tóm tắt luận án tiến sĩ tiếng anh phương trình parabolic ngược thời gian

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tóm tắt luận án tiến sĩ  tiếng anh  phương trình parabolic ngược thời gian

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MINISTRY OF EDUCATION AND TRAINING VINH UNIVERSITY  —————— ——————– —————– ——————  NGUYEN VAN DUC PARABOLIC EQUATIONS BACKWARD IN TIME Subject: Mathematical Analysis Code: 62 46 01 01 PhD THESIS SUMMARY VINH - 2011 The thesis is completed at Vinh University Supervisors: 1. Prof. Dr. Sc. Dinh Nho H`ao 2. Assoc. Prof. Dr. Dinh Huy Hoang Referee 1: Prof. Dr. Nguyen Huu Du Hanoi University of Science, VNU Referee 2: Assoc. Prof. Dr. Ha Tien Ngoan Institute of Mathematics - Vietnam Academy of Science and Technology Referee 3: Assoc. Prof. Dr. Nguyen Xuan Thao Hanoi University of Science and Technology The thesis will be defended at the exam Committee at . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . time. . . . . . date . . . . . . month . . . . . . 2011 The thesis is available at: - National Library of Vietnam - Library of Vinh University INTRODUCTION Parabolic equations backward in time appear frequently in the heat transfer theory, geophysics, groundwater problems, materials science, hydrodynamics, im- age processing This is the problem, when the initial condition is not known and we must determine it from the final condition. These problems have been intensively studied, but only for some special classes. Moreover, finding efficient numerical methods for them is always desired. Parabolic equations backward in time are ill-posed in sense Hadamard. A problem is called well-posed if it fulfills the following properties: a) For all admissible data, a solution exists. b) If a solution exists, it is unique. c) The solution depends continuously on the data. If one of the above properties is not satisfied, then the problem is called ill- posed. Hadamard supposed that ill-posed problems have no physical meaning. However, many practical problems of science and technology have led to ill-posed problems. Therefore, since 1950 many papers concerned ill-posed problems have been published. Mathematicians such as A. N. Tikhonov, M. M. Lavrent’ev, F. John, C. Pucci, V. K. Ivanov are pioneers in this field. In 1955, John reported some results about a method to numerically solve the Cauchy problem for the heat equation backward in time. Then, Krein and cowork- ers also published some results on stability estimates and backward uniqueness for parabolic equations backward in time. In 1963, Tikhonov proposed a regular- ization method which is applicable for almost all inverse and ill-posed problems. Especially, this method was applied successfully to the backward heat equation in 1974 by Franklin. Addition to these, many authors also use another methods such as: QR method, SQR method, the backward beam equation method, the method 1 2 of non-local boundary value problems, iterative methods, finite difference meth- ods, mollification method for parabolic equations backward in time. However, no method is universal for all problems. For example, Tikhonov method or QR method require to solve a equation of double of that of the original equation and choosing regularization parameters is not easy. Further, it is very difficult to use Tikhonov method in Banach spaces. Until now, hundreds of papers devoted to parabolic equations backward in time there have been published which focused mainly on 1) backward uniqueness, 2) stability estimates, 3) regularization methods, stable and efficient numerical methods. In this thesis, we focus on obtaining stability estimates and regularization meth- ods for parabolic equations backward in time. We regularize the problem  u t + Au = 0, 0 < t < T, u(T ) − f  ε (0.1) by the well-posed non-local boundary value problem  v αt + Av α = 0, 0 < t < aT, αv α (0) + v α (aT ) = f with a  1 being given and α > 0, the regularization parameter. We suggest a priori and a posteriori parameter choice rules in order to yield order-optimal regularization metho ds. Furthermore, the method were tested on the computer and the results are very encouraging. To our knowledge, Vabishchevich is one of the first using this method for the parabolic equations backward in time in 1981. He proposed an a priori method for (0.1) but without giving the convergence rates as we do in this thesis. Further, he suggested the following a posteriori metho d for (0.1). Solve the well-posed problem  u αt + Au α = 0, 0 < t < T, αu α (0) + u α (T ) = f, α > 0, and choose α such that u α (T ) − f = ε. 3 However, we could not prove that such an approach yields an order-optimal method. Therefore, in this thesis we propose to use the following method. Let a > 1 be a fixed number. Consider the well-posed problem  v αt + Av α = 0, 0 < t < aT, αv α (0) + v α (aT ) = f, α > 0, (0.2) and take v α ((a − 1)T + t) as an approximation to u(t). We suggest a priori and a posteriori strategies for choosing the parameter α and prove that these yield order-optimal regularization methods for (0.1). The a priori method is given in Theorems 1.2.1, 1.2.3, 1.2.5, 1.3.1, the a posteriori method is as follows. Suppose ε < f and let τ > 1 satisfy τε < f . Choose α > 0 such that v α (aT ) − f = τε. We note that, since f −v α (aT ) = αv α (0), the above discrepancy principle has the very simple form: Choose α > 0 such that αv α (0) = τε. Showalter, Clark and Oppenheimer and Mel’nikova also regularized the problem (0.1) by the non-local boundary value problem  u t + Au = 0, 0 < t < T, αu(0) + u(T) = f, α > 0. (0.3) Our results in case a = 1 are better than theirs. We note that Denche and Bessila approximated the problem (0.1) by the problem  u t + Au = 0, 0 < t < T, −αu t (0) + u(T ) = f, α > 0. (0.4) They obtained an error estimate at t = 0 of logarithmic type with a strong con- dition that Au(0) is bounded. It means u(0) has to be in the domain of A that is not frequently met in practice. We will show that we do not need to require u t exist at t = 0 as these authors required but again by the problem (1.6) we can establish stability estimates which are comparable to theirs. The problem becomes much more difficult if the operator A depends on time and there are very few results in this case. In this thesis, we improve the related 4 results by Krein, Agmon and Nirenberg. Furthermore, we also suggest a regu- larization method. Our regularization method with a priori and a posteriori pa- rameter choice yields error estimates of H¨older type. This is the only result when a regularization metho d for backward parabolic equations with time-dependent coefficients provides a convergence rate. In the last part of the thesis, we use the mollification method to regularize for the heat equation backward in time in Banach space L p (R) 1 < p < ∞ . Namely, we study the following problem: Let p ∈ (1, ∞), ϕ ∈ L p (R) and ε, E be given constants such that 0 < ε < E < ∞. Consider the heat equation backward in time  u t = u xx , x ∈ R, t ∈ (0, T), u(·, T ) − ϕ(·) L p (R)  ε, (0.5) subject to the constraint u(·, 0) L p (R)  E. (0.6) We note that the case p = 2 is much more difficult, since we do not have the Parseval equality and in general the Fourier transform of a function in L p (R) with p > 2 is a distribution. This problem has been considered by the first author. He gave a stability estimate of H¨older type for the case p ∈ (1, ∞]: if u 1 and u 2 are two solutions of the problem, there is a constant c ∗ such that u 1 (·, t)−u 2 (·, t) L p (R) ≤ 4 √ 3((c ∗ E) 1−t/T ε t/T +(c ∗ E) 1−t/(4T ) ε t/(4T ) ), ∀t ∈ [0, T ]. One of the aims of the thesis is to improve this estimate for p ∈ (1, ∞). Namely, for p ∈ (1, ∞), we show that there is a constant c > 0 such that u 1 (·, t) −u 2 (·, t) L p (R)  cε t/T E 1−t/T , ∀t ∈ [0, T ]. The heat equation backward in time is well-known to be ill-posed: a small perturbation in the Cauchy data may cause a very large error in solution. To overcome this difficulty, Dinh Nho H`ao proposed a mollification method for solving the problem in a stable way and proved stability estimates of H¨older type for the solutions. In this thesis we shall follow this technique to regularize the problem 5 (0.5)–(0.6). However, instead of using the de la Vall´ee Poussin kernel for mollifying the Cauchy data ϕ, we use the Dirichlet kernel and thus work with mollified data generated by the convolution of this kernel with ϕ. The mollified data belong to the space of band-limited functions, in which the Cauchy problem is well-posed, and with appropriate choices of mollification parameter we obtain error estimates of H¨older type. Stability estimates for the solutions of the problem (0.5)–(0.6) is the direct consequence of these error estimates and the triangle inequality. In this thesis, supplementally to the result of Dinh Nho H`ao for p = 2, we establish stability estimates of H¨older type for all derivatives with respect to x and t of the solutions. It is worth to note that such estimates are very seldom in the literature of ill-posed problems. It is well known that with only the condition (0.6), we cannot expect any con- tinuous dependence of the solution at t = 0. This can be recovered if an additional condition on the smoothness of u(x, 0) is available (see Theorem 3.2.7). To this purpose, in the literature the regularization parameters are chosen dependently on the parameters of this ”source condition” which are in general not known. To overcome this shortcoming, in Theorems 3.2.6 and 3.2.10 we propose a choice of mollification parameters using only the condition (0.6) which guarantees error es- timates of H¨older type in (0, T ] and a continuous dependence at t = 0 when a source condition is available but without knowing its parameters. This choice of mollification parameters seems to be quite interesting for the numerical treatment of the problem (0.5)–(0.6). For p = 2, since the Fourier transform of mollified data has compact support, one has at least two equivalent forms of the mollification method: one in its original form, another uses the frequency cut-off technique. These two forms lead to two different numerical schemes which can be easily implemented numerically using the fast Fourier transform technique (FFT). For p = 2, these schemes do not work and we propose a stable marching difference scheme for (0.5). We test the methods for different numerical examples and see that they are very stable and fast. The thesis consists of an introduction, three chapters, conclusion and references. Chapter 1 presents results on the regularization of parabolic equations backward 6 in time with time-independent coefficients in Hilbert Spaces. Theorems 1.2.1, 1.2.3 and 1.2.5 provide the results on an a priori parameter choice rule in case a = 1. Theorems 1.3.1, 1.3.3 provide the results on a priori and a posteriori parameter choice rules in case a > 1. At the end of Chapter 1, numerical results are presented and discussed to confirm the theory. Chapter 2 presents results on stability estimates and regularization of parabolic equations backward in time with time-dependent coefficients in Hilbert spaces. Chapter 3 presents stability results for the heat equation backward in time in Hilbert and Banach spaces, namely, in L p (R) for p ∈ (1, +∞). Theorems 3.2.1, 3.2.3 present stability and regularization results in Banach spaces with the same convergence rate as in case L 2 (R). A slightly modifying the choice of ν in Theorem 3.2.1 gives a stability estimate of H¨older type for t ∈ (0, T ] which guarantees a continuous dependence of logarithmic type at t = 0 without explicitly knowing ˜ E and γ is the main result of Theorem 3.2.6. In case p = 2, Theorem 3.2.7 provides error estimates of H¨older type for all derivatives with respect to x and t of the solutions.Theorem 3.2.10 shows that a slightly modifying the choice of ν in Theorem 3.2.7 guarantees a continuous dependence of logarithmic type at t = 0 without explicitly knowing ˜ E and γ. At the end of Chapter 3, a stable marching difference scheme and numerical examples are presented and discussed. CHAPTER 1 PARABOLIC EQUATIONS BACKWARD IN TIME WITH TIME-INDEPENDENT COEFFICIENTS Consider the ill-posed parabolic equation backward in time  u t + Au = 0, 0 < t < T, u(T ) − f  ε (1.1) with the positive self-adjoint unbounded operator A that admits an orthonormal eigenbasis {φ i } i1 in Hilbert space H with norm  · , associated with the eigen- values {λ i } i1 such that 0 < λ 1  λ 2  . . . , and lim i→+∞ λ i = +∞. In order to regularize the problem, we suppose that there is a positive constant E > ε > 0 such that u(0)  E. (1.2) In this chapter, we regularize the problem (1.1), (1.2) by the well-posed non-local boundary value problem  v αt + Av α = 0, 0 < t < aT, αv α (0) + v α (aT ) = f (1.3) with a  1 being given and α > 0, the regularization parameter. A priori and a posteriori parameter choice rules are suggested which yield order-optimal regular- ization methods. The results of this chapter are published in Journal of Mathe- matical Analysis and Applications and IMA Journal of Applied Mathematics. 1.1 Some concepts and basic lemmas Definition 1.1.1. Let H be a Hilbert space with the inner product ·, · and the norm ·, a and T are positive number. The space C([0, aT ]; H) consist of all con- tinuous functions u : [0, aT ] → H with the norm u C([0,aT ];H) = max 0taT u(t) < ∞. 7 8 The space C 1 ((0, aT ); H) consist of all continuously differentiable functions u : (0, aT) → H. D(A) ⊂ H is domain of the operator A : D(A) ⊂ H → H. Definition 1.1.2. A function v α : [0, aT ] → H is called a solution of (2.19) if v α ∈ C 1 ((0, aT ), H) ∩C([0, aT ], H), v α (t) ∈ D(A), ∀t ∈ (0, aT ), and satisfies the equation v αt + Av α = 0 in the interval (0, aT ) and the boundary value condition αv α (0) + v α (aT ) = f. Definition 1.1.3. The function H(η) is defined by H(η) =  η η (1 −η) 1−η , η ∈ (0, 1), 1, η = 0 and 1. (1.4) It is clear that H(η) ≤ 1. The function C(x, y) with 1 > x  0, y > 0 is defined by C(x, y) =  y 1 −x  y e 1−x−y . (1.5) 1.2 Regularization of parabolic equations backward in time by a non-local boundary value problem in case a=1 In this section, we regularize the problem (1.1), (1.2) by the non-local boundary value problem  v αt + Av α = 0, 0 < t < aT, αv α (0) + v α (aT ) = f (1.6) We denote the solution of (1.1) by u(t), and the solution of (1.6) by v(t). Theorem 1.2.1. The following inequality holds v(t) −u(t)  Q(t, α)  α t/T −1 ε + α t/T E  , ∀t ∈ [0, T ]. (1.7) If we choose α = ε E , then  v ( t ) − u ( t )   2 Q  t, ε E  ε t/T E 1−t/T , ∀ t ∈ [0 , T ] . Here, Q(t, α) = min{H(t, α), K(t)}, t ∈ [0, T ], H(t, α) :=  (t/T ) t/T (1 −t/T) 1−t/T √ 2α + 1 −t/T ∈ (0, 1), ∀t ∈ (0, T ), ∀α > 0, H(0) = 1, H(T ) = 1/ √ 2α + 1, K(t) := (t/T ) t/T (1 −t/T) 1−t/T ∈ (0, 1), ∀t ∈ (0, T ), K(0) = K(T ) = 1. [...]... Recommendations In the near future we will study the following issues: 1 Parabolic equations backward in time in Banach spaces 2 Nonlinear parabolic equations backward in time 28 List of publications of the doctoral student related to the thesis 1 Dinh Nho H`o, Nguyen Van Duc and H Sahli (2008), ”A non-local boundary a value problem method for parabolic equations backwards in time”, Journal of Mathematical... on computer 2 Improve Agmon and Nirenberg’s results on stability estimates for parabolic equations backward in time with time-dependent coefficients and use a non-local boundary value problem method to regularize this problem which yields a convergence rate This is the only result when a regularization method for backward parabolic equations with time-dependent coefficients provides a convergence rate... methods - Numerical results based on the boundary element method are presented and discussed to confirm the theory CHAPTER 2 PARABOLIC EQUATIONS BACKWARD IN TIME WITH TIME-DEPENDENT COEFFICIENTS In this chapter, we present results on stability estimates and regularization for parabolic equations backward in time with time-dependent coefficients ut + A(t)u = 0, u(T ) − f ε 0 < t < T, (2.1) subject to the... No 25, 055002, 27 pp 4 Dinh Nho H`o, Nguyen Van Duc and D Lesnic (2010), ”Regularization of a parabolic equations backward in time by a non-local boundary value problem method”, IMA Journal of Applied Mathematics, No 75, pp 291-315 5 Dinh Nho H`o and Nguyen Van Duc (2011), ”Stability results for backward a parabolic equations with time dependent coefficients”, Inverse Problems, Vol 27, No 2, 025003, 20... condition ξ p apq (x, t)ξ q ≥ δ|ξ|2m , |p|,|q|=m ∀(x, t) ∈ Q, ξ ∈ Rn \ {0} (2.23) 19 for a positive constant δ, independent of t 2.4 Conclusion of Chapter 2 We prove new stability estimates for backward parabolic equations with timedependent coefficients (Theorem 2.1.5) Our stability estimates improve the related results by Krein, Agmon and Nirenberg Furthermore, we propose a regularization method for the... and test several related numerical methods for it 27 General conclusions and recommendations I General conclusions Results of the thesis 1 Use a non-local boundary value problem method to regularize parabolic equations backward in time with time-independent and time-dependent coefficients In the time-independent coefficient cases, a priori and a posteriori parameter choice rules are suggested which yield... 1.4 Numerical examples We tested on the computer for the a posteriori parameter choice rule in §1.3 with two examples and find that that the method is stable and efficient 1.5 Conclusion of Chapter 1 The parabolic equation backward in time ut + Au = 0, 0 < t < T, u(T ) − f ε subject to the constraint u(0) E (E > ε > 0) is regularized by the well-posed non-local boundary value problem vαt + Avα = 0, 0 1 We denote the solution of (1.1) by u(t), and the solution of (1.3) by vα (t) 1.3.1 A priori parameter choice rule... := Ω × (0, T ) For any multi-index p = (p1 , p2 , , pn ), we define |p| = p1 + p2 + · · · + pn and ∂ |p| ∂xp1 xp2 · · · xpn n 1 2 The results and methods in §2.1 and §2.2 are applicable to following parabolic Dp = equations of order 2m (m ≥ 1) (−1)|p| Dp (apq (x, t)Dq u), ut = − (x, t) ∈ Q (2.22) |p|,|q|≤m with real functions apq ∈ C 1 ([0, T ], L∞ (Ω)) satisfying apq = aqp and the uniform ellipticity . examples are presented and discussed. CHAPTER 1 PARABOLIC EQUATIONS BACKWARD IN TIME WITH TIME-INDEPENDENT COEFFICIENTS Consider the ill-posed parabolic equation backward in time  u t + Au =. the theory. CHAPTER 2 PARABOLIC EQUATIONS BACKWARD IN TIME WITH TIME-DEPENDENT COEFFICIENTS In this chapter, we present results on stability estimates and regularization for parabolic equations. for some special classes. Moreover, finding efficient numerical methods for them is always desired. Parabolic equations backward in time are ill-posed in sense Hadamard. A problem is called well-posed

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