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

Inverse Problems in Science and Engineering ISSN: (Print) (Online) Journal homepage: https://www.tandfonline.com/loi/gipe20 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 Submit your article to this journal Article views: 15 View related articles View Crossmark data Full Terms & Conditions of access and use can be found at https://www.tandfonline.com/action/journalInformation?journalCode=gipe20 INVERSE PROBLEMS IN SCIENCE AND ENGINEERING https://doi.org/10.1080/17415977.2021.1902516 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 ARTICLE HISTORY 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 convergence rate by using an a priori regularization parameter choice rule Numerical examples are provided to show the effectiveness of the proposed method Received 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 Introduction Inverse source problem is of great importance in many branches of engineering and science; 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 variety 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 Mel nikova and Filinkov [16] The idea consists in replacing the final boundary value problem with an approximate solution of the final boundary value problem In the initial method of the quasi-reversibility, the author [17] replaced the heat operator ∂/∂t − ∂ /∂x2 by a perturbed operator Pε = (∂/∂t) + A − εAA∗ , perturbing the final condition we get an approximate solution from the final boundary value problem with a small parameter ε The authors [18] take f (A) = A − A2 using 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 problem 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 © 2021 Informa UK Limited, trading as Taylor & Francis Group J WEN ET AL 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 := c(x) u = f , constructing Tikhonov functional firstly Jε (u) = ∂tt u − Lu − f + ε2 u , it is equivalent with the abstract Euler equation Jε (uε )(v) = 0, for all v ∈ H02 (QT ), then through a minimizer uε of calculation for above equation, they obtain perturb term to approximate the solution of ill-posed problem with a small parameter ε In [22], from the original quasi-reversibility method, the mixed quasi-reversibility method with variable parameter λ is extended in a system of two second-order equations involving two functions u and λ, the aim is to find an approximation (uε , λ) of (u, λ) as a solution of the weak formulation and (δ, ε) denotes α for small ε > and δ > The method of quasi-reversibility proposed by [23] is a particular case of Tikhonov regularization and A = or + k2 , which provides corresponding error estimate with a priori choice for ε as a function of δ In [24], the article adds μ2 fxx (x) to the left-hand side of the equation u(x, y) − k2 u(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: ⎧ −uxx − uyy = f (x), < x < π , < y < ∞, ⎪ ⎪ ⎨ u(0, y) = u(π , y) = 0, ≤ y < ∞, (1) u(x, 0) = 0, u(x, y)| bounded, ≤ x ≤ π, ⎪ y→∞ ⎪ ⎩ u(x, 1) = g(x), ≤ x ≤ π, to find a pair of function (u(x, y), f (x)) which satisfies the Poisson equation on above conditions 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 g − gδ ≤ δ (2) We obtain that the solution of problem (1) using separation of variables has the following form: ∞ u(x, y) = n=1 − e−ny (f , Xn )Xn , n2 (3) where {Xn = sin nx, π (n = 1, 2, )}, (f , Xn ) = π π f (x) sin nxdx We define the operator K : f → g, then we have +∞ g(x) = Kf (x) = ∞ (g, Xn )Xn = n=1 n=1 − e−n (f , Xn )Xn n2 (4) INVERSE PROBLEMS IN SCIENCE AND ENGINEERING The singular values {σn }∞ n=1 of K satisfy σn = − e−n , n2 correspondingly (g, Xn ) = − e−n (f , Xn )(Xn , Xn ), n2 i.e (f , Xn ) = σn−1 (g, Xn ), then, f (x) = K −1 g(x) = +∞ n=1 (g, Xn )Xn = σn +∞ n=1 n2 (g, Xn )Xn − e−n From [30], the solution does not depend on the data continuously, the problem (1) is illposed Several articles impose regularization method to deal with ill-posed problem (1): for examples, the Tikhonov regularization method [31,32], the super order regularization method [33], the quasi-boundary value regularization method [34,35], the quasireversibility 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, f (x) where E > is a constant and defined as follows [43]: · H p (0,π ) H p (0,π ) ≤ E, p > 0, denotes the norm in Sobolev space which is +∞ f (x) H p (0,π ) (5) (1 + n2 )p | f , Xn |2 = n=1 Figure The exact and approximate solutions with M = 50, N = and various noise level for Example where x ∈ [0, π] (a): heat source for p = 1; (b): heat source for p = 4 J WEN ET AL Figure The exact and approximate solutions with M = 50, N = and various noise level for Example where x ∈ [0, π] (a): heat source for p = 1; (b): heat source for p = Figure The exact and approximate solutions with M = 50, N = and various noise level for Example where x ∈ [0, π] (a): heat source for p = 1; (b): heat source for p = This article is organized as follows Section gives some preliminary results In Section 3, a regularization solution and error estimation of the inverse problem are provided by a modified quasi-reversibility method Section gives some examples to illustrate the accuracy and efficiency of the proposed method in problem (1) Section puts an end to this paper with a brief conclusion Some auxiliary results In this section, we give four important lemmas as follows Lemma 2.1: For n ≥ 1, < − e−n Lemma 2.2: If μ > 0, n ≥ 1, 1−e −n 1+μ2 n2 √ ≤ (1 + μ) · e μ INVERSE PROBLEMS IN SCIENCE AND ENGINEERING Proof: 1−e −n 1+μ2 n2 √ ≤ + μ2 n2 · e n ≤ + μ2 · e n2 √ n 1+μ2 n2 1 +μ2 n2 ≤ (1 + μ) · e1/μ Lemma 2.3 ([30]): If μ > 0, n ≥ 1, p > 0, μ2 n2 p (1 + μ2 n2 )(1 + n2 ) < max {μ2 , μp } Lemma 2.4: If μ > 0, n ≥ 1, e −n 1+μ2 n2 √ μ2 n2 + μ2 n2 − e−n < n · Proof: e −n 1+μ2 n2 √ n − e−n = (n − ≤n· 0, if selecting ⎧ ⎪ < p ≤ 2, ⎪ , ⎪ ⎨ ln ( E ) p+2 δ μ= ⎪ ⎪ ⎪ p > 2, ⎩ , ln ( Eδ ) then we obtain the following error estimate: fμδ − f ⎞ ⎧ ⎛ ⎪ ⎪ ⎪ ⎠ E ⎝1 + ⎪ ⎪ ⎪ δ ⎪ ln ( Eδ ) p+2 ⎪ ⎪ ⎪ ⎨ < p ≤ 2, ≤ ⎪ ⎪ ⎪ E ⎪ ⎪ ⎪ 1+ ⎪ ⎪ δ ⎪ ln ( Eδ ) ⎪ ⎩ p > p+2 ⎛ ⎛ ⎝δ + δ + E( E (ln ( Eδ ) p+2 )p ln ( Eδ ) ⎞⎞ ⎝1 + )2 + 1 (ln ( Eδ ) p+2 )2 1 (ln ( Eδ ) )2 Proof: By the triangle inequality, we know fμδ − f ≤ fμδ − fμ + fμ − f Firstly, we give an estimate for the first term as follows: fμδ − fμ +∞ (1 + μ2 )n2 = n=1 (1 + μ2 n2 )(1 − e +∞ ≤ sup n≥1 (1 + μ2 n2 )(1 − e −n 1+μ2 n2 √ (g, Xn )Xn ) +∞ (1 + μ2 )n2 (1 + μ2 n2 )(1 − e ≤ δ(1 + μ2 ) sup ⎧ ⎨n · n≥1 ⎩ ≤δ ) (gδ , Xn )Xn (1 + μ2 )n2 − n=1 −n 1+μ2 n2 √ (1 + μ2 ) · e μ + μ2 n2 ≤ δ + μ2 · e μ ≤ δ(1 + μ) · e μ −n 1+μ2 n2 √ (g, Xn )Xn ) + μ2 n2 e n=1 √ + μ2 n2 n 1+μ2 n2 ⎫ ⎬ ⎭ , ⎠⎠ , INVERSE PROBLEMS IN SCIENCE AND ENGINEERING According to Lemmas 2.1–2.4 and an a priori bound condition of unknown source, we obtain fμ − f +∞ n=1 (1 + μ2 n2 )(1 − e +∞ μ2 n2 (e = +∞ (1 + μ2 )n2 = −n 1+μ2 n2 √ −n 1+μ2 n2 √ − 1) + e (g, Xn )Xn − ) −n 1+μ2 n2 √ (1 + μ2 n2 )(1 − e n=1 n=1 n2 (g, Xn )Xn − e−n − e−n + μ2 (1 − e−n ) √ −n 1+μ2 n2 p n2 (1 + n2 ) (g, Xn )Xn −n 1−e ⎧ ⎨ −μ2 n2 = sup p + n≥1 ⎩ (1 + μ2 n2 )(1 + n2 ) p )(1 + n2 ) · e −n 1+μ2 n2 √ − e−n √ −n p (1 + μ2 n2 )(1 − e 1+μ2 n2 )(1 + n2 ) ⎫ ⎬ +∞ n2 −n p μ (1 − e ) 2 + (1 + n ) (g, Xn )Xn −n √ p ⎭ − e−n n=1 (1 + μ2 n2 )(1 − e 1+μ2 n2 )(1 + n2 ) ⎧ ⎞⎫ ⎛ √ −n ⎬ ⎨ n2 2 −n −n 1+μ 1−e μ n −e ⎠ ·E ⎝ e + ≤ sup p −n −n √ √ ⎭ n≥1 ⎩ (1 + μ2 n2 )(1 + n2 ) μ2 n2 (1 − e 1+μ2 n2 ) n2 (1 − e 1+μ2 n2 ) ≤ E sup n≥1 μ2 n2 (1 + μ2 n2 )(1 + n2 ) ≤ E max {μ2 , μp }e μ (1 + μ) + μ μ1 n · (1 + μ) μ1 ·e ·e + 2 1+μ n n p +1 μ2 Combining above two estimates, we have fμδ − f 1 ≤ δ(1 + μ) · e μ + E max {μ2 , μp }e μ (1 + μ) ≤ ⎧ ⎪ ⎪ ⎪ ⎨ 1+ ⎪ ⎪ ⎪ ⎩ 1+ E δ ln ( Eδ ) p+2 1 ln ( Eδ ) E δ p+2 δ+ δ + E( E (ln ( Eδ ) p+2 )p 1 ln ( Eδ ) )2 + +1 μ2 1+ 1 (ln ( Eδ ) p+2 )2 1 (ln ( Eδ ) )2 , , < p ≤ 2, p > Based on the above discussion, we need to illustrate them with some examples in the next section 8 J WEN ET AL 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 = π − e−n (f , Xn )Xn n2 π +∞ (9) − e−n f (s) sin ns sin nxds = g(x) n2 n=1 (10) We use the rectangle formula to approach the integral and an approximate truncation for the series by choosing the sum of the front N terms By considering an equidistant grid i−1 = x1 < · · · < xM = π, (xi = M−1 π, i = 1, , M), we get π where h = M N i=1 n=1 − e−n f (xi ) sin nxi sin nxj h = g(xj ), n2 π M−1 Correspondingly, fμδ (xj ) = π M (11) we obtain N (1 + μ2 )n2 i=1 n=1 (1 + μ2 n2 )(1 − e −n 1+μ2 n2 √ g δ (xi ) sin nxi sin nxj 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 level δ can be measured in the sense of root mean square error(RMSE) according to δ δ = g −g = M M δ 2 (gn − g ) n=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 δ x ) − f (¯ xi ))2 i i=1 (fμ (¯ M xi ))2 i=1 (f (¯ , (13) where {¯xi }M i=1 is 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 normally distributed perturbation to each data function giving vector g δ Finally we obtain the regularization solutions using (12) INVERSE PROBLEMS IN SCIENCE AND ENGINEERING Table δ, μ, RRMSE(f ) with respect to various values of ε while p = 1, M = 50, N = and for Example ε μ δ RRMSE(f ) 0.01 0.0.001 0.0001 0.00001 0.0279 0.0113 0.0092 0.0202 0.0011 0.0016 0.0152 0.0001 0.0001 0.0127 8.7 × 10−6 1.05 × 10−5 Table δ, μ, RRMSE(f ) (relative error of the source term) with respect to various values of ε while p = 3, M = 50, N = and for Example ε μ δ RRMSE(f ) 0.01 0.0.001 0.0001 0.00001 0.1065 0.0110 0.0070 0.0985 0.0012 0.0006 0.0919 0.0001 0.0004 0.0878 1.03 × 10−5 3.557 × 10−4 Table δ, μ, RRMSE(f ) with respect to various values of ε while p = 1, M = 50, N = and the RRMSE(f ) approaches 0.0093 when ε ≤ × 10−4 for Example ε μ δ RRMSE(f ) 0.01 0.001 0.0001 0.00001 0.0952 0.0099 0.0169 0.0844 0.0010 0.0095 0.0774 0.0001 0.0093 0.0722 9.709 × 10−6 0.0093 Table δ, μ, RRMSE(f ) with respect to various values of ε while p = 3, M = 50, N = and the RRMSE(f ) approaches 0.1053 when ε ≤ × 10−9 for Example ε μ δ RRMSE(f ) 0.01 0.001 0.0001 0.00001 0.1280 0.0099 0.1663 0.1100 0.0009 0.1063 0.1010 0.0001 0.1059 0.0942 1.036 × 10−5 0.1056 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) = where α = π 4,β = 2 √ e−(x−β) /(2α ) α 2π π Example 3: Consider the reconstruction of a piecewise smooth source: ⎧ 0, ≤ x < π4 ⎪ ⎪ ⎪ π π ⎨ x − 1, π