150 research outputs found
Two numerical methods for solving a backward heat conduction problem
Abstract We introduce a central difference method and a quasi-reversibility method for solving a backward heat conduction problem (BHCP) numerically. For these two numerical methods, we give the stability analysis. Meanwhile, we investigate the roles of regularization parameters in these two methods. Numerical results show that our algorithm is effective
Fourier regularization for a backward heat equation
AbstractIn this paper a simple and convenient new regularization method for solving backward heat equation—Fourier regularization method is given. Meanwhile, some quite sharp error estimates between the approximate solution and exact solution are provided. A numerical example also shows that the method works effectively
A source identification problem in a bi-parabolic equation: convergence rates and some optimal results
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 revise
Fully discrete finite element data assimilation method for the heat equation
We consider a finite element discretization for the reconstruction of the
final state of the heat equation, when the initial data is unknown, but
additional data is given in a sub domain in the space time. For the
discretization in space we consider standard continuous affine finite element
approximation, and the time derivative is discretized using a backward
differentiation. We regularize the discrete system by adding a penalty of the
-semi-norm of the initial data, scaled with the mesh-parameter. The
analysis of the method uses techniques developed in E. Burman and L. Oksanen,
Data assimilation for the heat equation using stabilized finite element
methods, arXiv, 2016, combining discrete stability of the numerical method with
sharp Carleman estimates for the physical problem, to derive optimal error
estimates for the approximate solution. For the natural space time energy norm,
away from , the convergence is the same as for the classical problem with
known initial data, but contrary to the classical case, we do not obtain faster
convergence for the -norm at the final time
A final value problem with a non-local and a source term: regularization by truncation
This paper is concerned with recovering the solution of a final value problem
associated with a parabolic equation involving a non linear source and a
non-local term, which to the best of our knowledge has not been studied
earlier. It is shown that the considered problem is ill-posed, and thus, some
regularization method has to be employed in order to obtain stable
approximations. In this regard, we obtain regularized approximations by solving
some non linear integral equations which is derived by considering a truncated
version of the Fourier expansion of the sought solution. Under different Gevrey
smoothness assumptions on the exact solution, we provide parameter choice
strategies and obtain the error estimates. A key tool in deriving such
estimates is a version of Gr{\"o}nwalls' inequality for iterated integrals,
which perhaps, is proposed and analysed for the first time.Comment: Comments are welcome
- …