Complexity of initial-value problems for ordinary differential equations of order k
We study the worst-case ɛ -complexity of nonlinear initial-value problems u ( k ) ( x ) = g x , u ( x ) , u ′ ( x ) , … , u ( q ) ( x ) , x ∈ [ a , b ] , 0 ⩽ q < k , with given initial conditions. We assume that function g has r ( r ⩾ 1 ) continuous bounded partial derivatives. We consider two ty...
Saved in:
| Published in: | Journal of Complexity Vol. 22; no. 4; pp. 514 - 532 |
|---|---|
| Main Author: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
Elsevier Inc
01.08.2006
|
| Subjects: | |
| ISSN: | 0885-064X, 1090-2708 |
| Online Access: | Get full text |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | We study the worst-case
ɛ
-complexity of nonlinear initial-value problems
u
(
k
)
(
x
)
=
g
x
,
u
(
x
)
,
u
′
(
x
)
,
…
,
u
(
q
)
(
x
)
,
x
∈
[
a
,
b
]
,
0
⩽
q
<
k
, with given initial conditions. We assume that function
g has
r (
r
⩾
1
) continuous bounded partial derivatives. We consider two types of information about
g: standard information defined by values of
g or its partial derivatives, and linear information defined by the values of linear functionals on
g. For standard information, we show that the worst-case complexity is
Θ
(
1
/
ɛ
)
1
/
r
, which is independent of
k and
q. By defining an algorithm using integral information, we show that the complexity is
O
(
1
/
ɛ
)
1
/
(
r
+
k
-
q
)
if linear information is used. Hence, linear information is more powerful than standard information. For
q
=
0
for instance, the complexity decreases from
Θ
(
1
/
ɛ
)
1
/
r
to
O
(
1
/
ɛ
)
1
/
(
r
+
k
)
. We also give a lower bound on the
ɛ
-complexity for linear information. We show that the complexity is
Ω
(
1
/
ɛ
)
1
/
(
r
+
k
)
, which means that upper and lower bounds match for
q
=
0
. The gap for the remaining values of
q is an open problem. |
|---|---|
| ISSN: | 0885-064X 1090-2708 |
| DOI: | 10.1016/j.jco.2006.03.002 |