A Parallel Method for Backward Parabolic Problems Based on the Laplace Transformation

Lee, Jun Young; Sheen, Dongwoo

doi:10.1137/050624649

Cited by 22 publications

(11 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [16,17], an approximate solution for the BHCP have been presented by using the Fourier truncated methods. Many numerical schemes have been also developed to solve the BHCP including Tikhonov regularization [18], fundamental solution [19], meshless [20], central difference and quasi-reversibility [21], parallel [22], quasi-reversibility [23], boundary element [24], operator marching [25], convolution regularization [26] and mollification [27] methods. The nonhomogeneous case of the BHCP has been considered by Trong et al [28,29].…”

Section: Introductionmentioning

confidence: 99%

On regularization and error estimates for the backward heat conduction problem with time-dependent thermal diffusivity factor

Karimi

Moradlou

Hajipour

2018

Communications in Nonlinear Science and Numerical Simulation

View full text Add to dashboard Cite

In this work, an accurate regularization technique based on the Meyer wavelet method is developed to solve the ill-posed backward heat conduction problem with time-dependent thermal diffusivity factor in an infinite "strip". In principle, the extremely ill-posedness of the considered problem is caused by the amplified infinitely growth in the frequency components which lead to a blow-up in the representation of the solution. Using the Meyer wavelet technique, some new stable estimates are proposed in the Hölder and Logarithmic types which are optimal in the sense of given by Tautenhahn. The stability and convergence rate of the proposed regularization technique are proved. The good performance and the high-accuracy of this technique is demonstrated through various one and two dimensional examples. Numerical simulations and some comparative results are presented.

show abstract

Section: Introductionmentioning

confidence: 99%

On regularization and error estimates for the backward heat conduction problem with time-dependent thermal diffusivity factor

Karimi

Moradlou

Hajipour

2018

Communications in Nonlinear Science and Numerical Simulation

View full text Add to dashboard Cite

show abstract

“…For a general problem with variable coefficients, Manselli and Miller proposed certain modifications of known least squares methods and eigenfunction expansion methods. A parallel method for backward parabolic problems based on the Laplace transformation was considered by Lee and Sheen . Several regularization methods for parabolic equations backward in time together with the usual additional constraints for their solution are considered, for instance, the finite element method is introduced by Hohn .…”

Section: Introductionmentioning

confidence: 99%

TV‐like regularization for backward parabolic problems

Peng

et al. 2016

Math Methods in App Sciences

View full text Add to dashboard Cite

This paper investigates an inverse problem for parabolic equations backward in time, which is solved by total‐variation‐like (TV‐like, in abbreviation) regularization method with cost function ∥ux∥2. The existence, uniqueness and stability estimate for the regularization problem are deduced in the linear case. For numerical illustration, the variational adjoint method, which presents a simple method to derive the gradient of the optimization functional, is introduced to reconstruct the unknown initial condition for both linear and nonlinear parabolic equations. The conjugate gradient method is used to iteratively search for the optimal approximation. Numerical results validate the feasibility and effectiveness of the proposed algorithm. Copyright © 2016 John Wiley & Sons, Ltd.

show abstract

“…In image science, images blurred by Gaussian point spread functions are a common occurrence. Deblurring Gaussian blur is mathematically equivalent to solving the heat conduction equation backward in time, [8,9,16]. More recently, in [7] and references therein, striking enhancements were obtained when time-reversed fractional and/or logarithmic diffusion equations were applied in blind deconvolution of Hubble space telescope galaxy images, as well as scanning electron microscope imagery of interest in nanotechnology.…”

Section: Introductionmentioning

confidence: 99%

Hazardous Continuation Backward in Time in Nonlinear Parabolic Equations, and an Experiment in Deblurring Nonlinearly Blurred Imagery

Carasso

2013

J. RES. NATL. INST. STAN.

View full text Add to dashboard Cite

Identifying sources of ground water pollution, and deblurring nanoscale imagery as well as astronomical galaxy images, are two important applications involving numerical computation of parabolic equations backward in time. Surprisingly, very little is known about backward continuation in nonlinear parabolic equations. In this paper, an iterative procedure originating in spectroscopy in the 1930’s, is adapted into a useful tool for solving a wide class of 2D nonlinear backward parabolic equations. In addition, previously unsuspected difficulties are uncovered that may preclude useful backward continuation in parabolic equations deviating too strongly from the linear, autonomous, self adjoint, canonical model.This paper explores backward continuation in selected 2D nonlinear equations, by creating fictitious blurred images obtained by using several sharp images as initial data in these equations, and capturing the corresponding solutions at some positive time T. Successful backward continuation from t=T to t = 0, would recover the original sharp image. Visual recognition provides meaningful evaluation of the degree of success or failure in the reconstructed solutions.Instructive examples are developed, illustrating the unexpected influence of certain types of nonlinearities. Visually and statistically indistinguishable blurred images are presented, with vastly different deblurring results. These examples indicate that how an image is nonlinearly blurred is critical, in addition to the amount of blur. The equations studied represent nonlinear generalizations of Brownian motion, and the blurred images may be interpreted as visually expressing the results of novel stochastic processes.

show abstract

A Parallel Method for Backward Parabolic Problems Based on the Laplace Transformation

Cited by 22 publications

References 23 publications

On regularization and error estimates for the backward heat conduction problem with time-dependent thermal diffusivity factor

On regularization and error estimates for the backward heat conduction problem with time-dependent thermal diffusivity factor

TV‐like regularization for backward parabolic problems

Hazardous Continuation Backward in Time in Nonlinear Parabolic Equations, and an Experiment in Deblurring Nonlinearly Blurred Imagery

Contact Info

Product

Resources

About