Inverse Estimation of the Initial Condition for the Heat Equation

Tao, Min; Geng, Bei; Ren, Jucheng

doi:10.12732/ijpam.v82i4.7

Cited by 6 publications

(4 citation statements)

References 13 publications

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“…The forward problem maps u 0 to the solution u T at time t = T , and the inverse problem is then to reconstruct the initial condition from u T , cf. [30].…”

Section: Inverse Diffusionmentioning

confidence: 99%

IR Tools: a MATLAB package of iterative regularization methods and large-scale test problems

2018

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This paper describes a new MATLAB software package of iterative regularization methods and test problems for large-scale linear inverse problems. The software package, called IR TOOLS, serves two related purposes: we provide implementations of a range of iterative solvers, including several recently proposed methods that are not available elsewhere, and we provide a set of large-scale test problems in the form of discretizations of 2D linear inverse problems. The solvers include iterative regularization methods where the regularization is due to the semi-convergence of the iterations, Tikhonov-type formulations where the regularization is explicitly formulated in the form of a regularization term, and methods that can impose bound constraints on the computed solutions. All the iterative methods are implemented in a very flexible fashion that allows the problem's coefficient matrix to be available as a (sparse) matrix, a function handle, or an object. The most basic call to all of the various iterative methods requires only this matrix and the right hand side vector; if the method uses any special stopping criteria, regularization parameters, etc., then default values are set automatically by the code. Moreover, through the use of an optional input structure, the user can also have full control of any of the algorithm parameters. The test problems represent realistic large-scale problems found in image reconstruction and several other applications.

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“…The forward problem maps u 0 to the solution u T at time t = T , and the inverse problem is then to reconstruct the initial condition from u T , cf. [30].…”

Section: Inverse Diffusionmentioning

confidence: 99%

IR Tools: a MATLAB package of iterative regularization methods and large-scale test problems

2018

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“…In this example, we train a convolutional neural network to learn the mapping from observation to optimal stopping iteration. We consider a linear inverse diffusion example described in [26,64] where the goal is to determine an initial function, given measurements obtained at some later time. The solution is represented on a finite-element mesh and the forward computation involves the solution of a time-dependent PDE.…”

Section: Learning the Stopping Iteration For Iterative Regularizationmentioning

confidence: 99%

Learning regularization parameters of inverse problems via deep neural networks

Afkham¹,

Chung²,

Chung³

2021

Inverse Problems

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In this work, we describe a new approach that uses deep neural networks (DNN) to obtain regularization parameters for solving inverse problems. We consider a supervised learning approach, where a network is trained to approximate the mapping from observation data to regularization parameters. Once the network is trained, regularization parameters for newly obtained data are computed by efficient forward propagation of the DNN. We show that a wide variety of regularization functionals, forward models, and noise models may be considered. The network-obtained regularization parameters can be computed more efficiently and may even lead to more accurate solutions compared to existing regularization parameter selection methods. We emphasize that the key advantage of using DNNs for learning regularization parameters, compared to previous works on learning via bilevel optimization or empirical Bayes risk minimization, is greater generalizability. That is, rather than computing one set of parameters that is optimal with respect to one particular design objective, DNN-computed regularization parameters are tailored to the specific features or properties of the newly observed data. Thus, our approach may better handle cases where the observation is not a close representation of the training set. Furthermore, we avoid the need for expensive and challenging bilevel optimization methods as utilized in other existing training approaches. Numerical results demonstrate that trained DNNs can predict regularization parameters faster and better than existing methods, hence resulting in more accurate solutions to inverse problems.

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“…Problems with forward heat management aim to determine the temperature range of the medium, when the boundaries and initial conditions-the heat source / sink term (if any)-and the physical properties of the material are known [5]. On the other hand, the problem of inverse heat conduction concerns the estimation of the unknown initial temperature distribution, from the knowledge of the measured temperatures or the heat flux at time t > 0 [6]. In this study, the use of thermal initial conditions is demonstrated using numerical examples of simple structures-particularly experimental external test walls.…”

Section: Introductionmentioning

confidence: 99%

The Influence of the Initial Condition in the Transient Thermal Field Simulation Inside a Wall

Zozulák¹,

Vertaľ²,

Katunský³

2019

Buildings

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The envelope structures of buildings are exposed to heat-humidity conditions. The heat and humidity flow through these structures depends on the boundary conditions of the indoor and outdoor environments. This paper shows different initial conditions for the determination of temperature spread. The aim is to bring certain results of temperature calculated considering the initial conditions. When temperature changes, heat flow also rapidly changes. In certain specific cases, it is necessary to consider the initial conditions of temperature—for example; when transient energy simulations of real buildings are carried out. The reasons as to why it is necessary to consider the initial conditions, are shown in the examples of one-layer assemblies. The test walls exposed to the transient hygrothermal conditions are placed in the outdoor test cell. The cell has a stable temperature and relative humidity, and outdoor weather conditions change. The measured data on these walls and the calculated values of the temperatures in the wall structure, according to the different initial conditions, are compared. The average difference of the mean by the simulation and the measured values is significant. For a simulation time of about five days, the initial condition for calculating the temperature in the center of the masonry is necessary.

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Inverse Estimation of the Initial Condition for the Heat Equation

Cited by 6 publications

References 13 publications

IR Tools: a MATLAB package of iterative regularization methods and large-scale test problems

IR Tools: a MATLAB package of iterative regularization methods and large-scale test problems

Learning regularization parameters of inverse problems via deep neural networks

The Influence of the Initial Condition in the Transient Thermal Field Simulation Inside a Wall

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