Research purposes: We consider some inverse source problems for parabolic equations, focus on three topics: First, we give stability estimates. Second, we propose regularization methods to solve these problems. Third, we set up algorithms and give numerical examples to illustrate the performance of the proposed regularization methods in this thesis.
MINISTRY OF EDUCATION AND TRAINING VINH UNIVERSITY LUONG DUY NHAT MINH ON SOME PROBLEMS OF IDENTIFYING UNKNOWN SOURCE TERM FOR PARABOLIC EQUATIONS CODE: 46 01 02 A SUMMARY OF DOCTORAL THESIS IN MATHEMATICS Nghe An – 2021 The work is accomplished at Vinh University, Nghe An Supervisors:: Assoc Prof Dr Nguyen Van Duc Dr Nguyen Trung Thanh Reviewer 1: Reviewer 2: Reviewer 3: Thesis will be defended at School-level thesis vealuating Council at: On the hour day month year Thesis is stored in at: National Library of Vietnam Nguyen Thuc Hao Center of Information and Library, Vinh University INTRODUCTION Rationale The problems of identifying unknown source terms for parabolic equations has been studied by many scientists from 1960s Mathematicians who have works on this problem are Cannon, Dinh Nho Hao, Dang Duc Trong, Hasanov, Isakov, Li, Savateev, Prilepko, Yang, Fu, The above problem is usually ill-posed in Hadamard’s sense A problem is called well-posed in the Hadamard’s sense if it satisfies all of the following conditions: i) The solution of the problem exists ii) The solution of the problem is unique iii) The solution depends continuously on the data of the problem If at least one of the three above conditions is not satisfied, the problem is called ill-posed In ill-posed problems, a small error of data can also lead to large deviation of the solution Therefore, ill-posed problems are often more difficult to solve than well-posed problems because the data used in ill-posed problems are often generated by measurements, so there is inevitable error Furthermore, many calculations are only performed approximately To solve an ill-posed problem, scientists often propose regularization methods, that is to use the solution of a well-posed problem to make the approximate solution to the original ill-posed problem Research on the problems of identifying unknown source terms for parabolic equations often focuses on three main topics: i) The uniqueness of solutions ii) Stability estimates iii) Regularization methods and the numerical methods There has been a lot of research on the problems of identifying unknown source terms of parabolic equations because they are the mathematical models of practical problems such as determining the heat source in the heat transfer equation, identifying the polluted water source, Currently, there are many open problems related to the problem of determining the source terms in parabolic equations that need to be studied, in which the results of stability estimates and regularization for parabolic equations with time-dependent coefficients has been fully studied, with only a few results on the uniqueness of the solution of this type of problem The research direction of the problem of determining the source of the fractional parabolic equations has received the attention of many scientists However, most of the results listed above are for fraction parabolic equations by time or spatial variable, with only a few results for fractional parabolic equations for both spatial and time variables Regarding the results of stability estimates and regularization for the problem of determining the source of parabolic equations in Banach space, according to our search, there are only a few relevant results For the above reasons, we choose research topics for our thesis as: "On some problems of identifying unknown source term for parabolic equations" Research purposes We consider some inverse source problems for parabolic equations, focus on three topics: First, we give stability estimates Second, we propose regularization methods to solve these problems Third, we set up algorithms and give numerical examples to illustrate the performance of the proposed regularization methods in this thesis Research subjects We focus on identifying unknown source terms of parabolic equations in three cases: i) Parabolic equations with time dependent coefficients in Hilbert space L2 (Rn ); ii) Time-space fractional parabolic equations in Hilbert space L2 (Rn ); iii) Parabolic equations in Banach spaces Research scopes We research stability estimates, regularization methods and numerical methods to solve some problems of identifying unknown source terms for parabolic equations Research methods We used the logical reasoning method based on the previously known results We also use numerical methods to solve these inverse source problems Scientific and practical meaning The thesis contributes to enriching the research results in the field of inverse problems The thesis has achieved some results on stability estimates, regularization methods and numerical methods to solve the inverse source problem of parabolic equations This thesis is the reference for students, graduated students Overview and structure of the thesis 7.1 Overview of some issues related to the thesis To facilitate the introduction of research results related to the inverse source problems for parabolic equations, we take a concrete example of the linear parabolic equation in Hilbert space Let T be a positive real number, X is a Hilbert space · , u : [0, T ] → X is a function from [0, T ] to X and F ∈ X We consider the initial value problem u (t) + Au(t) = F, t ∈ (0, T ), (1) u(0) = 0, with the norm where A is the unbounded linear operator on X The problem (1) is forward problem, in which we need to find u when F is known The problem of identifying an unknown source for (1) is to find the source function F from the measurements of the function u This is an inverse problem There are many different types of measurements in use, for example: boundary measurements, at the final time measurements, or measurements at some discrete points, Therefore, there are many types of problems of identifying an uknown source for parabolic equations Due to the ill-posedness, the solution of the problem does not always exist and if it exists, the solution may not depend continuously on the data of the problem This makes the inverse source proplem difficult to solve Usually, mathematicians have to propose regularization methods to solve the ill-posed problems There are many regularization methods to solve the inverse source problems for parabolic equations, such as: quasi-reversibility method, Tikhonov method, finite element method, variational method, conjugate gradient method, mollification method, Here, we would like to summarize the types of inverse source problems for parabolic equations that we investigate and summarize the main results that we has achieved i) First, we considered the inverse source problem for parabolic equations with time dependent coefficients in Hilbert space L2 (Rn ) Find a pair of functions {u(x, t), f (x)} satisfies: ∂u (x, t) = a(t)∆u(x, t) + f (x)h(t), x ∈ Rn , t ∈ (0, T ), ∂t u(x, 0) = 0, x ∈ Rn , u(x, T ) = g(x), x ∈ Rn , (2) with ∆ is the Laplace operator, a(t), h(t) are given and g(x) is the data at the final time T > For the case when the coefficient a(t) is time independent, several results have been obtained concerning the uniqueness; stability estimates and regularization methods Results concerning the case of time-dependent coefficient as in (2) are less popular Some uniqueness results for this kind of problems were reported by Slodiˇcka and Johansson; D’haeyer et al In 2014, D’haeyer et al considered the inverse source problem for the parabolic heat equation, where the coefficients that depend both on space and time With domain Rn , they considered problem ∂u + Lu = f (x) in (0, T ), u=0 u(x, 0) = u (x) on ∂ (0, T ), for x , (3) T > is the final time, L is a linear elliptic operator of the second order with differentiable coefficients depending both on space and time They proved that this inverse source problem has a unique solution, and they also proposed a stable iterative procedure to find the heat source In 2016, Slodiˇcka and Johansson investigated the uniqueness of a solution for inverse source problems arising in linear parabolic equations with time dependent coefficients They determined f (x) which satisfies ∂u(x, t) + L(t)u(x, t) = f (x)h(t) in (0, T ), (4) u=0 on ∂ (0, T ), u(x, 0) = u (x) for x , with function h(t) given and the information of L(t), T, u(x, T ), , ∂ being the same as (3) For the inverse source problem for a parabolic equation with time dependent coefficients in Hilbert space L2 (Rn ) in our thesis, we proved the Hăolder-type stability estimates (Theorem 2.2.1) To regularize this problem, we used the mollification method, we established the Hăolder-type error estimates for the regularized solutions using both a priori and a posteriori parameter choice rules (Theorem 2.3.2 and 2.3.5) We illustrated the above theoretical results by using numerical examples ii) Second, we considered the inverse source problem for a time-space fractional parabolic equation in Hilbert space L2 (Rn ): Find a pair functions fu(x, t), f (x)g satisfies γ ∂tα u + ( ) u = f (x)h(t), x Rn , t (0, T ), u(x, 0) = 0, x Rn , u(x, T ) = g(x), x Rn (5) ∂α Here, T is a positive real number, α (0, 1), x = (x1 , , xn ) R , = α is the ∂t γ Caputo fractional derivative of order α respect to t, γ > , ( ) is the fractional Laplacian, g(x) is the data at the final time Most of published works consider the fractional derivative with respect to one variable only, either time or space variable Results concerning the case of equations with fractional derivatives with respect to both variables as in (5) are less popular In 2014, Tartar et al considered the problem n ∂tα of finding the function u(t, x) and f (x) with t ∈ (0, T ) and x ∈ Ω = (−1, 1), which satisfies ∂β α β u(t, x) + f (x)h(t, x), u(t, x) = −r (−∆) β ∂t u(t, −1) = u(t, 1) = 0, u(0, x) = 0, u(T, x) = ϕ(x), (t, x) ∈ ΩT , < t < T, (6) x ∈ Ω, ¯ x ∈ Ω, here, ΩT := (0, T ) × Ω, r > is a parameter, f (x) ∈ L2 (Ω), h(t, x) is given, β ∈ (0, 1), α ∈ (1, 2) are fractional order of the time and the space derivatives, ∂β respectively, T > is the final time and β is the Caputo fractional derivative With ∂t the data at the final time T , the authors proved the existance and the uniqueness of the solution In a research, Li and Wei considered the problem determining time dependent term p(t) of the source function f (x)p(t), which satisfies β α u(x, t) + f (x)p(t), (x, t) ∈ ΩT , u(x, t) = −(−∆) ∂ 0+ u(x, 0) = φ(x), ¯ x ∈ Ω, (7) u(x, t) = 0, x ∈ ∂Ω, t ∈ (0, T ], u(x , t) = g(t), x0 ∈ Ω, < t ≤ T, here, ΩT := Ω × (0, T ], Ω ⊂ Rn ; T > is the final time; α ∈ (0, 1), β ∈ (1, 2) α are fractional order of the time and the space derivatives, respectively; ∂0+ is the Caputo fractional derivative They proved the uniqueness and proposed the sta- bility estimate In 2014, Tuan et al considered an inverse problem to identify an unknown source term in a space time fractional diffusion equation In addition to the similar information of problem (6), they also replaced the first equation in (6) by ∂β u(t, x) = −rβ (−∆)α/2 u(t, x) + f (x)h(t) At this time, function h depends only β ∂t on the time variable t Moreover, in equation u(T, x) = ϕ(x), then x ∈ Ω instead of ¯ They proposed the Fourier truncation method to solve this problem They x∈Ω also proposed a priori and a posteriori parameter choice rules and analyzed them But no numerical examples were given For the inverse source problem for a time-space fractional parabolic equation in Hilbert space L2 (Rn ) in our thesis, we proved a Hăolder-type stability estimate of optimal order (Theorem 3.2.3 and Remark 3.2.4) We regularized this problem by the mollification method and we established the Hăolder-type error estimates for the regularized solutions using both a priori and a posteriori parameter choice rules (The- orem 3.3.2 and 3.3.6) We also demonstrated the above results by using numerical examples iii) Third, we considered the inverse source problem for parabolic equation in Banach spaces Suppose X is the Banach space with norm · , A : D(A) ⊂ X → X is the unbounded linear operator such that −A generates an analytic semigroup {S(t)}t≥0 on X , with D(A) is the domain of A and assume that D(A) is dense in X For t ∈ [0, T ], denote u(t) is a function from [0, T ] to X and F ∈ X , we identify source function F from u (t) + Au(t) = F, t ∈ (0, T ), (8) u(0) = 0, u(T ) = g, with g ∈ X is the data at the final time T To the best of our knowledge, the results about this inverse source problem are not popular Some of the earliest works on inverse source problems for parabolic equations in Banach spaces were due to Iskenderov and Tagiev; Rundell The uniqueness of (8) is proven by Eidelman; Tikhonov In case of F is the time dependent function, the authors also proved the uniqueness In 2005, Tikhonov and Eidelman considered an inverse source problem in Banach space E , with A is a closed linear operator, and the domain D(A) ⊂ E (the domain may not dense in E ) Let T > and function ϕ = is continous on [0, T ] They found {u(t), p} satisfied ∂u(t) = Au(t) + ϕ(t)p, ∂t u(0) = u0 , u(T ) = u1 , ≤ t ≤ T, (9) u0 , u1 ∈ D(A) They proved the uniqueness of the solution of the inverse problem For a Banach space X , · is the norm of X , in 1980, Rundell considered a problem find a pair functions {u(t), f } satisfy du + Au = f, dt u(0) = u0 , u(T ) = u1 , (10) here A is a linear operator in X and f ∈ X , additional information is of two values of u at two fixed points (t = and t = T > 0) With assumption u0 , u1 ∈ D(A) (D(A) is the domain of A, A−1 : X → D(A) exists and A generates a strongly continuous semigroup of operators {S(t)}t≥0 such that S(t) < 1, the authors proved the existance and uniqueness of this problem and (u(t), f ) were described as t follow: u(t) = S(t)u0 + S(t − τ )f dτ and f = (I − S(T ))−1 (Au1 + AS(T )u0 ), with I is the identity operator in X In 1991, Eidelman used the theory of semigroups of operators to prove the uniqueness of an inverse source problem in Banach space E , the author considered the problem of finding a pair of functions {v(t), p} which ∂v satisfied = Av + f (t) + p, here A is an unbounded linear operator, f (t) is a ∂t continous function on [0, t1 ] with values in E , p ∈ E is an unknown parameter and they set the boundary conditions: v(0) = v0 , v(t1 ) = v1 In 2007, Prilepko et al regularized an inverse source problem in Banach space, but they did not propose the converate rates and numerical examples were not considered In 2013, Hasanov et al presented a semigroup approach to propose a representation formula for a solution of an inverse source problem for the heat equation ut = Au + F with measured data at the final time uT (x) := u(x, T ), and they proved the uniquess of this problem In our thesis, to regularize the inverse source problem for a parabolic equation in Banach spaces, base on the theory of semigroups of operators, we proposed a new regularization method and proved Hăolder-type error estimates for the regularized solutions are proved for both a priori and a posteriori regularization parameter choice rules (Theorem 4.2.7 and 4.2.9) We also demonstrated the above theoretical results by using numerical examples 7.2 Organization of the thesis The main content of the thesis is presented in 04 chapters In Chapter 1, we present the auxilary results, which are used in thesis In Chapter 2, we present the new results of stability estimates and the obtained new regularization for an inverse source problem for parabolic equation with time dependent coefficients in Hilbert space L2 (Rn ) by mollification method In Chapter 3, we present the new result of stability estimate and the obtained new regularization for an inverse source problem for time-space fraction parabolic equation in Hilbert space L2 (Rn ) by mollification method In Chapter 4, we present the obtained new regularization results of the inverse source problem for parabolic equation in Banach spaces The main results of the thesis were presented at the seminar of the Analysis Department, Institute of Natural Pedagogy - Vinh University, Scientific seminar "Researching and teaching Mathematics to meet the current education innovation requirements" at Vinh University, Nghe An on September 21st, 2019 These results have published in 03 articles, including 01 article on Inverse Problems in Science and Engineering (SCIE, IF: 1.314), 01 article on Applicable Analysis (SCIE, IF: 1.107) and 01 article on Applied Numerical Mathematics (SCIE, IF: 1.979) CHAPTER AUXILIARY RESULTS 1.1 Some results in Analysis Definition 1.1.1 For v : Rn → R is a measurable function, and p is a real number which satisfies ≤ p < ∞, Lp (Rn ) is defined by Lp (Rn ) = v : Rn → R : |v|p dx < ∞ Rn with norm v Lp |v|p dx := p Rn Definition 1.1.3 For k ∈ L1 (Rn ) and f ∈ L2 (Rn ), we define the convolution of k and f as follow (k ∗ f )(x) = √ ( 2π)n k(x − y)f (y)dy = √ ( 2π)n Rn k(y)f (x − y)dy, Rn with x ∈ Rn Definition 1.1.4 For every v ∈ L2 (Rn ), we define the function F and F−1 from L2 (Rn ) to L2 (Rn ) as follow ( 2π)n e−iξ·x v(x)dx, F(v)(ξ) := v(ξ) := √ ξ ∈ Rn , Rn eiξ·x v(ξ)dξ, x ∈ Rn , n ( 2π) Rn here ξ · x is the inner product of two vectors ξ and x in Rn We call F(v)(ξ) and F−1 (v)(x) are Fourier transform and inverse Fourier transform of v F−1 (v)(x) := v ˇ(x) := √ Definition 1.1.5 For p > 0, H p (Rn ) space is defined by H p (Rn ) = v ∈ L2 (Rn ) : |v(ξ)|2 (1 + |ξ|2 )p dξ < ∞ Rn with norm v Hp 2 p |v(ξ)| (1 + |ξ| ) dξ := Rn 10 ii) The set σ(A) = C \ ρ(A) is called the spectrum set of A Definition 1.2.4 Let Ω = {z : ϕ1 < arg z < ϕ2 , ϕ1 < < ϕ2 } be a sector For z ∈ Ω, let S(z) be a bounded linear operator The family {S(z)}z∈Ω is an analytic semigroup in Ω if i) For all z ∈ Ω, z → S(z)x is analytic in Ω; ii) S(0) = I and for every x ∈ X , we have lim z→0, z∈Ω S(z)x = x; iii) For z1 , z2 ∈ Ω, S(z1 + z2 ) = S(z1 )S(z2 ) A semigroup {S(t)}t≥0 will be analytic if it is analytic in a sector Ω containing the nonnegative real axis Definition 1.2.5 We will call the (possibly unbounded) operator, A, a generator if A generates a uniformly bounded strongly continous holomorphic semigroup {e−zA }Re z By switching to the equivalent norm | x | = sup e−zA , Re z if necessary, we may assume that e−zA ≤ 1, whenever Re z Definition 1.2.6 If A is a generator and s G(s, A) := 0, then − cos(sr) irA dr e r2 π R (1.1) Definition 1.2.10 Let A be a densely-defined closed linear operator which satisfies Definition 1.2.5 such that ρ(A) ⊃ Σ+ := {λ : < ω < |argλ| ≤ π} ∪ V, where ω is a given positive real number and V is a neighborhood of zero in the complex plane C For b > 0, the power A−b of A is defined by: A−b := 2πi z −b (A − Iz)−1 dz, (1.2) C where C is a path running in the resolvent set of A from ∞e−iv to ∞eiv , with −1 ω < v < π We also define Ab := A−b and A0 = I For positive integral values of b the definition (1.2) coincides with the classical definition of (A−1 )n 1.3 1.3.1 Ill-posed problem Definition of ill-posed problem In this section, we present the ill-posed problem 11 1.3.2 Regularization of ill-posed problem In this section, we present the regularization for ill-posed problem and the regularization operator 1.3.3 Optimal oder In this section, we present some content on optimal order 1.4 The mollification method In order to solve the ill-posed problems, scientists must propose regularization methods In our thesis, we used the mollification method to solve two inverse source problems for parabolic equations In this section, we summarize some content related to mollification method 12 CHAPTER IDENTIFYING AN UNKNOWN SOURCE TERM OF A PARABOLIC EQUATION WITH TIME-DEPENDENT COEFFICIENTS An inverse source problem for an n-dimensional heat equation with a time-varying coefficient is investigated The spatially-dependent component of a source function is determined from a measurement at the final time The inverse problem is regularized by a mollification method These results were published in N V Duc, L D Nhat Minh and N T Thanh (2020), "Identifying an unknown source term in a heat equation with time-dependent coefficients", Inverse Problems in Science and Engineering (https://doi.org/10.1080/17415977.2020.1798421)(SCIE, IF: 1.314) 2.1 Introduction We consider the following inverse source problem ISP1: Find a pair functions {u(x, t), f (x)} satisfying: ∂u (x, t) = a(t)∆u(x, t) + f (x)h(t), x ∈ Rn , t ∈ (0, T ), ∂t u(x, 0) = 0, x ∈ Rn , u(x, T ) = g(x), x ∈ Rn (2.2) ∂u with is the change in temperature over time, a(t) is the time dependent coef∂t ficient, f (x)h(t) is the heat source, ∆ is the Laplace operator Instead of g(x) we assume that a “noisy” data g δ ∈ L2 (Rn ) is given, which satisfies g − g δ L2 ≤ δ , δ is a positive constant representing the level of measurement error Throughout this chapter, we make use of the following assumptions Assumption 1: Coefficient a(t) is continuous on [0, T ] and there exist positive constants a and a such that a ≤ a(t) ≤ a Assumption 2: Function h(t) is integrable over [0, T ] In addition, it satisfies one of the following conditions: a) There exist positive constants h and h such that h ≤ h(t) ≤ h, ∀t ∈ [0, T ] 13 b) h(t) ≥ 0, ∀t ∈ [0, T ] and T θ h(t)dt > for each θ ∈ [0, T ) Assumption 3: The space-dependent term f (x) of the source function satisfies one of the following conditions: a) f Hp ≤ Ep for some positive constants p and Ep b) |||f |||q ≤ Eq for some positive constants q and Eq We define the following functions for θ ∈ [0, T ] and τ ∈ R: T H(θ) := T h(t)dt, b := θ 2.2 T a(t)dt, I(θ, τ ) := e −τ T s a(t)dt h(s)ds (2.5) θ Stability estimates Theorem 2.2.1 Suppose that Assumption holds and g L2 ≤ δ i) If Assumptions 2(a) and 3(a) are satisfied, with Ep > δ , then f p 2(p+2) L2 ≤ C1 p+2 p Ep δ p+2 , (2.17) where C1 is a positive constant ii) If Assumptions 2(b) and 3(b) are satisfied, with Eq > δ , then −q f 2.3 L2 q b ≤ (H(0)) q+b δ q+b Eqq+b (2.18) The regularization for the inverse source problem The first step is to mollify the noisy data g δ to obtain the following function δ Sν (g )(x) := π n (Dν ∗ g δ )(x) = πn Dν (y)g δ (x − y)dy (2.23) Rn In the second step, an approximation f ν of f (x) is determined by solving the following problem: Find f ν satisfies ν ∂v (x, t) = a(t)∆v ν (x, t) + f ν (x)h(t), x ∈ Rn , t ∈ (0, T ), ∂t (2.24) v ν (x, 0) = 0, x ∈ Rn , ν v (x, T ) = Sν (g δ (x)), x ∈ Rn Now, we state our results on error estimate for a priori and a posteriori parameter choice rules in the following theorem 14 Theorem 2.3.2 Assume that Assumption holds Then, i) If Assumptions 2(a) and 3(a) are satisfied, with Ep > δ , and the mollification E parameter ν is chosen as ν = ( δp ) p+2 , then there exists a constant Cp > such that the following error estimate holds f −f ν L2 ≤ Cp δ p p+2 p+2 Ep (2.31) ii) If Assumptions 2(b) and 3(b) are satisfied, with Eq > δ , and the mollification Eq 1/2 parameter ν is chosen as ν = ln , then the following error estimate q+b δ holds b q q+b ν q+b f − f L2 ≤ + δ Eq , (2.32) H(θ1 ) where θ1 ∈ [0, T ) In a posteriori mollification parameter choice rules Assume that δ is a positive number such that δ < g δ L2 Let τ be a positive number slightly larger than such that τ δ < g δ L2 That there exists a number νδ > depending on δ such that G(νδ ) = Sνδ (g δ ) − g δ L2 = τ δ (2.41) Theorem 2.3.5 Assume that Assumption holds and νδ is chosen by (2.41) i) If Assumptions 2(a) and 3(a) are satisfied, with Ep > δ , then there exists a constant Cp∗ > such that f −f νδ L2 ≤ p Cp∗ δ p+2 Epp+2 (2.42) ii) If Assumptions 2(b) and 3(b) are satisfied, with Eq > δ , then there exists a constant Cq∗ > such that f −f 2.4 νδ b L2 ≤ q Cq∗ δ q+b Eqq+b (2.43) Numerical algorithm and examples In this section, we propose a non-iterative algorithm for solving the inverse source problem Then we demonstrate the performance of the proposed algorithm using numerical examples in one and two dimensions 2.5 Conclusions We proved Hăolder-type stability estimates for an inverse source problem for the heat equation with a time-dependent coefficient in Hilbert space L2 (Rn ) 15 To regularize the ISP, we used the mollification method Error estimates of the same types were also obtained for both a priori and a posteriori mollification parameter choice rules We also proposed a fast non-iterative algorithm for solving the inverse source problem 16 CHAPTER IDENTIFYING AN UNKNOWN SOURCE TERM OF A TIME-SPACE FRACTIONAL PARABOLIC EQUATION An inverse problem of identifying an unknown spatial-dependent source term in a time-space fractional parabolic equation in Hilbert space L2 (Rn ) is considered Under reasonable boundedness assumptions about the source function, a Hăolder-type stability estimate of optimal order is proved To regularize the inverse source problem, a mollification regularization method is applied These results were published in N V Thang, N V Duc, L D Nhat Minh and N T Thanh (2021), "Identifying an unknown source term of a time-space fractional parabolic equation", Applied Numerical Mathematics, 166, 313-332 https://doi.org/10.1016/j.apnum.2021.04.016 (SCIE, IF: 1.979) 3.1 Introduction Let T > 0, α ∈ (0, 1), x = (x1 , , xn ) ∈ Rn For γ > 0, we define the operator γ (−∆) by γ (−∆) v(x) := F−1 (|ξ|γ F(v)(ξ))(x), γ where v ∈ D((−∆) ) = H γ (Rn ) and H γ (Rn ) is the homogeneous Sobolev space of order γ Here F and F−1 denote the Fourier transform and inverse Fourier transform operators, respectively We consider the following inverse source problem for the time-space fractional parabolic equation: ISP2: Find a pair of functions {u(x, t), f (x)} satisfying γ α u = f (x)h(t), ∂ u + (−∆) x ∈ Rn , t ∈ (0, T ), t u(x, 0) = 0, x ∈ Rn , u(x, T ) = g(x), x ∈ Rn , (3.1) where α and γ are positive constants and < α < Function g(x) in (3.1) is considered as the measured data at the final time instant, t = T As usual, to account for measurement error, we assume that instead of the 17 “exact” data g , we have a “noisy” data g δ ∈ L2 (Rn ) which satisfies g − gδ ≤ δ, L2 (3.2) where δ is a positive constant representing the level of measurement error and · L2 is the L2 -norm in Rn Here, function h(t) in (3.1) is a continuous differential function on [0, T ], which satisfies Assumption in Chapter 2, it means T h(t) ≥ 0, t ∈ [0, T ]; h(s)ds > 0, s ∈ [0, T ) (3.3) In this chapter, we further assume that h(t) satisfies the following condition,that is: there exists a constant T0 ∈ [0, T ) such that h(t) ≥ η > 0, t ∈ [T0 , T ] Moreover, for p > 0, we assume that there exists a positive constant E > δ such that the following condition holds f 3.2 Hp ≤ E (3.4) Stability estimate Theorem 3.2.3 If f is a solution of problem (3.1) and f with δ ≤ E , then there exists a constant C > such that f p L2 p ≤ E, p > 0, g ≤ δ γ ≤ Cδ γ+p E γ+p (3.8) Remark 3.2.4 In Theorem 3.2.3, the estimate (3.8) is of optimal order 3.3 The regularization for the inverse source problem We use the mollification method Problem (3.1) find a pair functions {u(x, t), f (x)} is switching to the problem find a pair functions γ ∂tα v ν + (−∆) v ν = f ν (x)h(t), v ν (x, 0) = 0, v ν (x, T ) = S (g δ )(x), ν δ with Sν (g )(x) := π n {v ν (x, t), f ν } satisfies x ∈ Rn , t ∈ (0, T ) x ∈ Rn , (3.12) x ∈ Rn , (Dν ∗ g δ )(x) Theorem 3.3.1 Problem (3.12) is stable Furthermore, if {viν (x, t), fiν } are solutions of (3.12) associated with data functions giδ for i = 1, 2, then there exists a constant C¯ > such that f1ν − f2ν L2 ¯ + ν γ ) g1δ − g2δ ≤ C(1 L2 18 The following theorem states an error estimate for an a priori mollification parameter choice rule Theorem 3.3.2 Assume that condition (3.4) is satisfied, with E > δ , and the E 1/(p+γ) Then there exists a constant mollification parameter ν is chosen as ν = δ C ∗ > such that p γ f − f ν L2 ≤ C ∗ δ p+γ E p+γ To state an error estimate for the a posteriori mollification parameter choice rule, we assume that < δ < g δ L2 Let τ be a real number sightly larger than such that < τ δ < g δ L2 There exists a number νδ > depending on δ such that G(νδ ) := Sνδ (g δ ) − g δ L2 = τ δ (3.23) With the mollification parameter νδ being chosen by the aposteriori rule (3.23), we obtain the following error estimate Theorem 3.3.6 Assume that condition (3.4) is satisfied, and νδ is chosen by (3.23), with E > δ , then there exists a constant C ∗∗ > such that f − f νδ 3.4 L2 ≤ C ∗∗ δ p/(p+γ) E γ/(p+γ) Numerical algorithm and examples We propose in this paper a direct method for reconstructing the regularized source function f ν (x) We demonstrate the performance of the proposed algorithm using numerical examples in one and two dimensions 3.5 Conclusions We proved Hăolder-type stability estimates of optimal order for an inverse source problem for the time-space fractional parabolic equation in Hilbert space L2 (Rn ) We proposed a mollification regularization method with Dirichlet kernel and also obtained Hăolder-type error estimates for both a priori and a posteriori mollification parameter choice rules We also proposed a fast non-iterative algorithm for solving the inverse problem 19 CHAPTER IDENTIFYING AN UNKNOWN SOURCE TERM OF A PARABOLIC EQUATION IN BANACH SPACES In this chapter, a new regularization method base on semigroup theory for an inverse source problem for a parabolic equation in a Banach space is proposed Hăolder-type error estimates for the regularized solutions are proved for both a priori and a posteriori regularization parameter choice rules Some numerical examples are presented for illustrating the efficiency of the method These above results were published in N V Duc, N V Thang, L D Nhat Minh and N T Thanh (2020), "Identifying an unknown source term of a parabolic equation in Banach spaces", Applicable Analysis (https://doi.org/10.1080/00036811.2020.1800650) (SCIE, IF: 1.107) 4.1 Introduction · Let A : D(A) ⊂ X → X be a densely-defined linear operator in X such that −A generates an analytic semigroup {S(t)}t≥0 on X Here D(A) is the domain of A and we assume D(A) X We assume that the following condition holds: Let T be a positive real number and X be a Banach space with norm S(T ) < (4.1) Let u : [0, T ] → X be a function from [0, T ] to X and F be an element in X We consider the inverse problem ISP3: Determining the element F in the following problem u (t) + Au(t) = F, t ∈ (0, T ), u(0) = 0, u(T ) = g, (4.3) with g ∈ X is the data at the final time T , we assume that only a noisy datum g δ is given, which is merely in X and satisfies g − g δ ≤ δ , where δ > represents the noise level Throughout this chapter, we assume that there exists a positive constant 20 E > δ such that the following condition holds: (I + Ap )F ≤ E , where p is some positive real number such that p ≥ and Ap is a fractional power of A 4.2 The regularization for the inverse source problem In this section, we propose a regularization method, which approximates (4.3) by finding the source function Fα in the following problem v (t) + Av(t) = Fα , t ∈ (0, T ), v(0) = 0, v(T ) = (I + αAb )−1 g δ , (4.6) where α > is a regularization parameter, b is a positive integer, and I is the identity operator in X First, we represent the solutions of the inverse source problem (4.3) and the regularized problem (4.6) via the semigroup S(t) Since u(0) = 0, the solution u(t) of (4.3) is given by t u(t) = S(t)u(0) + t S(s)F ds = S(s)F ds Hence, T S(s)F ds = (S(T ) − I)F Au(T ) = A Since u(T ) = g , we have F = (S(T ) − I)−1 Ag (4.8) In our approach, the solution of the regularized problem (4.6) is given by Fα = (S(T ) − I)−1 A(I + αAb )−1 g δ (4.9) Error estimates for the regularized solution Fα in case of a priori parameter choice rules are given in the following theorem Theorem 4.2.7 Suppose that b is a positive integer and condition (I +Ap )F ≤ E is satisfied for some p ≥ Then there exist positive constants C ∗ , C ∗∗ such that C ∗ (α−1/b δ + αE) if p ≥ b, F − Fα ≤ (4.16) C ∗∗ α−1/b δ + αp/b E if ≤ p < b In particular, i) for p ≥ b and α = δ E b/(b+1) , we obtain F − Fα ≤ 2C ∗ δ b/(b+1) E 1/(b+1) ; (4.17) 21 ii) for p < b and α = δ E b/(1+p) , we obtain F − Fα ≤ 2C ∗∗ δ p/(1+p) E 1/(1+p) (4.18) For a posteriori parameter choice rules, we define the function ρ(α) by ρ(α) := (I + αAb )−1 g δ − g δ (4.21) We assume that < δ < g δ This assumption is practically reasonable since the measured datum is useless when its error is too high ρ(α) is a continuous function on (0, ∞) and lim ρ(α) → and lim ρ(α) → g δ α→∞ α→0 Let τ be a positive constant such that τ > and τ δ < g δ Then, there exists a parameter α = α(δ, τ ) such that ρ(α) = τ δ (4.28) Error estimates for the regularized solution Fα with α being chosen using the a posteriori rule (4.28) are stated in the following theorem Theorem 4.2.9 Suppose that b > is an integer and condition (I + Ap )F ≤ E is satisfied for some p ≥ Let Fα be the solution of the regularized problem (4.6) with regularization parameter α given by (4.28) Then, there exist positive constants C¯ and C such that 4.3 ¯ (b−1)/b E 1/b , if p ≥ b − 1, F − Fα ≤ Cδ (4.29) F − Fα ≤ Cδ p/(1+p) E 1/(1+p) , if ≤ p < b − (4.30) Numerical algorithm and examples To demonstrate how the proposed regularization method works, we consider here the one dimensional problem 4.4 Conclusions We proposed a regularization method for an inverse source problem in the Banach space setting and proved the Hăolder-type error estimates for the regularized solution using both a priori and a posteriori parameter choice rules Numerical examples showed good reconstruction results for a simple case Numerical realization in more general cases is under consideration and will be reported in a future work 22 GENERAL CONCLUSIONS AND RECOMMENDATIONS General conclusions We research about stability estimates, regularization methods, numerical methods to solve problems of identifying unknown source term of parabolic equation The main results are: An inverse source problem for an n-dimensional heat equation with a timevarying coefficient is investigated We obtained - Hăolder-type stability estimates are proved - The inverse problem is regularized by a mollification method Error estimates of Hăolder type are also proved for regularized solutions for both a priori and a posteriori mollification parameter choice rules - A non-iterative reconstruction algorithm is proposed Numerical examples in one and two dimensions are shown to illustrate the performance of the proposed method An inverse problem of identifying an unknown spatial-dependent source term in a time-space fractional parabolic equation in Hilbert space L2 (Rn ) is considered We obtained - A Hăolder-type stability estimate of optimal order is proved - To regularize the inverse source problem, a mollification regularization method is applied Error estimates of the regularized solution are proved for both a priori and a posteriori rules for choosing the mollification parameter - A direct numerical method for solving the regularized problem is proposed and numerical exam- ples are presented to illustrate its effectiveness An inverse source problem for a parabolic equation in a Banach space is considered We obtained - By using a new regularization method base on semigroup theory, Hăolder-type error estimates for the regularized solutions are proved for both a priori and a posteriori regularization parameter choice rules 23 - Some numerical examples are presented for illustrating the efficiency of the method Recommendations In the future, we look forward to continuing to study the following issues: Stability estimates, regularization methods, numerical methods for solving problems of identifying unknown source terms of parabolic equations in Banach spaces with time-dependent coefficients, and more complicated forms of coefficients, Stability estimates, regularization methods, numerical methods for solving problems of identifying an unknown source term of a time-space fractional parabolic equation with time-dependent coefficients, and more complicated forms of coefficients, 24 LIST OF PUBLICATIONS RELATED TO THE THESIS Nguyen Van Duc, Luong Duy Nhat Minh and Nguyen Trung Thanh (2020), "Identifying an unknown source term in a heat equation with time-dependent coefficients", Inverse Problems in Science and Engineering https://doi.org/10.1080/17415977.2020.1798421 (SCIE, IF: 1.314) Nguyen Van Duc, Nguyen Van Thang, Luong Duy Nhat Minh and Nguyen Trung Thanh (2020), "Identifying an unknown source term of a parabolic equation in Banach spaces", Applicable Analysis https://doi.org/10.1080/00036811.2020.1800650 (SCIE, IF: 1.107) Nguyen Van Thang, Nguyen Van Duc, Luong Duy Nhat Minh and Nguyen Trung Thanh (2021), "Identifying an unknown source term of a time-space fractional parabolic equation", Applied Numerical Mathematics, 166, 313-332 https://doi.org/10.1016/j.apnum.2021.04.016 (SCIE, IF: 1.979) ... results For the above reasons, we choose research topics for our thesis as: "On some problems of identifying unknown source term for parabolic equations" Research purposes We consider some inverse source. .. solution to the original ill-posed problem Research on the problems of identifying unknown source terms for parabolic equations often focuses on three main topics: i) The uniqueness of solutions... performance of the proposed regularization methods in this thesis Research subjects We focus on identifying unknown source terms of parabolic equations in three cases: i) Parabolic equations