13,548 research outputs found
The discretized discrepancy principle under general source conditions
AbstractWe discuss adaptive strategies for choosing regularization parameters in Tikhonov–Phillips regularization of discretized linear operator equations. Two rules turn out to be based entirely on data from the underlying regularization scheme. Among them, only the discrepancy principle allows us to search for the optimal regularization parameter from the easiest problem. This potential advantage cannot be achieved by the standard projection scheme. We present a modified scheme, in which the discretization level varies with the successive regularization parameters, which has the advantage, mentioned before
A Mathematica Program for heat source function of 1D heat equation reconstruction by three types of data
We solve an inverse problem for the one-dimensional heat diffusion equation.
We reconstruct the heat source function for the three types of data: 1) single
position point and different times, 2) constant time and uniformly distributed
positions, 3) random position points and different times. First we demonstrate
reconstruction using simple inversion of discretized Kernel matrix. Then we
apply Tikhonov regularization for two types of the parameter of regularization
estimation. The first one, which is in fact exemplary simulation, is based on
minimization of the distance in C space of reconstructed function to the
initial source function. Second rule is known as Discrepancy principle. We
generate the data from the chosen source function. In order to get some measure
of accuracy of reconstruction we compare the result with the function from
which data was generated. We also deliver corresponding application in symbolic
computation environment of Mathematica. The program has a lot of flexibility,
it can perform reconstruction for much more general input then one considered
in the paper.Comment: 10 pages, 7 figure
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