<s>
The	O
mathematical	O
term	O
well-posed	B-Algorithm
problem	I-Algorithm
stems	O
from	O
a	O
definition	O
given	O
by	O
20th-century	O
French	O
mathematician	O
Jacques	O
Hadamard	O
.	O
</s>
<s>
the	O
solution	O
's	O
behaviour	O
changes	O
continuously	O
with	O
the	O
initial	B-Algorithm
conditions	I-Algorithm
.	O
</s>
<s>
Examples	O
of	O
archetypal	O
well-posed	B-Algorithm
problems	I-Algorithm
include	O
the	O
Dirichlet	O
problem	O
for	O
Laplace	O
's	O
equation	O
,	O
and	O
the	O
heat	O
equation	O
with	O
specified	O
initial	B-Algorithm
conditions	I-Algorithm
.	O
</s>
<s>
Problems	O
that	O
are	O
not	O
well-posed	B-Algorithm
in	O
the	O
sense	O
of	O
Hadamard	O
are	O
termed	O
ill-posed	B-Algorithm
.	O
</s>
<s>
Inverse	O
problems	O
are	O
often	O
ill-posed	B-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
the	O
inverse	O
heat	O
equation	O
,	O
deducing	O
a	O
previous	O
distribution	O
of	O
temperature	O
from	O
final	O
data	O
,	O
is	O
not	O
well-posed	B-Algorithm
in	O
that	O
the	O
solution	O
is	O
highly	O
sensitive	O
to	O
changes	O
in	O
the	O
final	O
data	O
.	O
</s>
<s>
Continuum	O
models	O
must	O
often	O
be	O
discretized	B-Algorithm
in	O
order	O
to	O
obtain	O
a	O
numerical	O
solution	O
.	O
</s>
<s>
While	O
solutions	O
may	O
be	O
continuous	O
with	O
respect	O
to	O
the	O
initial	B-Algorithm
conditions	I-Algorithm
,	O
they	O
may	O
suffer	O
from	O
numerical	B-Algorithm
instability	I-Algorithm
when	O
solved	O
with	O
finite	O
precision	O
,	O
or	O
with	O
errors	O
in	O
the	O
data	O
.	O
</s>
<s>
Even	O
if	O
a	O
problem	O
is	O
well-posed	B-Algorithm
,	O
it	O
may	O
still	O
be	O
ill-conditioned	B-Algorithm
,	O
meaning	O
that	O
a	O
small	O
error	O
in	O
the	O
initial	O
data	O
can	O
result	O
in	O
much	O
larger	O
errors	O
in	O
the	O
answers	O
.	O
</s>
<s>
An	O
ill-conditioned	B-Algorithm
problem	O
is	O
indicated	O
by	O
a	O
large	O
condition	B-Algorithm
number	I-Algorithm
.	O
</s>
<s>
If	O
the	O
problem	O
is	O
well-posed	B-Algorithm
,	O
then	O
it	O
stands	O
a	O
good	O
chance	O
of	O
solution	O
on	O
a	O
computer	O
using	O
a	O
stable	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
If	O
it	O
is	O
not	O
well-posed	B-Algorithm
,	O
it	O
needs	O
to	O
be	O
re-formulated	O
for	O
numerical	O
treatment	O
.	O
</s>
<s>
Tikhonov	O
regularization	O
is	O
one	O
of	O
the	O
most	O
commonly	O
used	O
for	O
regularization	O
of	O
linear	O
ill-posed	B-Algorithm
problems	I-Algorithm
.	O
</s>
<s>
A	O
method	O
to	O
determine	O
the	O
well-posedness	B-Algorithm
of	O
a	O
problem	O
is	O
the	O
energy	O
method	O
.	O
</s>
<s>
From	O
this	O
energy	O
estimate	O
one	O
can	O
conclude	O
that	O
the	O
problem	O
is	O
well-posed	B-Algorithm
.	O
</s>
