inverse problem:

⎧

⎪

⎨

⎪

−u xx − u yy = f (x), 0< x < π, 0 < y < ∞,

u(0, y) = u(π, y) = 0, 0≤ y < ∞,

u(x, 0) = 0, u(x, y)| y→∞bounded, 0≤ x ≤ π,

u(x, 1) = g(x), 0≤ x ≤ π,

(1)

to find a pair of function(u(x, y), f (x)) which satisfies the Poisson equation on above

con-ditions Subsequently we will study the above problem, where we perturb the equation to form an approximate problem depending on a small parameter, before that we need to give the following preparations

Generally, the input data g (x) with a noise level δ is merely measured in L2(0, π), and

we give that

We obtain that the solution of problem (1) using separation of variables has the following form:

u(x, y) =

∞

n=1

1− e −ny

n2 (f , X n )X n, (3) where

{X n=

2

π sin nx, (n = 1, 2, )}, (f , X n ) =

2

π

0

f (x) sin nxdx.

We define the operator K : f → g, then we have

g(x) = Kf (x) =

+∞

n=1

(g, X n )X n=

∞

n=1

1− e −n

n2 (f , X n )X n (4)

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The singular values{σ n}∞

n=1of K satisfy

σ n= 1− e −n

n2 , correspondingly

(g, X n ) = 1− e −n

n2 (f , X n )(X n , X n ),

i.e

(f , X n ) = σ−1

n (g, X n ),

then,

f (x) = K−1g(x) =

+∞

n=1

1

σ n (g, X n )X n =

+∞

n=1

n2

1− e −n (g, X n )X n From [30], the solution does not depend on the data continuously, the problem (1) is ill-posed Several articles impose regularization method to deal with ill-posed problem (1): for examples, the Tikhonov regularization method [31,32], the super order regulariza-tion method [33], the quasi-boundary value regularization method [34,35], the quasi-reversibility method [36], the modified regularization method [30,37], the truncation method [38] Recently, Boussetila and Rebbani [39] propose a modified quasi-reversibility method, and it is employed by Huang [40] and Fury [41] and Trong and Tuan [42] in the case of the autonomous Cauchy problem

In this article, we will use a modified quasi-reversibility method to deal with identifying the unknown source of the problem (1) Before doing that, we need to define an a priori bound on unknown source,

where E > 0 is a constant and · H p (0,π) denotes the norm in Sobolev space which is

defined as follows [43]:

f (x) H p (0,π)=

+∞

n=1

(1 + n2) p

n|2

1

Figure 1.The exact and approximate solutions withM = 50, N = 5 and various noise level for

Exam-ple 1 wherex ∈ [0, π] (a): heat source for p = 1; (b): heat source for p = 3.

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This article is organized as follows Section2gives some preliminary results In Section3, a regularization solution and error estimation of the inverse problem are provided by a mod-ified quasi-reversibility method Section4gives some examples to illustrate the accuracy and efficiency of the proposed method in problem (1) Section5puts an end to this paper with a brief conclusion

2 Some auxiliary results

In this section, we give four important lemmas as follows

Lemma 2.1: For n ≥ 1,

1

1− e −n < 2.

Lemma 2.2: If μ > 0, n ≥ 1,

1

1− e√1+μ2n2−n

≤ (1 + μ) · e μ1

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1

1− e√1+μ2n2−n

≤

1+ μ2n2· e√1+μ2n2n

n

≤

1

n2 + μ2· e

1 1

n2 +μ2

≤ (1 + μ) · e1/μ.

Lemma 2.3 ([ 30]): If μ > 0, n ≥ 1, p > 0,

μ2n2 (1 + μ2n2)(1 + n2) p2

< max {μ2,μ p}

Lemma 2.4: If μ > 0, n ≥ 1,

e

−n

√ 1+μ2n2 − e −n < n · μ2n2

1+ μ2n2

Proof:

e

−n

√ 1+μ2n2 − e −n = (n − n

1+ μ2n2) · e ζ

≤ n · (

1+ μ2n2− 1)(1+ μ2n2+ 1)

1+ μ2n2(1+ μ2n2+ 1)

< n · μ2n2

1+ μ2n2, whereζ ∈ [−n,√ −n

3 A modified regularization method and convergence estimates

We will investigate the following problem:

⎧

⎪

−u xx − u yy + μ2u xxxx = f (x) − μ2/(1 + μ2)(f xx (x) + f (x)), 0< x < π, 0 < y < ∞,

(6)

By separation of variables, we obtain that

f μ δ (x) =+∞

n=1

(1 + μ2)n2

(1 + μ2n2)(1 − e√1+μ2n2−n )

(g δ , X n )X n, (7) which is called the modified regularized solution of problem (1), correspondingly

f μ (x) =+∞

n=1

(1 + μ2)n2

(1 + μ2n2)(1 − e√1+μ2n2−n )

(g, X n )X n (8)

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Theorem 3.1: Let g δ be measured data at y = 1 satisfying (2) and the a priori condition (5)

hold for p > 0, if selecting

μ =

⎧

⎪

⎩

1

ln( E

δ ) +21 , 0< p ≤ 2,

1

ln( E

δ )1, p > 2, then we obtain the following error estimate:

f μ δ − f 

≤

⎧

⎪

⎨

⎪

⎛

ln( E

δ ) +21

⎞

⎠E

δ

1 +2

⎛

(ln ( E

δ ) +21 ) p

⎛

(ln ( E

δ ) +21 )2

⎞

⎠

⎞

⎠ ,

0< p ≤ 2,

ln( E

δ )1

E δ

1

δ + E( 1

ln( E

δ )1)2

(ln ( E

δ )1

)2 ,

p > 2.

Proof: By the triangle inequality, we know

f μ δ − f ≤ f μ δ − f μ + f μ − f .

Firstly, we give an estimate for the first term as follows:

f μ δ − f μ

=

+∞

n=1

(1 + μ2)n2

(1 + μ2n2)(1 − e√1+μ2n2−n )

(g δ , X n )X n

−+∞

n=1

(1 + μ2)n2

(1 + μ2n2)(1 − e√1+μ2n2−n )

(g, X n )X n

≤ sup

n≥1

(1 + μ2)n2

(1 + μ2n2)(1 − e√1+μ2n2−n )

+∞

n=1

(g, X n )X n

≤ δ(1 + μ2) sup

n≥1

⎧

⎨

⎩

n·1+ μ2n2e

n

√ 1+μ2n2

1+ μ2n2

⎫

⎬

⎭

≤ δ (1 + μ2) · e

1

μ

1+ μ2n2

≤ δ1+ μ2· e μ1

≤ δ(1 + μ) · e μ1

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According to Lemmas 2.1–2.4 and an a priori bound condition of unknown source, we obtain

f μ − f 

=

+∞

n=1

(1 + μ2)n2

(1 + μ2n2)(1 − e√1+μ2n2−n )

(g, X n )X n−+∞

n=1

n2

1− e −n (g, X n )X n

=

+∞

n=1

μ2n2(e√1+μ2n2−n − 1) + e√1+μ2n2−n − e −n + μ2(1 − e −n )

(1 + μ2n2)(1 − e√1+μ2n2−n )(1 + n2) p2

· n2

1− e −n (1 + n2) p2(g, X n )X n

= sup

n≥1

⎧

⎨

⎩

−μ2n2 (1 + μ2n2)(1 + n2) p2

−n

√ 1+μ2n2 − e −n

(1 + μ2n2)(1 − e√1+μ2n2−n )(1 + n2) p2

+ μ2(1 − e −n )

(1 + μ2n2)(1 − e√1+μ2n2−n )(1 + n2) p2

⎫

⎬

⎭

+∞

n=1

n2

1− e −n (1 + n2) p2(g, X n )X n

≤ sup

n≥1

⎧

⎨

⎩

μ2n2 (1 + μ2n2)(1 + n2) p2

⎛

⎝ e

−n

√ 1+μ2n2 − e −n

μ2n2(1 − e√1+μ2n2−n )

+ 1− e −n

n2(1 − e√1+μ2n2−n )

⎞

⎠

⎫

⎬

⎭· E

≤ E sup

n≥1

μ2n2 (1 + μ2n2)(1 + n2) p2

n · (1 + μ)

1+ μ2n2 · e μ1 + 1+ μ

n · e μ1

≤ E max {μ2,μ p }e μ1(1 + μ)

1

μ2 + 1

Combining above two estimates, we have

f μ δ − f 

≤ δ(1 + μ) · e μ1 + E

max{μ2,μ p }e μ1(1 + μ)

1

μ2+ 1

≤

⎧

⎪

⎩

ln( E

δ ) +21

E

δ

1

p+2

δ + E (ln ( E

δ ) +21 ) p

(ln ( E

δ ) +21 )2 , 0< p ≤ 2,

ln( E

δ )1

E

δ

1

δ + E( 1

ln( E

δ )1)2

(ln ( E

δ )1)2

, p > 2.

Based on the above discussion, we need to illustrate them with some examples in the next section

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4 Numerical verification

In this section, we give some different examples on the basis of the following preparation process

From (4), we know that

(Kf )(x) =

+∞

n=1

1− e −n

= 2

π

0

+∞

n=1

1− e −n

n2 f (s) sin ns sin nxds = g(x). (10)

We use the rectangle formula to approach the integral and do an approximate truncation

for the series by choosing the sum of the front N terms By considering an equidistant grid

0= x1< · · · < x M = π, (x i = i−1

M−1π, i = 1, , M), we get

2

π

M

i=1

N

n=1

1− e −n

n2 f (x i ) sin nx i sin nx j h = g(x j ), (11)

where h= M π−1 Correspondingly, we obtain

f μ δ (x j ) = 2

π

M

i=1

N

n=1

(1 + μ2)n2

(1 + μ2n2)(1 − e√1+μ2n2−n )

g δ (x i ) sin nx i sin nx j h. (12)

Adding a random distribute perturbation to each data function, we obtain g δ, i.e

g δ = g + εrandn(size(g)).

The total noise levelδ can be measured in the sense of root mean square error(RMSE)

according to

δ = g δ − g2=

1

M

n=1

(g n − g δ )2

1

In order to research the effect of numerical computations, we compute the relative root

mean squares error (RRMSE) of f (x) by

RRMSE(f ) =

M

i=1(f δ

μ (¯x i ) − f (¯x i ))2

M

i=1(f (¯x i ))2

where{¯x i}M

i=1is a set of discrete points in internal [0,π].

The numerical examples are constructed in the following way: First we select the exact

solution f (x) and obtain the exact data function g(x) using (11) Then we add a

nor-mally distributed perturbation to each data function giving vector g δ Finally we obtain the regularization solutions using (12)

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Table 1.δ, μ, RRMSE(f) with respect to various values of ε while p = 1, M = 50, N = 5 and for

Example 1

Table 2.δ, μ, RRMSE(f) (relative error of the source term) with respect to various values of ε while p = 3,

M = 50, N = 5 and for Example 1.

Table 3.δ, μ, RRMSE(f) with respect to various values of ε while p = 1, M = 50, N = 5 and the RRMSE(f) approaches 0.0093 when ε ≤ 1 × 10−4for Example 2.

Example 1: We suppose that the solution of equation u(x, y) = (1 − exp(−y)) sin(x)

and the source function f (x) = sin(x), easily know that the data function g(x) = (1 −

exp(−1)) sin(x), we choose x ∈ [0, π] in this example.

Example 2: Consider the reconstruction of a Gaussian normal distribution:

f (x) = 1

α√2π e −(x−β)

2/(2α2)

whereα = π

4,β = π

2

Example 3: Consider the reconstruction of a piecewise smooth source:

f (x) =

⎧

⎪

⎨

⎪

⎩

0, 0≤ x < π4

4

π x− 1, π4 < x ≤ π2

3− 4

π x, π2 < x ≤ 3π

4

0, 34π < x ≤ π.

For Examples 1–3, we illustrate the comparisons about exact solutions and regularized solutions by a priori regularization parameter choice rule with different noise levels and

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different cases of p We can find that the smaller ε is, the better the computed

approxi-mation is For both continuous and discontinuous cases in Examples 2–3, it can also be seen that the well-known Gibbs phenomenon occurs and the approximate solutions near non-smooth and discontinuous points are less ideal

5 Conclusion

In this article, we use a modified quasi-reversibility method to identify an unknown source term depending only on one variable in two dimensional Poisson equation It is shown that with a certain choice of the parameterμ, a stability estimate is obtained Meanwhile, three

examples verify the efficiency and accuracy of our proposed method

Acknowledgments

The authors gratefully thank the referees for their valuable constructive comments which improve greatly the quality of the paper The work described in this article was supported by the NNSF

of China (11326234), NSF of Gansu Province (145RJZA099), Scientific research project of Higher School in Gansu Province (2014A-012), and Project of NWNU-LKQN2020-08

Disclosure statement

No potential conflict of interest was reported by the authors

ORCID

Jin Wen http://orcid.org/0000-0001-9141-2681

References

[1] Cannon JR Determination of an unknown heat source from overspecified boundary data SIAM J Numer Anal.1968;5:275–286

[2] Shi C, Wang C, Wei T Numerical reconstruction of a space-dependent heat source term

in a multi-dimensional heat equation CMES Comput Model Eng Sci 2012;86(2):71–92

doi:10.1002/qua.23236

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