Basic notions in constructive 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
Basic notions in constructive 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
Basic notions in constructive approximation
Goals and problems
Examples: some norms and piecewise polynomial spaces
Outline
1
Basic notions in constructive approximation
Goals and problems
Examples: some norms and piecewise polynomial
spaces
A. Veeser
FE tree approximation
Basic notions in constructive approximation
Goals and problems
Examples: some norms and piecewise polynomial spaces
Introduction
Replace a complicated function by a simple one, striving for a
optimal trade-off.
An example: approximate
a continuous function
with piecewise constants over intervals of same length
in the maximum norm
A. Veeser
FE tree approximation
Basic notions in constructive approximation
Goals and problems
Examples: some norms and piecewise polynomial spaces
Applications
Such replacements are used in, e.g.,
numerical integration
lossy compression of an acoustic signal, image
numerical solution of PDEs
...
A. Veeser
FE tree approximation
Basic notions in constructive approximation
Goals and problems
Examples: some norms and piecewise polynomial spaces
Formalization – 1
V : ‘target functions’, linear space
(Sn )n : with Sn ⊂ Sn+1 , the functions in each Sn are
determined by n parameters,
k·k: norm on all V − Sn
such that
∀v ∈ V , > 0 ∃n ∈ N, s ∈ Sn
kv − sk ≤ If all Sn are linear spaces, then
linear approximation, otherwise nonlinear approximation.
A. Veeser
FE tree approximation
Basic notions in constructive approximation
Goals and problems
Examples: some norms and piecewise polynomial spaces
Formalization – 2
For any algorithm
An : V → Sn ,
N ∈ N,
we have
kv − An v k ≥ E(v , Sn ) := inf kv − sk
s∈Sn
and
# of operations for An v ≥ n
A. Veeser
FE tree approximation
(" best error ")
Basic notions in constructive approximation
Goals and problems
Examples: some norms and piecewise polynomial spaces
The practical problem
Find an algorithm with the opposite inequalities
kv − An v k 4 E(v , Sn )
("An v is near best")
and
# of operations for An v 4 n,
up to constants.
Such algorithms are called instance optimal. Since the density
assumption yields limn→∞ E(v , Sn ) = 0, they are in particular
convergent. This brings us to . . .
A. Veeser
FE tree approximation
Basic notions in constructive approximation
Goals and problems
Examples: some norms and piecewise polynomial spaces
The theoretical problem
Which properties of v ∈ V determine the speed of
E(v , Sn ) → 0
as n → ∞?
For example: under which conditions do we have
E(v , Sn ) ≤ Cn−r
with r > 0 and C ≥ 0? What determines r and C?
A. Veeser
FE tree approximation
Basic notions in constructive approximation
Goals and problems
Examples: some norms and piecewise polynomial spaces
Approximation spaces – linear case
Introduce
Ar := {v ∈ V | |v |Ar < ∞}
with
|v |Ar := sup nr E(v , Sn ),
n∈N
which is a seminorm for linear approximation, since each
E(·, Sn ) is one.
Clearly,
|v |Ar < ∞ is decided for big n
s ≤ r =⇒ Ar ⊂ As
A. Veeser
FE tree approximation
Basic notions in constructive approximation
Goals and problems
Examples: some norms and piecewise polynomial spaces
Approximation spaces – nonlinear case
For nonlinear approximation, we assume that, for some c ≥ 1,
Sn + Sn := {v1 + v2 | v1 , v2 ∈ Sn } ⊂ Scn
Then
|v |Ar := sup nr E(v , Sn ),
n∈N
is a quasi-seminorm, ie we have only
|v1 + v2 |Ar ≤ C (|v1 |Ar + |v2 |Ar )
with some C > 1.
A. Veeser
FE tree approximation
Basic notions in constructive approximation
Goals and problems
Examples: some norms and piecewise polynomial spaces
Rate optimality
An algorithm An : V → Sn , n ∈ N, is r -rate-optimal whenever
there is C 0 ≥ 0 such that
E(v , Sn ) ≤ Cn−r =⇒ kv − An v k ≤ C 0 Cn−r
(and # of operations for An 4 n)
Notice:
instance optimality implies rate optimality, but not vice
versa.
A. Veeser
FE tree approximation
Goals and problems
Examples: some norms and piecewise polynomial spaces
Basic notions in constructive approximation
The role of C 0
Rate optimality is more than
E(v , Sn ) = O(n−r ) =⇒ kv − An v k = O(n−r ) :
0
10
−1
local error in H1
10
1
2
−2
10
θ=0.000
θ=0.125
θ=0.250
θ=0.375
θ=0.500
−3
10
1
10
2
10
A. Veeser
3
10
DOFs
4
10
FE tree approximation
5
10
Basic notions in constructive approximation
Goals and problems
Examples: some norms and piecewise polynomial spaces
Outline
1
Basic notions in constructive approximation
Goals and problems
Examples: some norms and piecewise polynomial
spaces
A. Veeser
FE tree approximation
Basic notions in constructive approximation
Goals and problems
Examples: some norms and piecewise polynomial spaces
(Quasi-)norms
Given Ω ⊂ Rd and p ∈ (0, ∞], we define
Z
1/p


p
|v |
kv kLp (Ω) :=
Ω

sup |v |
Ω
p < ∞,
p = ∞,
1
which is a quasi-norm with constant max{1, 2 p
−1
}.
However, if p ∈ (0, 1), then
kv1 + v2 kpLp (Ω) ≤ kv1 kpLp (Ω) + kv2 kpLp (Ω)
often helps.
A. Veeser
FE tree approximation
Basic notions in constructive approximation
Goals and problems
Examples: some norms and piecewise polynomial spaces
One-dimensional meshes
Let Ω ⊂ R be an interval. A mesh M of Ω, is a partition
{K }K ∈M into intervals such that
Ω = ∪K ∈M K
and K̊ ∩ K̊ 0 = ∅ whenever K 6= K 0
We denote by M(Ω) the set of all meshes of Ω.
M is a uniform mesh whenever
|K | = hM :=
A. Veeser
|Ω|
.
#M
FE tree approximation
Basic notions in constructive approximation
Goals and problems
Examples: some norms and piecewise polynomial spaces
Linear approximation with piecewise polynomials
Fix a maximal polynomial degree ` ∈ N and set
P` (Mn ) := {v | ∀K ∈ Mn v|K ∈ P` }.
Let Mn be the equidistant mesh of Ω with n intervals; doubling
n halves the meshsize.
Consider the linear spaces
Sn = P` (Mn ),
A. Veeser
n ∈ N.
FE tree approximation
Basic notions in constructive approximation
Goals and problems
Examples: some norms and piecewise polynomial spaces
Nonlinear approximation with free breakpoints
Given n ∈ N, set
Mn (Ω) := {M ∈ M(Ω) | #M ≤ n}
and consider
Sn = P` Mn (Ω) :=
[
P` (M).
M∈Mn (Ω)
Observe
the functions in Sn can be described with the help of O(n)
parameters,
we have
Sn + Sn ⊂ S2n
A. Veeser
FE tree approximation
Basic notions in constructive approximation
Goals and problems
Examples: some norms and piecewise polynomial spaces
Nonlinear approximation with bisection
A bisection of an interval [a, b] is its replacement with the two
subintervals [a, m], [m, b] where m = 12 (a + b) is the midpoint.
Given n ∈ N and starting form the initial mesh {Ω}, set
Mbisect
(Ω) := M ∈ M(Ω) | M is obtained by n − 1 bisections
n
and consider
`
Sn = P` Mbisect
(Ω) := ∪M∈Mbisect
n
(Ω) P (M).
n
Also here
Sn + Sn ⊂ S2n .
A. Veeser
FE tree approximation
Basic notions in constructive approximation
Goals and problems
Examples: some norms and piecewise polynomial spaces
Comparison – uniform versus free breakpoints
Best errors in the maximum norm for various functions:
functions on (0, 1) E ·, P` (Mn )
E ·, P` (Mn )
sin(2πkx)
= 1 for n ≤ k
= 1 for n ≤ k
x ρ, ρ ≥ 1
≤ 2ρ n−1
≤ 12 n−1
x ρ , ρ ∈ (0, 1)
≤ 12 n−ρ
≤ 12 n−1
Notice
they may be no difference,
the bigger n, the bigger may be the difference
.
A. Veeser
FE tree approximation
Basic notions in constructive approximation
Goals and problems
Examples: some norms and piecewise polynomial spaces
Comparison – bisection
For n = 2k , we have
Mn ∈ Mbisect
(Ω) ⊂ Mn (Ω)
n
and so
P` (Mn ) ⊂ P` Mbisect
(Ω) ⊂ P` Mn (Ω)
n
In Mbisect
(Ω), the positions of the breakpoints are chosen from
n
a priori fixed set, which becomes dense as n → ∞.
A. Veeser
FE tree approximation