Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On a new notion of the solution to an ill-posed problem   A new understanding of the notion of the stable solution to ill-posed
problems is proposed. The new notion is more realistic than the old one and
better fits the practical computational needs. A method for constructing stable
solutions in the new sense is proposed and justified. The basic point is: in
the traditional definition of the stable solution to an ill-posed problem
$Au=f$, where $A$ is a linear or nonlinear operator in a Hilbert space $H$, it
is assumed that the noisy data $\{f_\delta, \delta\}$ are given,
$||f-f_\delta||\leq \delta$, and a stable solution $u_\d:=R_\d f_\d$ is defined
by the relation $\lim_{\d\to 0}||R_\d f_\d-y||=0$, where $y$ solves the
equation $Au=f$, i.e., $Ay=f$. In this definition $y$ and $f$ are unknown. Any
$f\in B(f_\d,\d)$ can be the exact data, where $B(f_\d,\d):=\{f:
||f-f_\delta||\leq \delta\}$.The new notion of the stable solution excludes the
unknown $y$ and $f$ from the definition of the solution.

26
The categories of this paper are: bayes-an
36
The categories of this paper are: cmp-lg
19
The categories of this paper are: eess
9
The categories of this paper are: nlin
11
The categories of this paper are: hep-lat
23
The categories of this paper are: solv-int
12
The categories of this paper are: nucl-th
16
The categories of this paper are: stat
32
The categories of this paper are: atom-ph
8
The categories of this paper are: hep-ex
29
The categories of this paper are: funct-an
0
The categories of this paper are: hep-ph
7
The categories of this paper are: hep-th
4
The categories of this paper are: cond-mat
18
The categories of this paper are: econ
33
The categories of this paper are: plasm-ph