<s>
The	O
adjoint	B-Algorithm
state	I-Algorithm
method	I-Algorithm
is	O
a	O
numerical	B-Algorithm
method	I-Algorithm
for	O
efficiently	O
computing	O
the	O
gradient	O
of	O
a	O
function	O
or	O
operator	O
in	O
a	O
numerical	O
optimization	O
problem	O
.	O
</s>
<s>
It	O
has	O
applications	O
in	O
geophysics	O
,	O
seismic	O
imaging	O
,	O
photonics	O
and	O
more	O
recently	O
in	O
neural	B-Architecture
networks	I-Architecture
.	O
</s>
<s>
The	O
adjoint	O
method	O
formulates	O
the	O
gradient	O
of	O
a	O
function	O
towards	O
its	O
parameters	O
in	O
a	O
constraint	B-Application
optimization	O
form	O
.	O
</s>
<s>
By	O
using	O
the	O
dual	O
form	O
of	O
this	O
constraint	B-Application
optimization	O
problem	O
,	O
it	O
can	O
be	O
used	O
to	O
calculate	O
the	O
gradient	O
very	O
fast	O
.	O
</s>
<s>
in	O
the	O
Landweber	B-Algorithm
iteration	I-Algorithm
method	O
.	O
</s>
<s>
The	O
name	O
adjoint	B-Algorithm
state	I-Algorithm
method	I-Algorithm
refers	O
to	O
the	O
dual	O
form	O
of	O
the	O
problem	O
,	O
where	O
the	O
adjoint	B-Algorithm
matrix	I-Algorithm
is	O
used	O
.	O
</s>
<s>
The	O
state	O
variable	O
is	O
often	O
implicitly	O
dependant	O
on	O
through	O
the	O
(	O
direct	O
)	O
state	O
equation	O
(	O
usually	O
the	O
weak	B-Algorithm
form	I-Algorithm
of	O
a	O
partial	O
differential	O
equation	O
)	O
,	O
thus	O
the	O
considered	O
objective	O
is	O
.	O
</s>
<s>
where	O
is	O
the	O
Gateaux	B-Algorithm
derivative	I-Algorithm
of	O
with	O
respect	O
to	O
in	O
the	O
direction	O
.	O
</s>
<s>
In	O
a	O
real	O
finite	O
dimensional	O
linear	B-Algorithm
programming	I-Algorithm
context	O
,	O
the	O
objective	O
function	O
could	O
be	O
,	O
for	O
,	O
and	O
,	O
and	O
let	O
the	O
state	O
equation	O
be	O
,	O
with	O
and	O
.	O
</s>
<s>
where	O
is	O
a	O
third	O
order	O
tensor	B-Device
,	O
is	O
the	O
dyadic	O
product	O
between	O
the	O
direct	O
and	O
adjoint	O
states	O
and	O
denotes	O
a	O
double	O
tensor	B-Device
contraction	O
.	O
</s>
