The Cauchy problem of coupled elliptic sine–Gordon equations with noise: Analysis of a general kernel-based regularization and reliable tools of computing

Khoa, Vo Anh; Nhat, Truong Thanh; Duy, Nguyen Ho Minh; Tuan, Nguyen Huy

doi:10.1016/j.camwa.2016.11.001

Cited by 27 publications

(15 citation statements)

References 23 publications

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“…The number of y n is not a matter here since we can apply some quadrature methods to get good accuracy for, e.g., N ≤ 10. This is already postulated in our previous work 23 and is not our main interest in this work.…”

Section: Numerical Resultsmentioning

confidence: 91%

“…As to the Dirichlet eigen‐elements we have mentioned in Remark 1, it is trivial to get that

ϕ_{j} (y) = \sqrt{2} \sin (j π y), μ_{j} = j^{2} π^{2} for j \in ℕ .

Prior to the derivation of the discrete version of (), we note that we are not concerned with highly oscillatory integrals usually met in infinite series (cf., e.g., Khoa et al 23 ) because of the very low Fourier domain after truncation. Indeed, for

ε = 1 0^{- 2}

one has

μ_{j} \leq \frac{1}{16} \log^{2} (\frac{1}{ε}) \approx 1.33,

which means j ≤ 0.37, i.e.,

j = 0

in this case.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…In the nonlinear context, we know that it is not easy to handle the practical error control since one struggles with the Fourier accumulation. A typical example is analyzed numerically in Khoa et al 23 Due to the inception of this new QR approach, we however postpone our analysis of the method in nonlinear cases in the upcoming work.…”

Section: Numerical Resultsmentioning

confidence: 99%

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Constructing a variational quasi‐reversibility method for a Cauchy problem for elliptic equations

Khoa

Nhan

2020

Math Methods in App Sciences

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In the recent developments of regularization theory for inverse and ill‐posed problems, a variational quasi‐reversibility (QR) method has been designed to solve a class of time‐reversed quasi‐linear parabolic problems. Known as a PDE‐based approach, this method relies on adding a suitable perturbing operator to the original problem and consequently, on gaining the corresponding fine stabilized operator, which leads us to a forward‐like problem. In this work, we establish new conditional estimates for such operators to solve a prototypical Cauchy problem for elliptic equations. This problem is based on the stationary case of the inverse heat conduction problem, where one wants to identify the heat distribution in a certain medium, given the partial boundary data. Using the new QR method, we obtain a second‐order initial value problem for a wave‐type equation, whose weak solvability can be deduced using a priori estimates and compactness arguments. Weighted by a Carleman‐like function, a new type of energy estimates is explored in a variational setting when we investigate the Hölder convergence rate of the proposed scheme. Besides, a linearized version of this scheme is analyzed. Numerical examples are provided to corroborate our theoretical analysis.

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Section: Numerical Resultsmentioning

confidence: 91%

“…As to the Dirichlet eigen‐elements we have mentioned in Remark 1, it is trivial to get that

ϕ_{j} (y) = \sqrt{2} \sin (j π y), μ_{j} = j^{2} π^{2} for j \in ℕ .

ε = 1 0^{- 2}

one has

μ_{j} \leq \frac{1}{16} \log^{2} (\frac{1}{ε}) \approx 1.33,

which means j ≤ 0.37, i.e.,

j = 0

in this case.…”

Section: Numerical Resultsmentioning

confidence: 99%

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Constructing a variational quasi‐reversibility method for a Cauchy problem for elliptic equations

Khoa

Nhan

2020

Math Methods in App Sciences

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“…That is, we discretize the system integral Equation (); a uniform grid of mesh points ( x i , t j ) is used

\begin{matrix} normalΔ x & = \frac{π}{N x}, x_{i} = i normalΔ x, i = 1, \dots, N x, \\ normalΔ t & = \frac{1}{N t}, t_{j} = j normalΔ t, j = 1, \dots, N t \end{matrix}

We herein use the Picard‐like procedure 23,24 to approximate the system of nonlinear Volterra‐type integral equations

({, \begin{matrix} left \\ array {\tilde{u}}_{1}^{N} (t_{j - 1}) = \sum_{i = 1}^{N} (\underset{A_{1} (i, t_{j})}{\underset{⏟}{Λ_{i}^{G_{1} ({\tilde{u}}^{N})} (t_{j}, T)}} ⟨ \tilde{ρ}, e_{i} ⟩ - \int_{t_{j}}^{T} Λ_{i}^{G_{i} ({\tilde{u}}^{N})} (t_{j}, s) ⟨ f (\cdot, s), e_{i} ⟩ d s) \end{matrix})

…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…We herein use the Picard-like procedure 23,24 to approximate the system of nonlinear Volterra-type integral equations…”

Section: Approximate For Integral System Equationmentioning

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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p-Adic Analogue of the Wave Equation

Khrennikov

2019

J Fourier Anal Appl

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The Cauchy problem of coupled elliptic sine–Gordon equations with noise: Analysis of a general kernel-based regularization and reliable tools of computing

Cited by 27 publications

References 23 publications

Constructing a variational quasi‐reversibility method for a Cauchy problem for elliptic equations

Constructing a variational quasi‐reversibility method for a Cauchy problem for elliptic equations

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

p-Adic Analogue of the Wave Equation

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