Page 102 - Fundamentals of Magnetic Thermonuclear Reactor Design

86 Fundamentals of Magnetic Thermonuclear Reactor Design

For classical problems in mathematical physics, these conditions provide
mathematical definiteness of a problem and the possibility to solve related op-
erator equations with standard numerical methods when approximate inputs
are used. According to a classical definition, a well-posed problem should have
the following properties: (1) a solution exists; (2) the solution is unique and
(3) the solution behaviour is stable, that is, changes continuously with the ini-
tial data. Otherwise, problems are said to be ill-posed. The main challenges of
solving the ill-posed inverse problems are generally related to failures to fulfil
the third requirement, that is, to provide a stable solution of Eq. (4.11). In this
case, the incorrectness of the initial problem is reflected by a poorly conditioned
SLAE (4.11), as its discrete analogue.
For simplicity, let us assume the operator A in (4.11) to be given precisely
and the right-hand side to be given with an error:

uδ−u/≤δ. u δ − u / ≤ δ. (4.15)
It would be only natural to look for approximate solutions of equation
Az = u in the class of elements z ∈ F, consistent with error δ, that is, satisfy-
δ
ing condition ρ = ||Az − u || ≤ δ. However, we have no criterion by which one
δ
U
approximate solution is better than another, if we do not take additional infor-
mation about the solution.
−1
In the general case, symbol A u does not make sense, and Eq. (4.11) has
δ
no solution that can be found with the help of an inverse operator. This happens,
for instance, in cases where A is a continuous operator and the right-hand side
u has no derivative in a number of points that does not satisfy the first correct-
δ
ness requirement.
To avoid difficulties arising from the absence of solution of an operator
equation with an approximate input, a notation is introduced for a quasi-solu-
tion z* ∈ M (a pseudo-solution for a system of algebraic equations), minimising
the ρ = (Az, u ) functional over the set M:
δ
U
*
,
,
ρU(Az*,uδ)=inf ρU(Az,uδ)ρU, z∈M ρ ( Az u ) = inf ρ ( Azu ) z , ∈M (4.16)
δ
δ ρU
U
U
A search for a quasi-solution is actually a search for a minimum residual
functional over the set M. If M is an n-dimensional set, the problem is reduced
to minimisation of a function of n variables.
Let z be a solution to equation Az = u , that is, Az u .
T
T
T
T
Therefore, the numerical parameter δ > 0 characterises the error in the right
side of Eq. (4.11). The approximate solution z of this equation can be derived
δ
using an operator that depends upon this parameter in such a way that the solu-
tion would converge ρ (z , z ) → 0 at δ → 0.
δ
T
F
Under the regularisation procedure, operator R (u, α) is a regulariser for
equation Az = u in the vicinity of element u if
T
l there are numbers α , δ such that R(u, α) is defined for any α and u ∈ U,
0
0
which satisfy the conditions 0 < α ≤ α , ρ (u, u ) ≤ δ ;
0
0
u
T