et us review.
We stop evolution in time face the problem of taking
where
and
are two known functions. These functions are given by the columns
and
:
where
are bases in
of manageable size and
is a rectangular domain in
.
We keep the size of
manageable by using sparse tensor product. However, in multiple dimensions
we are prohibited from taking the
sums
because these would have to be described as piecewise polynomials on straight
product
.
Such descriptions would have unmanageable size.
To overcome this difficulty we construct
instead
In the previous sections we construct a procedure that adaptively approximates
by a linear combination of
.
We make an observation that for corrected
the difference
should not have positive component. Then we subtract adaptively nonnegative
functions until we reach our goal.
If we use plain pyramid functions in multidimensional situation then we would
have to subtract a lot of them. It would not be efficient. How do we
efficiently select a good function to subtract? We can subtract several
pyramid functions with nonoverlapping support in parallel. We also can
subtract probing functions with good approximation properties. For example, we
could use the crudification operator, see the definitions
(
Crudification
operator
),(
Crudification operator
2
). The difficulty with taking
is not the size of the basis but the size the piecewise polynomial
representation. By using the crudification operator, we control the size of
representation. We crudify representations of
and
,
calculate the crudified version of
and use it as
in the procedure of the section
(
Mix
of backward induction and penalty term approaches II
).
The crudification of the coordinate columns is performed as follows. Let
be a relatively small crudified basis. We
introduce
We can efficiently calculate
for any
.
Indeed,
The matrixes
are of the form
see the sections (
Solving
Ndimensional PDEs
) and
(
Reduction
to system of linear algebraic equations for Black PDE
).
