scholarly journals Recovering the initial condition in the one-phase Stefan problem

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Chifaa Ghanmi ◽  
Saloua Mani Aouadi ◽  
Faouzi Triki
Keyword(s):  
Stefan Problem ◽  
Synthetic Data ◽  
Unique Continuation ◽  
Parabolic Type ◽  
Stability Estimate ◽  
Initial Condition ◽  
Unique Continuation Property ◽  
Ill Posed ◽  
The One ◽  
Representation Techniques

<p style='text-indent:20px;'>We consider the problem of recovering the initial condition in the one-dimensional one-phase Stefan problem for the heat equation from the knowledge of the position of the melting point. We first recall some properties of the free boundary solution. Then we study the uniqueness and stability of the inversion. The principal contribution of the paper is a new logarithmic type stability estimate that shows that the inversion may be severely ill-posed. The proof is based on integral equations representation techniques, and the unique continuation property for parabolic type solutions. We also present few numerical examples operating with noisy synthetic data.</p>

scholarly journals Stability estimate for the semi-discrete linearized Benjamin-Bona-Mahony equation

2021 ◽  
Vol 27 ◽  
pp. 93
Author(s):  
Rodrigo Lecaros ◽  
Jaime H. Ortega ◽  
Ariel Pérez
Keyword(s):  
Unique Continuation ◽  
Mesh Size ◽  
Stability Estimate ◽  
Finite Difference Approximation ◽  
Dispersive Media ◽  
Difference Approximation ◽  
Large Parameter ◽  
One Dimensional ◽  
Unique Continuation Property ◽  
Continuation Property

In this work we study the semi-discrete linearized Benjamin-Bona-Mahony equation (BBM) which is a model for propagation of one-dimensional, unidirectional, small amplitude long waves in non-linear dispersive media. In particular, we derive a stability estimate which yields a unique continuation property. The proof is based on a Carleman estimate for a finite difference approximation of Laplace operator with boundary observation in which the large parameter is connected to the mesh size.

Exact Solution to the One-Dimensional Inverse-Stefan Problem in Nonideal Biological Tissues

1995 ◽  
Vol 117 (2) ◽  
pp. 425-431 ◽  
Author(s):  
Y. Rabin ◽  
A. Shitzer
Keyword(s):  
Steady State ◽  
Phase Change ◽  
Stefan Problem ◽  
Blood Perfusion ◽  
Steady State Solution ◽  
Biological Tissues ◽  
Initial Condition ◽  
Quasi Steady State ◽  
Single Temperature ◽  
The One

A new analytic solution of the inverse-Stefan problem in biological tissues is presented. The solution, which is based on the enthalpy method, assumes that phase change occurs over a temperature range and includes the thermal effects of metabolic heat generation, blood perfusion, and density changes. As a first stage a quasi-steady-state solution is derived, defined by uniform velocities of the freezing fronts and thus by constant cooling rates at those interfaces. Next, the fixed boundary condition leading to the quasi-steady state is calculated. It is shown that the inverse-Stefan problem may not be solved exactly for a uniform initial condition, but rather for a very closely approximating exponential initial condition. Very good agreement is obtained between the new solution and an earlier one assuming biological tissues to behave as pure materials in which phase change occurs at a single temperature. A parametric study of the new solution is presented taking into account property values of biological tissues at low freezing rates typical of cryosurgical treatments.

Resolution in crosswell traveltime tomography: The dependence on illumination

Geophysics ◽  
2016 ◽  
Vol 81 (1) ◽  
pp. W1-W12 ◽  
Author(s):  
Renato R. S. Dantas ◽  
Walter E. Medeiros
Keyword(s):  
A Priori ◽  
Synthetic Data ◽  
Geometric Approach ◽  
Real Data ◽  
Angular Aperture ◽  
Traveltime Tomography ◽  
Sparsity Constraints ◽  
Ill Posed ◽  
Priori Information ◽  
The One

The key aspect limiting resolution in crosswell traveltime tomography is illumination, a well-known result but not well-exemplified. We have revisited resolution in the 2D case using a simple geometric approach based on the angular aperture distribution and the Radon transform properties. We have analytically found that if an isolated interface had dips contained in the angular aperture limits, it could be reconstructed using just one particular projection. By inversion of synthetic data, we found that a slowness field could be approximately reconstructed from a set of projections if the interfaces delimiting the slowness field had dips contained in the available angular apertures. On the one hand, isolated artifacts might be present when the dip is near the illumination limit. On the other hand, in the inverse sense, if an interface is interpretable from a tomogram, there is no guarantee that it corresponds to a true interface. Similarly, if a body is present in the interwell region, it is diffusely imaged, but its interfaces, particularly vertical edges, cannot be resolved and additional artifacts might be present. Again, in the inverse sense, there is no guarantee that an isolated anomaly corresponds to a true anomalous body, because this anomaly could be an artifact. These results are typical of ill-posed inverse problems: an absence of a guarantee of correspondence to the true distribution. The limitations due to illumination may not be solved by the use of constraints. Crosswell tomograms derived with the use of sparsity constraints, using the discrete cosine transform and Daubechies bases, essentially reproduce the same features seen in tomograms obtained with the smoothness constraint. Interpretation must be done taking into consideration a priori information and the particular limitations due to illumination, as we have determined with a real data case.

scholarly journals On the Two-phase Fractional Stefan Problem

2020 ◽  
Vol 20 (2) ◽  
pp. 437-458 ◽  
Author(s):  
Félix del Teso ◽  
Jørgen Endal ◽  
Juan Luis Vázquez
Keyword(s):  
Free Boundary ◽  
Stefan Problem ◽  
Free Boundary Problems ◽  
Classical Problem ◽  
Nonlocal Diffusion ◽  
Two Phase ◽  
Boundary Problems ◽  
Numerical Studies ◽  
The One ◽  
Evolution Type

AbstractThe classical Stefan problem is one of the most studied free boundary problems of evolution type. Recently, there has been interest in treating the corresponding free boundary problem with nonlocal diffusion. We start the paper by reviewing the main properties of the classical problem that are of interest to us. Then we introduce the fractional Stefan problem and develop the basic theory. After that we center our attention on selfsimilar solutions, their properties and consequences. We first discuss the results of the one-phase fractional Stefan problem, which have recently been studied by the authors. Finally, we address the theory of the two-phase fractional Stefan problem, which contains the main original contributions of this paper. Rigorous numerical studies support our results and claims.

scholarly journals Strict monotonicity and unique continuation of the biharmonic operator - doi: 10.5269/bspm.v30i2.14502

1970 ◽  
Vol 30 (2) ◽  
pp. 79-83
Author(s):  
Najib Tsouli ◽  
Omar Chakrone ◽  
Mostafa Rahmani ◽  
Omar Darhouche
Keyword(s):  
Unique Continuation ◽  
Biharmonic Operator ◽  
Strict Monotonicity ◽  
Unique Continuation Property ◽  
Continuation Property

In this paper, we will show that the strict monotonicity of the eigenvalues of the biharmonic operator holds if and only if some unique continuation property is satisfied by the corresponding eigenfunctions.

scholarly journals Gradient Profile Estimation Using Exponential Cubic Spline Smoothing in a Bayesian Framework

Entropy ◽  
2021 ◽  
Vol 23 (6) ◽  
pp. 674
Author(s):  
Kushani De De Silva ◽  
Carlo Cafaro ◽  
Adom Giffin
Keyword(s):  
State Of The Art ◽  
Estimation Method ◽  
Synthetic Data ◽  
Noisy Data ◽  
Bayesian Framework ◽  
Physical Systems ◽  
Gradient Profile ◽  
Ill Posed ◽  
Profile Estimation ◽  
Different Levels

Attaining reliable gradient profiles is of utmost relevance for many physical systems. In many situations, the estimation of the gradient is inaccurate due to noise. It is common practice to first estimate the underlying system and then compute the gradient profile by taking the subsequent analytic derivative of the estimated system. The underlying system is often estimated by fitting or smoothing the data using other techniques. Taking the subsequent analytic derivative of an estimated function can be ill-posed. This becomes worse as the noise in the system increases. As a result, the uncertainty generated in the gradient estimate increases. In this paper, a theoretical framework for a method to estimate the gradient profile of discrete noisy data is presented. The method was developed within a Bayesian framework. Comprehensive numerical experiments were conducted on synthetic data at different levels of noise. The accuracy of the proposed method was quantified. Our findings suggest that the proposed gradient profile estimation method outperforms the state-of-the-art methods.

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