Tree approximation
Adaptive tree approximation
with finite elements
Andreas Veeser
Università degli Studi di Milano (Italy)
July 2015 / Cimpa School / Mumbai
A. Veeser
FE tree approximation
Tree approximation
Outline
1
Basic notions in constructive approximation
2
Tree approximation
3
Mesh refinement with tree structure for PDEs
4
Approximation of gradients
5
Convergence rates for tree approximation of gradients
A. Veeser
FE tree approximation
Tree approximation
Abstract setting
An instance-optimal algorithm
Outline
1
Tree approximation
Abstract setting
An instance-optimal algorithm
A. Veeser
FE tree approximation
Tree approximation
Abstract setting
An instance-optimal algorithm
Motivation
Tree approximation provides a solution to the practical problem
for approximation with bisection.
A. Veeser
FE tree approximation
Tree approximation
Abstract setting
An instance-optimal algorithm
Literature
P. Binev, R. DeVore, Fast computation in adaptive tree
approximation, Numer. Math. 97 (2004), 193–217.
(ignore Section 7, apart from Remark 7.1)
P. Binev, Tree approximation for hp-adaptivity, to appear.
A. Veeser
FE tree approximation
Tree approximation
Abstract setting
An instance-optimal algorithm
Outline
1
Tree approximation
Abstract setting
An instance-optimal algorithm
A. Veeser
FE tree approximation
Tree approximation
Abstract setting
An instance-optimal algorithm
Master tree
Iterated interval bisection has the following abstract structure
upon identifying intervals and nodes.
Let T be a master tree, ie an infinite binary tree such that
each node has two children,
each node has a parent, except one which is called the
root (which corresponds to Ω).
A. Veeser
FE tree approximation
Tree approximation
Abstract setting
An instance-optimal algorithm
(Sub)trees and partitions
A subset T ⊂ T is a subtree whenever
T is finite,
each node in T has its parent in T , except one.
A subtree T is full whenever each node has either two or non
children. The nodes of a subtree T with no children are the
leaves L(T ).
A tree T is subtree with root Ω. The leaves L(T ) of a full tree T
form a partition of Ω and we write T ∈ MT .
A. Veeser
FE tree approximation
Tree approximation
Abstract setting
An instance-optimal algorithm
Subadditive error functionals
Given local error functionals : T → R+
0 , we define
P
E(T ) := K ∈L(T ) (K )
be the global error for any full tree T and assume subadditivity:
(K1 ) + (K2 ) ≤ (K )
whenever K1 and K2 are the children of K ∈ T .
Set MTn := {T ∈ MT | #L(T ) ≤ n} and
E(MTn ) := min E(T ).
T ∈MT
n
A. Veeser
FE tree approximation
Tree approximation
Abstract setting
An instance-optimal algorithm
Example with interval bisection
Let MT the master tree associated with interval bisection
Mbisect (Ω).
For 0 < p < ∞, the local ‘errors’
(K ) = inf kv − PkpLp (K )
P∈P`
are subadditive. Moreover,
p
E(T ) = E P` (L(T ))
and
p
E(MTn ) = E P` (Mbisect
) =
n
A. Veeser
min
M∈Mbisect
n
p
E P` (M) .
FE tree approximation
Tree approximation
Abstract setting
An instance-optimal algorithm
Outline
1
Tree approximation
Abstract setting
An instance-optimal algorithm
A. Veeser
FE tree approximation
Tree approximation
Abstract setting
An instance-optimal algorithm
Approximating with greedy (maximum) strategy
Starting from T0 = {Ω}, iterate
1
Compute tN := maxK ∈L(TN ) (K )
2
Bisect all leaves K ∈ TN with (K ) = tN to obtain TN+1
3
increment N
until, e.g., E(TN ) is sufficiently small.
(type of algorithm and first analysis goes back to
Birman/Solomyak ’67)
A. Veeser
FE tree approximation
Tree approximation
Abstract setting
An instance-optimal algorithm
No instance optimality – 1
Consider approximation with piecewise constants in L2 (0, 1).
Let H be the Haar function given by

1

+1, x ∈ [0, 2 ],
H(x) := −1, x ∈ [ 12 , 1],


0,
otherwise,
and, for any I = 2−m [k , k + 1] with m ∈ N0 and k ∈ {0, 2m − 1},
HI (x) := 2m/2 H 2m (x − k )
so that
Z
1
2
Z
|HI | = 1,
0
HI = 0.
I
A. Veeser
FE tree approximation
Tree approximation
Abstract setting
An instance-optimal algorithm
No instance optimality – 2
Given M, m ∈ N and ∈ (0, 12 ), let v ∈ L2 (0, 1) be zero except
for
√
v|[1−2−M ,1] = H[1−2−M ,1] ,
v|I = 1 − HI
for all I = 2−m [k , k + 1] with k = 0, . . . , 2m−1 − 1.
For M > 2m−1 , we have
E(TN )
= 2m−1 (1 − )
E(Tn∗ )
for
n
2m + 1
= m−1
N
2
+M
where Tn∗ has the leaves 2m+1 [k , k + 1], k = 0, . . . , 2m − 1 and
[ 12 , 1].
A. Veeser
FE tree approximation
Tree approximation
Abstract setting
An instance-optimal algorithm
Modified error functionals
Define ˜ : T → R+
0 by
˜(Ω) = (Ω)
and, if K ∗ is the parent of K ,
1
1
1
=
+
˜(K )
(K ) ˜(K ∗ )
with obvious conventions.
A. Veeser
FE tree approximation
Tree approximation
Abstract setting
An instance-optimal algorithm
Instance optimality with modified functionals
Then the greedy strategy on ˜ instead of yields, for each
n ≤ N,
N
E(MTn ).
E(T̃N ) ≤
N −n+1
The number of operations is O(N), apart from computing and
sorting.
Sorting can be avoided with the help of dyadic bins, at the cost
of a multiplicative factor 2.
A. Veeser
FE tree approximation
Tree approximation
Abstract setting
An instance-optimal algorithm
Bounding E(T̃N ) in terms of t̃N – 1
Given K ∈ T , denote by A(K ) its ancestors.
For any K ∈ T , an induction yields
1
1
=
+
˜(K )
(K )
X
K 0 ∈A(K )
1
(K 0 )
(?)
In particular, the modified error functionals ˜ reduce even if the
original ones do not.
A. Veeser
FE tree approximation
Tree approximation
Abstract setting
An instance-optimal algorithm
Bounding E(T̃N ) in terms of t̃N
Multiplying (?) with ˜(K )(K ) gives

X
X
E(T̃N ) ≤
˜(K ) 1 +
K 0 ∈A(K )
K ∈L(TN )

(K ) 
(K 0 )

X
≤ tN #L(TN ) +
X
K 0 ∈TN \L(TN ) K ∈L(TN ):K ⊂K 0

(K ) 
(K 0 )
≤ tN #TN ≤ 2N tN
where in the last but one step we have used subadditivity.
A. Veeser
FE tree approximation
Tree approximation
Abstract setting
An instance-optimal algorithm
Bounding t̃N in terms of E(MTn ) – 1
Let T be a subtree with root K ∗ ∈ T and such that ˜(K ) ≥ tN for
all nodes K ∈ T . Then
(K ∗ ) ≥ #T tN .
by induction. Key step: if K is the parent of K1 and K2 , then
multiplying
(K1 ) + (K2 )
(K1 ) + (K2 ) ≥ tN k1 + k2 +
˜(K )
with
(K )
(K1 )+(K2 )
≥ 1 yields
(K )
(K )
(K )
(K ) ≥ tN k1 + k2 +
+
= tN k1 + k2 +
˜(K )
(K ) ˜(K 0 )
where = is only possible if K has a parent, say K 0 .
A. Veeser
FE tree approximation
Tree approximation
Abstract setting
An instance-optimal algorithm
Bounding t̃N in terms of E(MTn ) – 2
Let T ∗ ∈ MTn such that E(T ∗ ) = E(MTn ).
For any K ∗ ∈ L(T ∗ ), consider
TK ∗ = {K ∈ TN \ L(TN ) | K ⊂ K ∗ }
and derive
E(T ∗ ) =
X
(K ∗ ) ≥ tN
K ∗ ∈L(T ∗ )
X
#TK ∗ ≥ tN (N − n).
K ∗ ∈L(T ∗ )
For the claimed constant, one has to reason more carefully, but
along these steps.
A. Veeser
FE tree approximation
Tree approximation
Abstract setting
An instance-optimal algorithm
PDE-motivated tree approximation?
PDE-motivated approximation settings differ from the preceding
examples by
‘Natural’ error norms involve derivatives, e.g.,
k∇v kL2 (Ω)
Often approximants have additional structure, reflecting in
their boundary values,
their global regularity and
the underlying meshes.
A. Veeser
FE tree approximation