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Recently, Nair and Danumjaya (2023) introduced a new regularization method for the homogeneous time-fractional backward heat conduction problem (TFBHCP) in a one-dimensional space variable, for determining the initial value function. In this paper, the authors extend the analysis done in the above referred paper to a more general setting of an inhomogeneous time-fractional heat equation involving the higher dimensional state variables and a general elliptic operator. We carry out the analysis for the newly introduced regularization method for the TFBHCP providing optimal order error estimates under a source condition by choosing the regularization parameter appropriately, and also carry out numerical experiments illustrating the theoretical results., Comment: 24 pages, 4 figures

This paper is concerned with identification of a spatial source function from final time observation in a bi-parabolic equation, where the full source function is assumed to be a product of time dependent and a space dependent function. Due to the ill-posedness of the problem, recently some authors have employed different regularization method and analysed the convergence rates. But, to the best of our knowledge, the quasi-reversibility method is not explored yet, and thus we study that in this paper. As an important implication, the H{\"o}lder rates for the apriori and aposteriori error estimates obtained in this paper improve upon the rates obtained in earlier works. Also, in some cases we show that the rates obtained are of optimal order. Further, this work seems to be the first one that has broaden the applicability of the problem by allowing the time dependent component of the source function to change sign. To the best of our knowledge, the earlier known work assumed the fixed sign of the time dependent component by assuming some bounded below condition., Comment: Comments are welcome. Typos and some mistakes with sign in the PDE are rectified. Section 4 and 5 are majorly revised

It is well-known that the backward heat conduction problem of recovering the temperature $u(\cdot, t)$ at a time $t\geq 0$ from the knowledge of the temperature at a later time, namely $g:= u(\cdot, \tau)$ for $\tau>t$, is ill-posed, in the sense that small error in $g$ can lead to large deviation in $u(\cdot, t)$. However, in the case of a time fractional backward heat conduction problem (TFBHCP), the above problem is well-posed for $t>0$ and ill-posed for $t=0$. We use this observation to obtain stable approximate solutions for the TFBHCP for $t=0$, and derive error estimates under suitable source conditions. We shall also provide some numerical examples to illustrate the approximation properties of the regularized solutions.

Light rare-earth antimonide LaSb2 is a material susceptible to nesting and exhibits a Charge Density Wave (CDW) at 355 K as well as superconductivity below 1.2 K. In the seek for additional CDW transitions, we have studied the temperature-dependent of LaSb2 at 200 and 13 K and infer CDW manifestations by combining angle-resolved photoemission spectroscopy (ARPES) measurements and density functional theory (DFT) calculations. ARPES measurements at 200 K show a metallic system while it appears to be semiconducting at 13 K, at odds with existing resistivity measurements. At 13 K, ARPES shows the band folding of the inner Fermi surface pockets, with considerable spectral weight on the folded band. We find a nesting vector at q = 0.25$\pm$0.02 {\AA} -1. In addition, we observe Umklapps of other bands due to the onset of the new periodicity, together with a semiconducting behavior in the whole reciprocal space. Calculations demonstrate that the folded band is associated with the La-Sb layer and that in-plane distortion is the most probable structural modification in the system, probably affecting the whole unit cell., Comment: In this work, the discussion presents two interpretations to explain the experimental results. We withdraw this work because after further discussions and data analysis we have concluded that only one of the two interpretations is consistent with the experimental results

This paper deals with the numerical approximation of the biharmonic inverse source problem in an abstract setting in which the measurement data is finite-dimensional. This unified framework in particular covers the conforming and nonconforming finite element methods (FEMs). The inverse problem is analysed through the forward problem. Error estimate for the forward solution is derived in an abstract set-up that applies to conforming and Morley nonconforming FEMs. Since the inverse problem is ill-posed, Tikhonov regularisation is considered to obtain a stable approximate solution. Error estimate is established for the regularised solution for different regularisation schemes. Numerical results that confirm the theoretical results are also presented., Comment: 26 pages, 6 figures, 5 tables

We consider an inverse problem of identifying the diffusion coefficient in matrix form in a parabolic PDE. In 2006, Cao and Pereverzev, used a \textit{natural linearisation} method for identifying a scalar valued diffusion coefficient in a parabolic PDE. In this paper, we make use of that idea for identifying a matrix valued coefficient, namely, using the notion of a weak solution for a parabolic PDE, we transform our non-linear inverse problem into a problem of solving an ill-posed operator equation where the operator depending on the data is linear. For the purpose of obtaining stable approximate solutions, Tikhonov regularization is employed, and error estimates under noisy data are derived. We have also showed the uniqueness of the solution of the inverse problem under some assumptions on the data and obtained explicit representation of adjoint of the linear operator involved. For the obtaining error estimates in the finite dimensional setting, Galerkin method is used, by defining orthogonal projections on the space of matrices with entries from $L^2(\O)$, by making use of standard orthogonal projections on $L^2(\O)$. For choosing the regularizing parameter, we used the adaptive technique, so that we have an order optimal rate of convergence. Finally, for the relaxed noisy data, we described a procedure for obtaining a smoothed version so as to obtain the error estimates., Comment: 22 pages

We consider a parameter identification problem related to a quasi-linear elliptic Neumann boundary value problem involving a parameter function $a(\cdot)$ and the solution $u(\cdot)$, where the problem is to identify $a(\cdot)$ on an interval $I:= g(\Gamma)$ from the knowledge of the solution $u(\cdot)$ as $g$ on $\Gamma$, where $\Gamma$ is a given curve on the boundary of the domain $\Omega \subseteq \mathbb{R}^3$ of the problem and $g$ is a continuous function. For obtaining stable approximate solutions, we consider new regularization method which gives error estimates similar to, and in certain cases better than, the classical Tikhonov regularization considered in the literature in recent past.

We study regularization of ill-posed equations involving multiplication operators when the multiplier function is positive almost everywhere and zero is an accumulation point of the range of this function. Such equations naturally arise from equations based on non-compact self-adjoint operators in Hilbert space, after applying unitary transformations arising out of the spectral theorem. For classical regularization theory, when noisy observations are given and the noise is deterministic and bounded, then non-compactness of the ill-posed equations is a minor issue. However, for statistical ill-posed equations with non-compact operators less is known if the data are blurred by white noise. We develop a regularization theory with emphasis on this case. In this context, we highlight several aspects, in particular we discuss the intrinsic degree of ill-posedness in terms of rearrangements of the multiplier function. Moreover, we address the required modifications of classical regularization schemes in order to be used for non-compact statistical problems, and we also introduce the concept of the effective ill-posedness of the operator equation under white noise. This study is concluded with prototypical examples for such equations, as these are deconvolution equations and certain final value problems in evolution equations.

In this note we consider the {\it spectral truncation} as the regularization for an ill-posed non-homogeneous parabolic final value problem, and obtain error estimates under a genral source condition when the data, which consist of the non-homogeneous term as well as the final value, are noisy. The resulting error estimate is compared with the corresponding estimate under the Lavrentieve method, and showed that the truncation method has no index of saturation., Comment: 10 pages

Zig-zag edge graphene ribbons grown on 6H-SiC facets are ballistic conductors. It has been assumed that zig-zag graphene ribbons grown on 4H-SiC would also be ballistic. However, in this work we show that SiC polytype matters; ballistic graphene ribbons only grow on 6H SiC. 4H and 4H-passivated ribbons are diffusive conductors. Detailed photoemmision and microscopy studies show that 6H-SiC sidewalls zig-zag ribbons are metallic with a pair of n-doped edge states associated with asymmetric edge terminations, In contrast, 4H-SiC zig-zag ribbons are strongly bonded to the SiC; severely distorting the ribbon's $\pi$-bands. $\text{H}_2$-passivation of the 4H ribbons returns them to a metallic state but show no evidence of edge states., Comment: 6 figure