Investigation of the Initial Inverse Problem in the Heat Equation

Masood, Khalid; Zaman, F. D.

doi:10.1115/1.1666886

Cited by 11 publications

(14 citation statements)

References 10 publications

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“…One can verify that the discretization of (3), considering x = mΔ1 −nΔ2, is equivalent to (6). We concatenate the Nt measurement vectors f and sampling operators Φ into a single vector f and single matrix Φ, respectively, to obtain f = Φf0.…”

Section: Sampling Of the Diffusion Fieldmentioning

confidence: 99%

“…More recently, many approaches have been considered, such as the hyperbolic heat equation [6], transform techniques [7], spatial superresolution [8] and an adaptive spatiotemporal sampling scheme [9]. A similar approach to ours, applied to the wave equation, has been proposed by Fannjiang et al [10] where the localization of remote sparse objects using electromagnetic pulses is solved by compressed sensing (CS) techniques.…”

Section: Introductionmentioning

confidence: 98%

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Sampling and reconstructing diffusion fields with localized sources

Ranieri

Chebira

et al. 2011

2011 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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We study the spatiotemporal sampling of a diffusion field generated by K point sources, aiming to fully reconstruct the unknown initial field distribution from the sample measurements. The sampling operator in our problem can be described by a matrix derived from the diffusion model. We analyze the important properties of the sampling matrices, leading to precise bounds on the spatial and temporal sampling densities under which perfect field reconstruction is feasible. Moreover, our analysis indicates that it is possible to compensate linearly for insufficient spatial sampling densities by oversampling in time. Numerical simulations on initial field reconstruction under different spatiotemporal sampling densities confirm our theoretical results.

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Section: Sampling Of the Diffusion Fieldmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Sampling and reconstructing diffusion fields with localized sources

Ranieri

Chebira

et al. 2011

2011 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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“…More recently, Al Masood et al considered a well conditioned damped heat equation [7] and a Bessel operator [8] to estimate the initial temperature distribution in a diffusion field. Nakamura et al [9] used transform techniques to solve the initial inverse problem in heat conduction, while Takeuchi et al [10] proved the existence of the solution and gave a numerical method to find point sources distributed on a 2-D domain.…”

Section: A Prior Artmentioning

confidence: 99%

Sensor networks for diffusion fields: Detection of sources in space and time

Ranieri

Chebira

Vetterli

2011

2011 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton)

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Abstract-We consider the problem of reconstructing a diffusion field, such as temperature, from samples collected by a sensor network. Motivated by the fast decay of the eigenvalues of the diffusion equation, we approximate the field by a truncated series. We show that the approximation error decays rapidly with time. On the other hand, the information content in the field also decays with time, suggesting the need for a proper choice of the sampling strategy. We propose two algorithms for sampling and reconstruction of the field. The first one reconstructs the distribution of point sources appearing at known times using the finite rate of innovation (FRI) framework. The second algorithm addresses a more difficult problem of estimating the unknown times at which the point sources appear, in addition to their locations and magnitudes. It relies on the assumption that the sources appear at distinct times. We verify that the algorithms are capable of reconstructing the field accurately through a set of numerical experiments. Specifically, we show that the second algorithm successfully recovers an arbitrary number of sources with unknown release times, satisfying the assumption. For simplicity, we develop the 1-D theory, noting the possibility of extending the framework to more general domains.

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“…Substitution of equation (7) in equations (1) and (6), and using orthonormality property of the eigenfunctions leads to the following ordinary differential…”

Section: Initial Inverse Problem In the Parabolic Heat Equationmentioning

confidence: 99%

“…The initial inverse problem based on the parabolic heat equation is extremely illposed [1]. One way to stabilize this problem is to consider a hyperbolic heat equation instead of a parabolic heat equation [2][3][4][5][6][7]. The hyperbolic reformulation is stable and well-posed and furthermore the numerical methods for such problems are efficient and accurate.…”

Section: Introductionmentioning

confidence: 99%

A smoothing spline-based regularization of the initial inverse problem in a two-dimensional heat equation

Masood

Mustafa

2008

Proceedings of the Institution of Mechanical Engineers, Part C:

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A smoothing spline-based method and a hyperbolic heat conduction model is applied to regularize the recovery of the initial profile from a parabolic heat conduction model in two-dimensions. An ill-posed inverse problem involving recovery of the initial temperature distribution from measurements of the final temperature distribution is investigated. A hyperbolic heat conduction model is considered instead of a parabolic model and smoothing splines are applied to regularize the recovered initial profile. The comparison of the proposed procedure and parabolic model is presented graphically by examples.

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Investigation of the Initial Inverse Problem in the Heat Equation

Cited by 11 publications

References 10 publications

Sampling and reconstructing diffusion fields with localized sources

Sampling and reconstructing diffusion fields with localized sources

Sensor networks for diffusion fields: Detection of sources in space and time

A smoothing spline-based regularization of the initial inverse problem in a two-dimensional heat equation

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