A two-dimensional backward heat problem with statistical discrete data

Minh, Nguyen Dang; Duc, Khanh To; Tuan, Nguyen Huy; Trong, Dang Duc

doi:10.1515/jiip-2016-0038

Cited by 19 publications

(18 citation statements)

References 24 publications

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“…Remark 5.1. The truncation method in this paper is similar to the method in [27,18]. The quasi-boundary value method in this section is more effective and useful than the one in [28].…”

Section: Convergence Resultsmentioning

confidence: 94%

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Final value problem for nonlinear time fractional reaction–diffusion equation with discrete data

Tuan

Băleanu

Thach

et al. 2020

Journal of Computational and Applied Mathematics

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“…Remark 5.1. The truncation method in this paper is similar to the method in [27,18]. The quasi-boundary value method in this section is more effective and useful than the one in [28].…”

Section: Convergence Resultsmentioning

confidence: 94%

“…The model in [18] is linear. Our problem is nonlinear and we use the Banach fixed point theorem to show the existence of the regularized solution in the space X T (note this space does not appear in [18]). Some new estimates of Mittag-Leffler type are used.…”

Section: Introductionmentioning

confidence: 99%

Final value problem for nonlinear time fractional reaction–diffusion equation with discrete data

Tuan

Băleanu

Thach

et al. 2020

Journal of Computational and Applied Mathematics

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“…According to some literature, 11–13 for random noise factors, the authors usually assume that they are drawn from the original normal distribution

scriptN false(0, σ^{2} false)

. This leads to being stuck in studying the existence and uniqueness of our regularized solution.…”

Section: Introductionmentioning

confidence: 99%

Recovering the initial value for a system of nonlocal diffusion equations with random noise on the measurements

Triet

Binh²,

Phương

et al. 2020

Math Methods in App Sciences

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In this work, we study the final value problem for a system of parabolic diffusion equations. In which, the final value functions are derived from a random model. This problem is severely ill‐posed in the sense of Hadamard. By nonparametric estimation and truncation methods, we offer a new regularized solution. We also investigate an estimate of the error and a convergence rate between a mild solution and its regularized solutions. Finally, some numerical experiments are constructed to confirm the efficiency of the proposed method.

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“…Recently, Hohage et al [9] applied spectral cut-off (truncation method) and Tikhonov-type methods for solving linear statistical inverse problems including backward heat equation (See p. 2625, [9]). In the linear inhomogenous case of (1), i.e, β = 1 and F = 0, the Problem (1) has been recently studied in [13] in two space dimensions.…”

Section: Introductionmentioning

confidence: 99%

Approximate Solutions of Inverse Problems for Nonlinear Space Fractional Diffusion Equations with Randomly Perturbed Data

Nane¹,

Tuan²

2018

SIAM/ASA J. Uncertainty Quantification

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This paper is concerned with backward problem for nonlinear space fractional diffusion with additive noise on the right-hand side and the final value. To regularize the instable solution, we develop some new regularized method for solving the problem. In the case of constant coefficients, we use the truncation methods. In the case of perturbed time dependent coefficients, we apply a new quasi-reversibility method. We also show the convergence rate between the regularized solution and the sought solution under some a priori assumption on the sought solution.Keywords: Inverse problem for fractional heat equation, truncation method, approximate solu-2 recently considered by Tuan and Nane [19].Next we give some details about our methods for the following two cases The first case: a(t) = 1. First, we transform the problem (1) into a nonlinear integral equation, then we apply the Fourier truncation method(using the eigenfunctions cos(px), p = 0, 1, 2 · · · of the Laplacian in the interval (0, π) with Neumann boundary conditions) associated with some techniques in nonparametric regression to establish a first regularized solution U Mn,n (x, t) which satisfies (33). To obtain the estimate between U Mn,n and u, we need some stronger assumptions on u, such as (27) and (29). The main result is Theorem 2.6. However, as pointed out in Remark 2.5, the assumptions (27) and (29) are difficult to come up in practice. Motivated by this, when g = 0 we develop a second regularized solution U Mn,n defined by (69) to obtain the estimate for u ∈ C([0, T ]; H γ (Ω)) (see Remark 2.5 for more details). The main result for the second type of regularization is given in Theorem 2.8. It is important to realize that the second regularized solution is a modification of the first regularized solution. Our methods in this paper can be applied to solve many ill-posed problems of nonlinear PDEs such as Cauchy problem for nonlinear elliptic, nonlinear ultraparabolic, nonlinear strongly damped wave equations, and many others. The second case: a(t) depend on t and is perturbed. Note that if the coefficient a in the main equation of (1) is not noisy then the Fourier truncation method in [21] can be applied to Problem (1). However, the difficulty occurs for (86) when the time dependent coefficient a(t) is noisy. Indeed, we assume that a(t) is noisy by observed random data a(t) which satisfy thatwhere ξ(t) is Brownian motion. If we have used a Fourier truncation solution for (1), then the regularized solution would contain some terms such as exp p 2β T t s t a(τ )dτ ds , which would lead to some complex computations. Hence, we don't follow the truncation method as in [21], instead we develop a new method to find a regularized solution. We will apply a new quasireversibility method for solving the problem. Further details of this method can be found in Tuan [21]. In this case our main results are Theorems 3.2 and 3.7.In this paper, we only study the upper bound of the convergence rate. In a future work, we will study the minimax rate of convergence for fi...

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A two-dimensional backward heat problem with statistical discrete data

Cited by 19 publications

References 24 publications

Final value problem for nonlinear time fractional reaction–diffusion equation with discrete data

Final value problem for nonlinear time fractional reaction–diffusion equation with discrete data

Recovering the initial value for a system of nonlocal diffusion equations with random noise on the measurements

Approximate Solutions of Inverse Problems for Nonlinear Space Fractional Diffusion Equations with Randomly Perturbed Data

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