Automatic differentiation: Point and interval; Automatic differentiation: Point and interval Taylor operators; Bounding
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VARIATIONAL
PROBLEMS,
IVP It is generally accepted that the notion 'ill-posed problem' originates from a considered concept of well-posedness: A problem is called ill-posed if it is not well-posed. There are a lot of different notions of well-posedness (cf. [15], [23], [27], [35], [38] and [40]), which correspond to certain classes of variational problems and numerical methods and take into account the 'quality' of the input data, in particular their exactness. For a comparison of different concepts of well-posedness see [12], [15] and [35]. For instance, Tikhonov well-posedness [35], [38] is convenient if we deal with methods generating feasible minimizing sequences, and it is not appropriate to analyse stability of exterior penalty methods. We shall proceed from two concepts of wellposedness which are suitable for wide classes of problems and methods. The first concept is destined to the problem min{J(u)'u
e K},
(1)
where K is a nonempty closed subset of a Banach space Y with the norm II'll and J" V --+ R U {+c~} is a proper lower-semicontinuous functional. DEFINITION 1 (cf. [27]) The sequence {u n} C V
is said to be a generalized minimizing sequence (Levitin-Polyak minimizing sequence) for ( 1 ) i f lim d(u n, K) - 0 n----~ o o
and lim J(u n) - inf J(u), n--+ c~
u CK
with
d(u, K) - inf IIu - vii vCK
a distance function.
[:3
DEFINITION 2 (1) is called well-posed (LevitinPolyak well-posed) if i) it is uniquely solvable, and ii) any generalized minimizing sequence converges to u* - arg min{J(u)" u e K}. [:] The second concept (cf. [20], [23])concerns (1) with
g-
{u e Uo" B(u) _ O,
i C I +,
xEX,
.-(0,...
,0) e
C "- Op x R ~ -p and R "- {x e X" g(x) e C}, with the stipulation that C - R ~ when p - 0 and C = Om := ( 0 , . . . , 0) C R m when p = m; m = 0 does not require to define C. A particular case of (P), call it (P)iso, is a classic isoperimetric type problem defined in the following way. Let ,4Cn(T) denote the class of absolutely continuous n-vector functions x(t) := ( x l ( t ) , . . . , x n ( t ) ) on T " - [a,b] C R with square integrable derivatives. By suitably defining the scalar product ~ and, consequently, the norm such a class is a Hilbert space; set H = ACn(T) and
f (x) - / ¢o(t , x(t), it(t)) dt,
458
where ¢i" R 1+2n ~ R, i E {0} U I, are given integrands. Fixed endpoints conditions can be included in the definition of X. We point out that the problems which can be reduced to the format of (P) share the characteristic of having a finitedimensional image. Hence, certain problems, for instance of geodesic type, are not covered by (P); the image analysis of t h e m is outside the scope of the present writing. The IS approach arises naturally in as much as an optimality condition for (P) is achieved through the impossibility of a system. More precisely, by paraphrasing the very definition of global minim u m we can say t h a t a feasible ~ C X is a global minimum point for (P) if and only if the system (in the unknown x):
(s)
.C,
f(x) > 0, x
x,
is impossible, or
where I ° "- {
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