<s>
They	O
have	O
wide	O
application	O
in	O
system	O
identification	O
,	O
optics	B-Algorithm
,	O
radar	B-Application
,	O
acoustics	O
,	O
communication	O
theory	O
,	O
signal	O
processing	O
,	O
medical	B-Application
imaging	I-Application
,	O
computer	B-Application
vision	I-Application
,	O
geophysics	O
,	O
oceanography	O
,	O
astronomy	O
,	O
remote	B-Application
sensing	I-Application
,	O
natural	B-Language
language	I-Language
processing	I-Language
,	O
machine	O
learning	O
,	O
nondestructive	O
testing	O
,	O
slope	O
stability	O
analysis	O
and	O
many	O
other	O
fields	O
.	O
</s>
<s>
A	O
historical	O
example	O
is	O
the	O
calculations	O
of	O
Adams	O
and	O
Le	O
Verrier	O
which	O
led	O
to	O
the	O
discovery	O
of	O
Neptune	B-Application
from	O
the	O
perturbed	O
trajectory	O
of	O
Uranus	O
.	O
</s>
<s>
Then	O
,	O
around	O
the	O
seventies	O
,	O
the	O
least-squares	B-Algorithm
and	O
probabilistic	O
approaches	O
came	O
in	O
and	O
turned	O
out	O
to	O
be	O
very	O
helpful	O
for	O
the	O
determination	O
of	O
parameters	O
involved	O
in	O
various	O
physical	O
systems	O
.	O
</s>
<s>
where	O
is	O
the	O
matrix	B-Architecture
that	O
characterizes	O
the	O
forward	O
map	O
.	O
</s>
<s>
To	O
solve	O
for	O
the	O
model	O
parameters	O
that	O
fit	O
our	O
data	O
,	O
we	O
might	O
be	O
able	O
to	O
invert	O
the	O
matrix	B-Architecture
to	O
directly	O
convert	O
the	O
measurements	O
into	O
our	O
model	O
parameters	O
.	O
</s>
<s>
In	O
general	O
,	O
the	O
numbers	O
of	O
data	O
and	O
unknowns	O
are	O
different	O
so	O
that	O
matrix	B-Architecture
is	O
not	O
square	O
.	O
</s>
<s>
However	O
,	O
even	O
a	O
square	O
matrix	B-Architecture
can	O
have	O
no	O
inverse	O
:	O
matrix	B-Architecture
can	O
be	O
rank	O
deficient	O
(	O
i.e.	O
</s>
<s>
This	O
is	O
usually	O
achieved	O
by	O
penalizing	B-Algorithm
the	O
norm	O
of	O
the	O
gradient	O
(	O
or	O
the	O
total	O
variation	O
)	O
of	O
the	O
parameters	O
(	O
this	O
approach	O
is	O
also	O
referred	O
to	O
as	O
the	O
maximization	O
of	O
the	O
entropy	O
)	O
.	O
</s>
<s>
This	O
approach	O
amounts	O
to	O
making	O
use	O
of	O
ordinary	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
,	O
an	O
approach	O
widely	O
used	O
in	O
statistics	O
.	O
</s>
<s>
Very	O
similar	O
to	O
the	O
least-squares	B-Algorithm
approach	O
is	O
the	O
probabilistic	O
approach	O
:	O
If	O
we	O
know	O
the	O
statistics	O
of	O
the	O
noise	O
that	O
contaminates	O
the	O
data	O
,	O
we	O
can	O
think	O
of	O
seeking	O
the	O
most	O
likely	O
model	O
m	O
,	O
which	O
is	O
the	O
model	O
that	O
matches	O
the	O
maximum	O
likelihood	O
criterion	O
.	O
</s>
<s>
If	O
the	O
noise	O
is	O
Gaussian	O
,	O
the	O
maximum	O
likelihood	O
criterion	O
appears	O
as	O
a	O
least-squares	B-Algorithm
criterion	O
,	O
the	O
Euclidean	O
scalar	O
product	O
in	O
data	O
space	O
being	O
replaced	O
by	O
a	O
scalar	O
product	O
involving	O
the	O
co-variance	O
of	O
the	O
noise	O
.	O
</s>
<s>
where	O
FT	O
denotes	O
the	O
matrix	B-Architecture
transpose	O
of	O
F	O
.	O
This	O
equation	O
simplifies	O
to	O
:	O
</s>
<s>
In	O
our	O
example	O
matrix	B-Architecture
turns	O
out	O
to	O
be	O
generally	O
full	O
rank	O
so	O
that	O
the	O
equation	O
above	O
makes	O
sense	O
and	O
determines	O
uniquely	O
the	O
model	O
parameters	O
:	O
we	O
do	O
not	O
need	O
integrating	O
additional	O
information	O
for	O
ending	O
up	O
with	O
a	O
unique	O
solution	O
.	O
</s>
<s>
Inverse	O
problems	O
are	O
typically	O
ill-posed	B-Algorithm
,	O
as	O
opposed	O
to	O
the	O
well-posed	B-Algorithm
problems	I-Algorithm
usually	O
met	O
in	O
mathematical	O
modeling	O
.	O
</s>
<s>
Of	O
the	O
three	O
conditions	O
for	O
a	O
well-posed	B-Algorithm
problem	I-Algorithm
suggested	O
by	O
Jacques	O
Hadamard	O
(	O
existence	O
,	O
uniqueness	O
,	O
and	O
stability	O
of	O
the	O
solution	O
or	O
solutions	O
)	O
the	O
condition	O
of	O
stability	O
is	O
most	O
often	O
violated	O
.	O
</s>
<s>
In	O
the	O
sense	O
of	O
functional	B-Application
analysis	I-Application
,	O
the	O
inverse	O
problem	O
is	O
represented	O
by	O
a	O
mapping	O
between	O
metric	O
spaces	O
.	O
</s>
<s>
In	O
this	O
case	O
the	O
inverse	O
problem	O
will	O
typically	O
be	O
ill-conditioned	B-Algorithm
.	O
</s>
<s>
In	O
these	O
cases	O
,	O
regularization	O
may	O
be	O
used	O
to	O
introduce	O
mild	O
assumptions	O
on	O
the	O
solution	O
and	O
prevent	O
overfitting	B-Error_Name
.	O
</s>
<s>
This	O
means	O
that	O
given	O
a	O
linear	O
combination	O
of	O
these	O
functions	O
,	O
the	O
coefficients	O
can	O
be	O
computed	O
by	O
arranging	O
the	O
vectors	O
as	O
the	O
columns	O
of	O
a	O
matrix	B-Architecture
and	O
then	O
inverting	O
this	O
matrix	B-Architecture
.	O
</s>
<s>
Concretely	O
,	O
this	O
is	O
done	O
by	O
inverting	O
the	O
Vandermonde	O
matrix	B-Architecture
.	O
</s>
<s>
Once	O
chosen	O
the	O
appropriate	O
algorithm	O
for	O
solving	O
the	O
forward	O
problem	O
(	O
a	O
straightforward	O
matrix-vector	O
multiplication	O
may	O
be	O
not	O
adequate	O
when	O
matrix	B-Architecture
is	O
huge	O
)	O
,	O
the	O
appropriate	O
algorithm	O
for	O
carrying	O
out	O
the	O
minimization	O
can	O
be	O
found	O
in	O
textbooks	O
dealing	O
with	O
numerical	O
methods	O
for	O
the	O
solution	O
of	O
linear	O
systems	O
and	O
for	O
the	O
minimization	O
of	O
quadratic	O
functions	O
(	O
see	O
for	O
instance	O
Ciarlet	O
or	O
Nocedal	O
)	O
.	O
</s>
<s>
Also	O
,	O
the	O
user	O
may	O
wish	O
to	O
add	O
physical	O
constraints	O
to	O
the	O
models	O
:	O
In	O
this	O
case	O
,	O
they	O
have	O
to	O
be	O
familiar	O
with	O
constrained	B-Application
optimization	I-Application
methods	I-Application
,	O
a	O
subject	O
in	O
itself	O
.	O
</s>
<s>
The	O
direction	O
of	O
the	O
largest	O
axis	O
of	O
this	O
ellipsoid	O
(	O
eigenvector	O
associated	O
with	O
the	O
smallest	O
eigenvalue	O
of	O
matrix	B-Architecture
)	O
is	O
the	O
direction	O
of	O
poorly	O
determined	O
components	O
:	O
if	O
we	O
follow	O
this	O
direction	O
,	O
we	O
can	O
bring	O
a	O
strong	O
perturbation	O
to	O
the	O
model	O
without	O
changing	O
significantly	O
the	O
value	O
of	O
the	O
objective	O
function	O
and	O
thus	O
end	O
up	O
with	O
a	O
significantly	O
different	O
quasi-optimal	O
model	O
.	O
</s>
<s>
We	O
clearly	O
see	O
that	O
the	O
answer	O
to	O
the	O
question	O
"	O
can	O
we	O
trust	O
this	O
model	O
"	O
is	O
governed	O
by	O
the	O
noise	O
level	O
and	O
by	O
the	O
eigenvalues	O
of	O
the	O
Hessian	O
of	O
the	O
objective	O
function	O
or	O
equivalently	O
,	O
in	O
the	O
case	O
where	O
no	O
regularization	O
has	O
been	O
integrated	O
,	O
by	O
the	O
singular	O
values	O
of	O
matrix	B-Architecture
.	O
</s>
<s>
For	O
example	O
,	O
a	O
naive	O
discretization	O
will	O
often	O
work	O
for	O
solving	O
the	O
deconvolution	B-Algorithm
problem	O
:	O
it	O
will	O
work	O
as	O
long	O
as	O
we	O
do	O
not	O
allow	O
missing	O
frequencies	O
to	O
show	O
up	O
in	O
the	O
numerical	O
solution	O
.	O
</s>
<s>
This	O
equation	O
is	O
an	O
extension	O
to	O
infinite	O
dimension	O
of	O
the	O
matrix	B-Architecture
equation	I-Architecture
given	O
in	O
the	O
case	O
of	O
discrete	O
problems	O
.	O
</s>
<s>
Thus	O
any	O
solution	O
of	O
this	O
equation	O
is	O
determined	O
up	O
to	O
an	O
additive	O
function	O
in	O
the	O
null-space	O
and	O
,	O
in	O
the	O
case	O
of	O
infinity	O
of	O
singular	O
values	O
,	O
the	O
solution	O
(	O
which	O
involves	O
the	O
reciprocal	O
of	O
arbitrary	O
small	O
eigenvalues	O
)	O
is	O
unstable	O
:	O
two	O
ingredients	O
that	O
make	O
the	O
solution	O
of	O
this	O
integral	O
equation	O
a	O
typical	O
ill-posed	B-Algorithm
problem	I-Algorithm
!	O
</s>
<s>
However	O
,	O
we	O
can	O
define	O
a	O
solution	O
through	O
the	O
pseudo-inverse	B-Algorithm
of	O
the	O
forward	O
map	O
(	O
again	O
up	O
to	O
an	O
arbitrary	O
additive	O
function	O
)	O
.	O
</s>
<s>
When	O
the	O
forward	O
map	O
is	O
compact	O
,	O
the	O
classical	O
Tikhonov	O
regularization	O
will	O
work	O
if	O
we	O
use	O
it	O
for	O
integrating	O
prior	O
information	O
stating	O
that	O
the	O
norm	O
of	O
the	O
solution	O
should	O
be	O
as	O
small	O
as	O
possible	O
:	O
this	O
will	O
make	O
the	O
inverse	O
problem	O
well-posed	B-Algorithm
.	O
</s>
<s>
Irregular	O
kernels	O
may	O
yield	O
a	O
forward	O
map	O
which	O
is	O
not	O
compact	O
and	O
even	O
unbounded	B-Algorithm
if	O
we	O
naively	O
equip	O
the	O
space	O
of	O
models	O
with	O
the	O
norm	O
.	O
</s>
<s>
A	O
mathematical	O
analysis	O
is	O
required	O
to	O
make	O
it	O
a	O
bounded	O
operator	O
and	O
design	O
a	O
well-posed	B-Algorithm
problem	I-Algorithm
:	O
an	O
illustration	O
can	O
be	O
found	O
in	O
.	O
</s>
<s>
The	O
goal	O
of	O
deconvolution	B-Algorithm
is	O
to	O
reconstruct	O
the	O
original	O
image	O
or	O
signal	O
which	O
appears	O
as	O
noisy	O
and	O
blurred	O
on	O
the	O
data	O
.	O
</s>
<s>
In	O
X-ray	O
computed	O
tomography	O
the	O
lines	O
on	O
which	O
the	O
parameter	O
is	O
integrated	O
are	O
straight	O
lines	O
:	O
the	O
tomographic	B-Algorithm
reconstruction	I-Algorithm
of	O
the	O
parameter	O
distribution	O
is	O
based	O
on	O
the	O
inversion	O
of	O
the	O
Radon	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
Although	O
from	O
a	O
theoretical	O
point	O
of	O
view	O
many	O
linear	O
inverse	O
problems	O
are	O
well	O
understood	O
,	O
problems	O
involving	O
the	O
Radon	B-Algorithm
transform	I-Algorithm
and	O
its	O
generalisations	O
still	O
present	O
many	O
theoretical	O
challenges	O
with	O
questions	O
of	O
sufficiency	O
of	O
data	O
still	O
unresolved	O
.	O
</s>
<s>
Solutions	O
explored	O
include	O
Algebraic	B-Algorithm
Reconstruction	I-Algorithm
Technique	I-Algorithm
,	O
filtered	O
backprojection	O
,	O
and	O
as	O
computing	O
power	O
has	O
increased	O
,	O
iterative	B-Algorithm
reconstruction	I-Algorithm
methods	O
such	O
as	O
iterative	B-Algorithm
Sparse	I-Algorithm
Asymptotic	I-Algorithm
Minimum	I-Algorithm
Variance	I-Algorithm
.	O
</s>
<s>
be	O
used	O
for	O
the	O
solving	O
the	O
wave	O
equation	O
,	O
these	O
methods	O
turn	O
out	O
to	O
be	O
closely	O
related	O
to	O
the	O
so-called	O
least-squares	B-Algorithm
migration	O
methods	O
derived	O
from	O
the	O
least-squares	B-Algorithm
approach	O
(	O
see	O
Lailly	O
,	O
Tarantola	O
)	O
.	O
</s>
<s>
Doppler	B-Algorithm
tomography	I-Algorithm
aims	O
at	O
converting	O
the	O
information	O
contained	O
in	O
spectral	O
monitoring	O
of	O
the	O
object	O
into	O
a	O
2D	O
image	O
of	O
the	O
emission	O
(	O
as	O
a	O
function	O
of	O
the	O
radial	O
velocity	O
and	O
of	O
the	O
phase	O
in	O
the	O
periodic	O
rotation	O
movement	O
)	O
of	O
the	O
stellar	O
atmosphere	O
.	O
</s>
<s>
A	O
variety	O
of	O
numerical	O
techniques	O
have	O
been	O
developed	O
to	O
address	O
the	O
ill-posedness	B-Algorithm
and	O
sensitivity	O
to	O
measurement	O
error	O
caused	O
by	O
damping	O
and	O
lagging	O
in	O
the	O
temperature	O
signal	O
.	O
</s>
<s>
The	O
spectrum	O
is	O
made	O
of	O
eigenvalues	O
and	O
eigenfunctions	B-Algorithm
,	O
forming	O
together	O
the	O
"	O
discrete	O
spectrum	O
"	O
,	O
and	O
generalizations	O
,	O
called	O
the	O
continuous	O
spectrum	O
.	O
</s>
<s>
Nonlinear	O
inverse	O
problems	O
are	O
also	O
currently	O
studied	O
in	O
many	O
fields	O
of	O
applied	O
science	O
(	O
acoustics	O
,	O
mechanics	O
,	O
quantum	O
mechanics	O
,	O
electromagnetic	O
scattering	O
-	O
in	O
particular	O
radar	B-Application
soundings	O
,	O
seismic	O
soundings	O
,	O
and	O
nearly	O
all	O
imaging	O
modalities	O
)	O
.	O
</s>
<s>
Practical	O
applications	O
,	O
using	O
the	O
least-squares	B-Algorithm
approach	O
,	O
were	O
developed	O
.	O
</s>
<s>
Realizing	O
how	O
difficult	O
is	O
the	O
inverse	O
problem	O
in	O
the	O
wave	O
equation	O
,	O
seismologists	O
investigated	O
a	O
simplified	O
approach	O
making	O
use	O
of	O
geometrical	O
optics	B-Algorithm
.	O
</s>
<s>
It	O
is	O
classically	O
solved	O
by	O
shooting	O
rays	B-Algorithm
(	O
trajectories	O
about	O
which	O
the	O
arrival	O
time	O
is	O
stationary	O
)	O
from	O
the	O
point	O
source	O
.	O
</s>
<s>
The	O
questions	O
concern	O
well-posedness	B-Algorithm
:	O
Does	O
the	O
least-squares	B-Algorithm
problem	I-Algorithm
have	O
a	O
unique	O
solution	O
which	O
depends	O
continuously	O
on	O
the	O
data	O
(	O
stability	O
problem	O
)	O
?	O
</s>
<s>
use	O
of	O
global	O
optimization	O
techniques	O
such	O
as	O
sampling	O
of	O
the	O
posterior	O
density	O
function	O
and	O
Metropolis	B-Algorithm
algorithm	I-Algorithm
in	O
the	O
inverse	O
problem	O
probabilistic	O
framework	O
,	O
genetic	O
algorithms	O
(	O
alone	O
or	O
in	O
combination	O
with	O
Metropolis	B-Algorithm
algorithm	I-Algorithm
:	O
see	O
for	O
an	O
application	O
to	O
the	O
determination	O
of	O
permeabilities	O
that	O
match	O
the	O
existing	O
permeability	O
data	O
)	O
,	O
neural	O
networks	O
,	O
regularization	O
techniques	O
including	O
multi	O
scale	O
analysis	O
;	O
</s>
<s>
reformulation	O
of	O
the	O
least-squares	B-Algorithm
objective	O
function	O
so	O
as	O
to	O
make	O
it	O
smoother	O
(	O
see	O
for	O
the	O
inverse	O
problem	O
in	O
the	O
wave	O
equations	O
.	O
)	O
</s>
<s>
Contrary	O
to	O
the	O
linear	O
situation	O
,	O
an	O
explicit	O
use	O
of	O
the	O
Hessian	O
matrix	B-Architecture
for	O
solving	O
the	O
normal	O
equations	O
does	O
not	O
make	O
sense	O
here	O
:	O
the	O
Hessian	O
matrix	B-Architecture
varies	O
with	O
models	O
.	O
</s>
<s>
The	O
linear	O
inverse	O
problem	O
is	O
also	O
the	O
fundamental	O
of	O
spectral	O
estimation	O
and	O
direction-of-arrival	B-Algorithm
(	O
DOA	O
)	O
estimation	O
in	O
signal	O
processing	O
.	O
</s>
<s>
Inverse	O
lithography	O
is	O
used	O
in	O
photomask	B-Algorithm
design	O
for	O
semiconductor	B-Architecture
device	I-Architecture
fabrication	I-Architecture
.	O
</s>
