1. No category

Reduced Basis Method - How to Best Sample the
Solution Manifold?
Wolfgang Dahmen
Institut fur
¨ Geometrie und Praktische Mathematik
RWTH Aachen
joint work with: Christian Plesken, Gerrit Welper
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
1 / 29
Outline
Outline
1
Introduction
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
2 / 29
Outline
Outline
1
Introduction
2
The Greedy Paradigm - Rate Optimality
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
2 / 29
Outline
Outline
1
Introduction
2
The Greedy Paradigm - Rate Optimality
3
Finding Good Projections - Stable Variational Formulations
Renormation
A Saddle Point Formulation
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
2 / 29
Outline
Outline
1
Introduction
2
The Greedy Paradigm - Rate Optimality
3
Finding Good Projections - Stable Variational Formulations
Renormation
A Saddle Point Formulation
4
Double Greedy Scheme
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
2 / 29
Outline
Outline
1
Introduction
2
The Greedy Paradigm - Rate Optimality
3
Finding Good Projections - Stable Variational Formulations
Renormation
A Saddle Point Formulation
4
Double Greedy Scheme
5
Numerical Experiments
Convection-Diffusion Equations
Transport Equation
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
2 / 29
Outline
Outline
1
Introduction
2
The Greedy Paradigm - Rate Optimality
3
Finding Good Projections - Stable Variational Formulations
Renormation
A Saddle Point Formulation
4
Double Greedy Scheme
5
Numerical Experiments
Convection-Diffusion Equations
Transport Equation
6
Summary and Outlook
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
2 / 29
Introduction
Background...Turbulent Drag Reduction - FOR 1779
Turbulence modeling
Central methodological topic in AICES
Aachen Institute for Advanced Study in Computational Engineering Science
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
3 / 29
Introduction
Background...Turbulent Drag Reduction - FOR 1779
Turbulence modeling
Geometric homogenization
Central methodological topic in AICES
Aachen Institute for Advanced Study in Computational Engineering Science
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
3 / 29
Introduction
Background...Turbulent Drag Reduction - FOR 1779
Turbulence modeling






Geometric homogenization 
⇒
Navier-Stokes equations in
“homogenized” domain
Central methodological topic in AICES
Aachen Institute for Advanced Study in Computational Engineering Science
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
3 / 29
Introduction
Background...Turbulent Drag Reduction - FOR 1779
Turbulence modeling






Geometric homogenization 
⇒
Navier-Stokes equations in
“homogenized” domain
Robin-type effective boundary
conditions depending on
frequency, amplitude,...= µ
Central methodological topic in AICES
Aachen Institute for Advanced Study in Computational Engineering Science
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
3 / 29
Introduction
Background...Turbulent Drag Reduction - FOR 1779
Turbulence modeling






Geometric homogenization 
⇒
Navier-Stokes equations in
“homogenized” domain
Robin-type effective boundary
conditions depending on
frequency, amplitude,...= µ
Parameter dependent family of PDEs: F (u, µ) = 0, µ ∈ P,
u(x, µ)
Central methodological topic in AICES
Aachen Institute for Advanced Study in Computational Engineering Science
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
3 / 29
Introduction
Background...Turbulent Drag Reduction - FOR 1779
Turbulence modeling






Geometric homogenization 
⇒
Navier-Stokes equations in
“homogenized” domain
Robin-type effective boundary
conditions depending on
frequency, amplitude,...= µ
Parameter dependent family of PDEs: F (u, µ) = 0, µ ∈ P,
online parameter optimization: I(µ)
u(x, µ)
→ opt
Central methodological topic in AICES
Aachen Institute for Advanced Study in Computational Engineering Science
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
3 / 29
Introduction
Background...Turbulent Drag Reduction - FOR 1779
Turbulence modeling






Geometric homogenization 
⇒
Navier-Stokes equations in
“homogenized” domain
Robin-type effective boundary
conditions depending on
frequency, amplitude,...= µ
Parameter dependent family of PDEs: F (u, µ) = 0, µ ∈ P,
u(x, µ)
online parameter optimization: I(µ) = `(u(µ)) → opt
is practically infeasible
Central methodological topic in AICES
Aachen Institute for Advanced Study in Computational Engineering Science
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
3 / 29
Introduction
Background...Turbulent Drag Reduction - FOR 1779




Turbulence modeling


Geometric homogenization 
⇒
Navier-Stokes equations in
“homogenized” domain
Robin-type effective boundary
conditions depending on
frequency, amplitude,...= µ
Parameter dependent family of PDEs: F (u, µ) = 0, µ ∈ P,
u(x, µ)
online parameter optimization: I(µ) = `(u(µ)) → opt
is practically infeasible
Model reduction needed...
Central methodological topic in AICES
Aachen Institute for Advanced Study in Computational Engineering Science
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
3 / 29
Introduction
Background...Turbulent Drag Reduction - FOR 1779
Turbulence modeling






Geometric homogenization 
Navier-Stokes equations in
“homogenized” domain
⇒
Robin-type effective boundary
conditions depending on
frequency, amplitude,...= µ
Parameter dependent family of PDEs: F (u, µ) = 0, µ ∈ P,
u(x, µ)
online parameter optimization: I(µ) = `(u(µ)) → opt
is practically infeasible
Model reduction needed... u(x, µ) ≈
Pm
j=1
cj (µ)u(x, µj )
RBM
Central methodological topic in AICES
Aachen Institute for Advanced Study in Computational Engineering Science
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
3 / 29
Introduction
Where do we Stand?
RBMs well understood for elliptic problems
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
4 / 29
Introduction
Where do we Stand?
RBMs well understood for elliptic problems
Classical techniques do not work well for indefinite, singularly perturbed,
unsymmetric ... transport dominated problems
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
4 / 29
Introduction
Where do we Stand?
RBMs well understood for elliptic problems
Classical techniques do not work well for indefinite, singularly perturbed,
unsymmetric ... transport dominated problems
Guiding examples:
Convection-diffusion-reaction:
−∆u + b(µ) · ∇u + cu = f
W. Dahmen (RWTH Aachen)
in Ω, u |∂Ω = 0, µ ∈ P
How to Best Sample a Solution Manifold?
July 2, 2013
4 / 29
Introduction
Where do we Stand?
RBMs well understood for elliptic problems
Classical techniques do not work well for indefinite, singularly perturbed,
unsymmetric ... transport dominated problems
Guiding examples:
Convection-diffusion-reaction:
−∆u + b(µ) · ∇u + cu = f
in Ω, u |∂Ω = 0, µ ∈ P
Transport/kinetic models:
b(µ) · ∇u(µ) + cu(µ) = f
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
in Ω, u |Γ− =0
July 2, 2013
4 / 29
Introduction
Where do we Stand?
RBMs well understood for elliptic problems
Classical techniques do not work well for indefinite, singularly perturbed,
unsymmetric ... transport dominated problems
Guiding examples:
Convection-diffusion-reaction:
−∆u + b(µ) · ∇u + cu = f
in Ω, u |∂Ω = 0, µ ∈ P
Transport/kinetic models:
b(µ) · ∇u(µ) + cu(µ) = f +
W. Dahmen (RWTH Aachen)
Z
K (µ, µ0 )u(µ0 )dµ0
P
How to Best Sample a Solution Manifold?
in Ω, u |Γ− =0
July 2, 2013
4 / 29
Introduction
Where do we Stand?
RBMs well understood for elliptic problems
Classical techniques do not work well for indefinite, singularly perturbed,
unsymmetric ... transport dominated problems
Guiding examples:
Convection-diffusion-reaction:
−∆u + b(µ) · ∇u + cu = f
in Ω, u |∂Ω = 0, µ ∈ P
Transport/kinetic models:
b(µ) · ∇u(µ) + cu(µ) = f +
Z
K (µ, µ0 )u(µ0 )dµ0
P
in Ω, u |Γ− =0
Nonlinear conservation laws:
∂t u + ∇ · f (u, µ) = 0
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
4 / 29
Introduction
Challenges...
Lack of stabilizing viscosity - pure Galerkin does not work!
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
5 / 29
Introduction
Challenges...
Lack of stabilizing viscosity - pure Galerkin does not work!
Non-smooth dependence on µ ∈ P
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
5 / 29
Introduction
Challenges...
Lack of stabilizing viscosity - pure Galerkin does not work!
Non-smooth dependence on µ ∈ P
Global coupling - collision operators
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
5 / 29
Introduction
Challenges...
Lack of stabilizing viscosity - pure Galerkin does not work!
Non-smooth dependence on µ ∈ P
Global coupling - collision operators
High-dimensional parameter space P
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
5 / 29
Introduction
Challenges...
Lack of stabilizing viscosity - pure Galerkin does not work!
Non-smooth dependence on µ ∈ P
Global coupling - collision operators
High-dimensional parameter space P
Nonlinearities
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
5 / 29
Introduction
Challenges...
Lack of stabilizing viscosity - pure Galerkin does not work!
Non-smooth dependence on µ ∈ P
Global coupling - collision operators
High-dimensional parameter space P
Nonlinearities
Here: Best-sampling for non-elliptic problems
find smallest possible reduced models !
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
5 / 29
Introduction
Challenges...
Lack of stabilizing viscosity - pure Galerkin does not work!
Non-smooth dependence on µ ∈ P
Global coupling - collision operators
High-dimensional parameter space P
Nonlinearities
Here: Best-sampling for non-elliptic problems
find smallest possible reduced models !
...in a rigorous sense!
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
5 / 29
Introduction
General Philosophy
Offline mode: precompute a “reduced basis” consisting of suitable
solution samples
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
6 / 29
Introduction
General Philosophy
Offline mode: precompute a “reduced basis” consisting of suitable
solution samples
heavy computational cost in the “truth space”
X
M
u(x, µj )
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
6 / 29
Introduction
General Philosophy
Offline mode: precompute a “reduced basis” consisting of suitable
solution samples
heavy computational cost in the “truth space”
Online mode: for each parameter query solve only a small
reduced system
X
M
Pn
j=1
W. Dahmen (RWTH Aachen)
cj (µ)u(x, µj )
How to Best Sample a Solution Manifold?
July 2, 2013
6 / 29
Introduction
General Philosophy
Offline mode: precompute a “reduced basis” consisting of suitable
solution samples
heavy computational cost in the “truth space”
Online mode: for each parameter query solve only a small
reduced system - with certified accuracy
X
M
Pn
j=1
W. Dahmen (RWTH Aachen)
cj (µ)u(x, µj ) ≈ u(x, µ)
How to Best Sample a Solution Manifold?
July 2, 2013
6 / 29
Introduction
A Guiding Model Problem - variational formulation
Convection-diffusion equation
−div(∇u(x)) + b(µ) · ∇u(x) + cu(x) = f (x),
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
in Ω,
u = 0 on ∂Ω,
July 2, 2013
7 / 29
Introduction
A Guiding Model Problem - variational formulation
Convection-diffusion equation
−div(∇u(x)) + b(µ) · ∇u(x) + cu(x) = f (x),
Find u ∈ H01 (Ω)
Z
Z
fv dx ,
Ω
Ω
W. Dahmen (RWTH Aachen)
u = 0 on ∂Ω,
such that
∇u · ∇v + (b(µ) · ∇u)v + cuv dx =
|
in Ω,
{z
}
v ∈ H01 (Ω)
| {z }
| {z }
How to Best Sample a Solution Manifold?
July 2, 2013
7 / 29
Introduction
A Guiding Model Problem - variational formulation
Convection-diffusion equation
−div(∇u(x)) + b(µ) · ∇u(x) + cu(x) = f (x),
in Ω,
u = 0 on ∂Ω,
Find u ∈ H01 (Ω) =: X such that
Z
∇u · ∇v + (b(µ) · ∇u)v + cuv dx =
W. Dahmen (RWTH Aachen)
fv dx ,
Ω
Ω
|
Z
{z
=:bµ (u,v )
}
v ∈ H01 (Ω)
| {z }
=:Y
| {z }
=:hf ,v i
How to Best Sample a Solution Manifold?
July 2, 2013
7 / 29
Introduction
A Guiding Model Problem - variational formulation
Convection-diffusion equation
−div(∇u(x)) + b(µ) · ∇u(x) + cu(x) = f (x),
in Ω,
u = 0 on ∂Ω,
Find u ∈ H01 (Ω) =: X such that
Z
∇u · ∇v + (b(µ) · ∇u)v + cuv dx =
Z
fv dx ,
Ω
Ω
|
{z
=:bµ (u,v )
}
v ∈ H01 (Ω)
| {z }
=:Y
| {z }
=:hf ,v i
hB µ u(µ), v i := bµ (u(µ), v )
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
7 / 29
Introduction
A Guiding Model Problem - variational formulation
Convection-diffusion equation
−div(∇u(x)) + b(µ) · ∇u(x) + cu(x) = f (x),
in Ω,
u = 0 on ∂Ω,
Find u ∈ H01 (Ω) =: X such that
Z
∇u · ∇v + (b(µ) · ∇u)v + cuv dx =
Z
fv dx ,
Ω
Ω
|
{z
=:bµ (u,v )
hB µ u(µ), v i := bµ (u(µ), v )
W. Dahmen (RWTH Aachen)
}
v ∈ H01 (Ω)
| {z }
=:Y
| {z }
=:hf ,v i
B µ u(µ) = f ,
How to Best Sample a Solution Manifold?
Bµ : X → Y 0
July 2, 2013
7 / 29
Introduction
A Guiding Model Problem - variational formulation
Convection-diffusion equation
−div(∇u(x)) + b(µ) · ∇u(x) + cu(x) = f (x),
in Ω,
u = 0 on ∂Ω,
Find u ∈ H01 (Ω) =: X such that
Z
∇u · ∇v + (b(µ) · ∇u)v + cuv dx =
Z
fv dx ,
Ω
Ω
|
{z
=:bµ (u,v )
hB µ u(µ), v i := bµ (u(µ), v )
}
v ∈ H01 (Ω)
| {z }
=:Y
| {z }
=:hf ,v i
B µ u(µ) = f ,
Bµ : X → Y 0
M := {u(µ) = B −1
µ f : µ ∈ P} ⊂ X
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
7 / 29
The Greedy Paradigm - Rate Optimality
The Greedy Paradigm...
Y. Maday, A. Patera,...
Given f ∈ Y 0 find u ∈ X s.t. for µ ∈ P
bµ (u(µ), v ) = hf , v i,
W. Dahmen (RWTH Aachen)
v ∈ Y,
How to Best Sample a Solution Manifold?
July 2, 2013
8 / 29
The Greedy Paradigm - Rate Optimality
The Greedy Paradigm...
Y. Maday, A. Patera,...
Given f ∈ Y 0 find u ∈ X s.t. for µ ∈ P
hB µ u(µ), v i := bµ (u(µ), v ) = hf , v i,
W. Dahmen (RWTH Aachen)
v ∈ Y , ⇔ B µ u(µ) = f ,
How to Best Sample a Solution Manifold?
Bµ : X → Y 0
July 2, 2013
8 / 29
The Greedy Paradigm - Rate Optimality
The Greedy Paradigm...
Y. Maday, A. Patera,...
Given f ∈ Y 0 find u ∈ X s.t. for µ ∈ P
hB µ u(µ), v i := bµ (u(µ), v ) = hf , v i,
Solution manifold:
v ∈ Y , ⇔ B µ u(µ) = f ,
Bµ : X → Y 0
−1
M := {u(µ) := Bµ
f : µ ∈ P}
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
8 / 29
The Greedy Paradigm - Rate Optimality
The Greedy Paradigm...
Y. Maday, A. Patera,...
Given f ∈ Y 0 find u ∈ X s.t. for µ ∈ P
hB µ u(µ), v i := bµ (u(µ), v ) = hf , v i,
Solution manifold:
v ∈ Y , ⇔ B µ u(µ) = f ,
Bµ : X → Y 0
−1
M := {u(µ) := Bµ
f : µ ∈ P}
Objective: construct Xn ⊂ X , n = n() as small as possible, s.t.
inf ku(µ) − v kX
v ∈Xn
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
8 / 29
The Greedy Paradigm - Rate Optimality
The Greedy Paradigm...
Y. Maday, A. Patera,...
Given f ∈ Y 0 find u ∈ X s.t. for µ ∈ P
hB µ u(µ), v i := bµ (u(µ), v ) = hf , v i,
Solution manifold:
v ∈ Y , ⇔ B µ u(µ) = f ,
Bµ : X → Y 0
−1
M := {u(µ) := Bµ
f : µ ∈ P}
Objective: construct Xn ⊂ X , n = n() as small as possible, s.t.
max inf ku(µ) − v kX =: σn (M)X
µ∈P v ∈Xn
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
8 / 29
The Greedy Paradigm - Rate Optimality
The Greedy Paradigm...
Y. Maday, A. Patera,...
Given f ∈ Y 0 find u ∈ X s.t. for µ ∈ P
hB µ u(µ), v i := bµ (u(µ), v ) = hf , v i,
Solution manifold:
v ∈ Y , ⇔ B µ u(µ) = f ,
Bµ : X → Y 0
−1
M := {u(µ) := Bµ
f : µ ∈ P}
Objective: construct Xn ⊂ X , n = n() as small as possible, s.t.
maxdist (M, Xn )X := max inf ku(µ) − v kX =: σn (M)X
µ∈P v ∈Xn
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
8 / 29
The Greedy Paradigm - Rate Optimality
The Greedy Paradigm...
Y. Maday, A. Patera,...
Given f ∈ Y 0 find u ∈ X s.t. for µ ∈ P
hB µ u(µ), v i := bµ (u(µ), v ) = hf , v i,
Solution manifold:
v ∈ Y , ⇔ B µ u(µ) = f ,
Bµ : X → Y 0
−1
M := {u(µ) := Bµ
f : µ ∈ P}
Objective: construct Xn ⊂ X , n = n() as small as possible, s.t.
!
maxdist (M, Xn )X := max inf ku(µ) − v kX =: σn (M)X ≤ µ∈P v ∈Xn
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
8 / 29
The Greedy Paradigm - Rate Optimality
The Greedy Paradigm...
Y. Maday, A. Patera,...
Given f ∈ Y 0 find u ∈ X s.t. for µ ∈ P
hB µ u(µ), v i := bµ (u(µ), v ) = hf , v i,
Solution manifold:
v ∈ Y , ⇔ B µ u(µ) = f ,
Bµ : X → Y 0
−1
M := {u(µ) := Bµ
f : µ ∈ P}
Objective: construct Xn ⊂ X , n = n() as small as possible, s.t.
!
maxdist (M, Xn )X := max inf ku(µ) − v kX =: σn (M)X ≤ µ∈P v ∈Xn
Surrogate: suppose one has
inf ku(µ) − v kX ≤ Rn (µ),
v ∈Xn
W. Dahmen (RWTH Aachen)
µ ∈ P,
How to Best Sample a Solution Manifold?
July 2, 2013
8 / 29
The Greedy Paradigm - Rate Optimality
The Greedy Paradigm...
Y. Maday, A. Patera,...
Given f ∈ Y 0 find u ∈ X s.t. for µ ∈ P
hB µ u(µ), v i := bµ (u(µ), v ) = hf , v i,
Solution manifold:
v ∈ Y , ⇔ B µ u(µ) = f ,
Bµ : X → Y 0
−1
M := {u(µ) := Bµ
f : µ ∈ P}
Objective: construct Xn ⊂ X , n = n() as small as possible, s.t.
!
maxdist (M, Xn )X := max inf ku(µ) − v kX =: σn (M)X ≤ µ∈P v ∈Xn
Surrogate: suppose one has
inf ku(µ) − v kX ≤ Rn (µ),
v ∈Xn
µ ∈ P,
Greedy Algorithm
X0 := {0};
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
8 / 29
The Greedy Paradigm - Rate Optimality
The Greedy Paradigm...
Y. Maday, A. Patera,...
Given f ∈ Y 0 find u ∈ X s.t. for µ ∈ P
hB µ u(µ), v i := bµ (u(µ), v ) = hf , v i,
Solution manifold:
v ∈ Y , ⇔ B µ u(µ) = f ,
Bµ : X → Y 0
−1
M := {u(µ) := Bµ
f : µ ∈ P}
Objective: construct Xn ⊂ X , n = n() as small as possible, s.t.
!
maxdist (M, Xn )X := max inf ku(µ) − v kX =: σn (M)X ≤ µ∈P v ∈Xn
Surrogate: suppose one has
inf ku(µ) − v kX ≤ Rn (µ),
v ∈Xn
µ ∈ P,
Greedy Algorithm
X0 := {0};
for n = 1, 2, . . ., given Xn−1
µn := argmaxµ∈P Rn−1 (µ),
W. Dahmen (RWTH Aachen)
Xn := span {Xn−1 , {u(µn )}}
How to Best Sample a Solution Manifold?
July 2, 2013
8 / 29
The Greedy Paradigm - Rate Optimality
How does the greedy search perform?...
X
W. Dahmen (RWTH Aachen)
M
How to Best Sample a Solution Manifold?
July 2, 2013
9 / 29
The Greedy Paradigm - Rate Optimality
Rate Optimality
Bench mark - Kolmogorov Widths:
W. Dahmen (RWTH Aachen)
(Buffa, Maday, Patera, Prud’homme, Turinici...)
How to Best Sample a Solution Manifold?
July 2, 2013
10 / 29
The Greedy Paradigm - Rate Optimality
Rate Optimality
Bench mark - Kolmogorov Widths:
(Buffa, Maday, Patera, Prud’homme, Turinici...)
dist (u, Vn )X
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
10 / 29
The Greedy Paradigm - Rate Optimality
Rate Optimality
Bench mark - Kolmogorov Widths:
(Buffa, Maday, Patera, Prud’homme, Turinici...)
sup dist (u, Vn )X
u∈M
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
10 / 29
The Greedy Paradigm - Rate Optimality
Rate Optimality
Bench mark - Kolmogorov Widths:
dn (M)X :=
W. Dahmen (RWTH Aachen)
inf
(Buffa, Maday, Patera, Prud’homme, Turinici...)
sup dist (u, Vn )X
dim(Vn )=n u∈M
How to Best Sample a Solution Manifold?
July 2, 2013
10 / 29
The Greedy Paradigm - Rate Optimality
Rate Optimality
Bench mark - Kolmogorov Widths:
dn (M)X :=
W. Dahmen (RWTH Aachen)
inf
(Buffa, Maday, Patera, Prud’homme, Turinici...)
sup dist (u, Vn )X ≤ σn (M)X
dim(Vn )=n u∈M
How to Best Sample a Solution Manifold?
July 2, 2013
10 / 29
The Greedy Paradigm - Rate Optimality
Rate Optimality
Bench mark - Kolmogorov Widths:
dn (M)X :=
inf
(Buffa, Maday, Patera, Prud’homme, Turinici...)
sup dist (u, Vn )X ≤ σn (M)X
dim(Vn )=n u∈M
THEOREM 1: Binev/Cohen/Dahmen/DeVore/Petrova/Wojtaszczyk
If the surrogate is tight, i.e., ∃ cS independent of µ ∈ P, s.t.
cS Rn (µ) ≤ inf ku(µ) − wkX ≤ Rn (µ),
w∈Xn
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
10 / 29
The Greedy Paradigm - Rate Optimality
Rate Optimality
Bench mark - Kolmogorov Widths:
dn (M)X :=
inf
(Buffa, Maday, Patera, Prud’homme, Turinici...)
sup dist (u, Vn )X ≤ σn (M)X
dim(Vn )=n u∈M
THEOREM 1: Binev/Cohen/Dahmen/DeVore/Petrova/Wojtaszczyk
If the surrogate is tight, i.e., ∃ cS independent of µ ∈ P, s.t.
cS Rn (µ) ≤ inf ku(µ) − wkX ≤ Rn (µ),
w∈Xn
then

 O(n−α ), n ∈ N,
dn (M)X =
 O(e−cnα ), n ∈ N,
W. Dahmen (RWTH Aachen)
⇒

 O(n−α ), n ∈ N,
σn (M)X =
 O(e−c˜nα ), n ∈ N,
How to Best Sample a Solution Manifold?
July 2, 2013
10 / 29
The Greedy Paradigm - Rate Optimality
Rate Optimality
Bench mark - Kolmogorov Widths:
dn (M)X :=
inf
(Buffa, Maday, Patera, Prud’homme, Turinici...)
sup dist (u, Vn )X ≤ σn (M)X
dim(Vn )=n u∈M
THEOREM 1: Binev/Cohen/Dahmen/DeVore/Petrova/Wojtaszczyk
If the surrogate is tight, i.e., ∃ cS independent of µ ∈ P, s.t.
cS Rn (µ) ≤ inf ku(µ) − wkX ≤ Rn (µ),
w∈Xn
κ(Rn ) ≤ 1/cS
then

 O(n−α ), n ∈ N,
dn (M)X =
 O(e−cnα ), n ∈ N,
⇒

 O(n−α ), n ∈ N,
σn (M)X =
 O(e−c˜nα ), n ∈ N,
C = C(α, c, κ(Rn ))
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
10 / 29
The Greedy Paradigm - Rate Optimality
Rate Optimality
Bench mark - Kolmogorov Widths:
dn (M)X :=
inf
(Buffa, Maday, Patera, Prud’homme, Turinici...)
sup dist (u, Vn )X ≤ σn (M)X
dim(Vn )=n u∈M
THEOREM 1: Binev/Cohen/Dahmen/DeVore/Petrova/Wojtaszczyk
If the surrogate is tight, i.e., ∃ cS independent of µ ∈ P, s.t.
cS Rn (µ) ≤ inf ku(µ) − wkX ≤ Rn (µ),
w∈Xn
κ(Rn ) ≤ 1/cS
then

 O(n−α ), n ∈ N,
dn (M)X =
 O(e−cnα ), n ∈ N,
⇒

 O(n−α ), n ∈ N,
σn (M)X =
 O(e−c˜nα ), n ∈ N,
C = C(α, c, κ(Rn ))
Central goal: find well conditioned feasible surrogates ...
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
10 / 29
The Greedy Paradigm - Rate Optimality
The only Conceivable Way...
Surrogate must be based on residuals
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
11 / 29
The Greedy Paradigm - Rate Optimality
The only Conceivable Way...
Surrogate must be based on residuals
One must have
surrogate
W. Dahmen (RWTH Aachen)
≈
best approximation error
How to Best Sample a Solution Manifold?
July 2, 2013
11 / 29
The Greedy Paradigm - Rate Optimality
The only Conceivable Way...
Surrogate must be based on residuals
One must have
f − Bµ un∗ (µ)
W. Dahmen (RWTH Aachen)
≈
u(µ) − un∗ (µ)
How to Best Sample a Solution Manifold?
July 2, 2013
11 / 29
The Greedy Paradigm - Rate Optimality
The only Conceivable Way...
Surrogate must be based on residuals
One must have
kf − Bµ un∗ (µ)kY 0
W. Dahmen (RWTH Aachen)
≈
ku(µ) − un∗ (µ)kX
How to Best Sample a Solution Manifold?
(1)
July 2, 2013
11 / 29
The Greedy Paradigm - Rate Optimality
The only Conceivable Way...
Surrogate must be based on residuals
One must have
kf − Bµ un∗ (µ)kY 0
≈
ku(µ) − un∗ (µ)kX
(1)
One must have
kbest approximation errorkX ≈ kmethod projection errorkX
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
(2)
11 / 29
The Greedy Paradigm - Rate Optimality
The only Conceivable Way...
Surrogate must be based on residuals
One must have
kf − Bµ un∗ (µ)kY 0
≈
ku(µ) − un∗ (µ)kX
(1)
One must have
kbest approximation errorkX ≈ kmethod projection errorkX
(2)
Works well for elliptic problems: X = Y = H01 (Ω)
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
11 / 29
The Greedy Paradigm - Rate Optimality
The only Conceivable Way...
Surrogate must be based on residuals
One must have
kf − Bµ un∗ (µ)kY 0
≈
ku(µ) − un∗ (µ)kX
(1)
One must have
kbest approximation errorkX ≈ kmethod projection errorkX
(2)
Works well for elliptic problems: X = Y = H01 (Ω)
Idea: find suitable (X , Y )-stable variational formulation of the problem
to ensure (3), (4)
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
11 / 29
Finding Good Projections
Renormation
Outline
1
Introduction
2
The Greedy Paradigm - Rate Optimality
3
Finding Good Projections - Stable Variational Formulations
Renormation
A Saddle Point Formulation
4
Double Greedy Scheme
5
Numerical Experiments
Convection-Diffusion Equations
Transport Equation
6
Summary and Outlook
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
12 / 29
Finding Good Projections
Renormation
Renormation...Dahmen/Huang/Schwab/Welper, Demkowicz et al., Cai et al, Manteuffel et al.....
Suppose Bµ u(µ) = f is well-posed, i.e.:
inf sup
u∈Xµ v ∈Yµ
bµ (u, v )
≥ β(µ),
kukXµ kv kYµ
W. Dahmen (RWTH Aachen)
sup sup
u∈Xµ v ∈Yµ
|bµ (u, v )|
≤ Cb (µ)
kukXµ kv kYµ
How to Best Sample a Solution Manifold?
July 2, 2013
13 / 29
Finding Good Projections
Renormation
Renormation...Dahmen/Huang/Schwab/Welper, Demkowicz et al., Cai et al, Manteuffel et al.....
Suppose Bµ u(µ) = f is well-posed, i.e.:
inf sup
u∈Xµ v ∈Yµ
bµ (u, v )
≥ β(µ),
kukXµ kv kYµ
sup sup
u∈Xµ v ∈Yµ
|bµ (u, v )|
≤ Cb (µ)
kukXµ kv kYµ
Re-define k · kXˆµ through (see also Nguyen/Patera/Rozza, Deparis, but 6=)
kukXˆµ := sup
v ∈Yµ
bµ (u, v )
kv kYµ
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
u ∈ Xµ , µ ∈ P
July 2, 2013
13 / 29
Finding Good Projections
Renormation
Renormation...Dahmen/Huang/Schwab/Welper, Demkowicz et al., Cai et al, Manteuffel et al.....
Suppose Bµ u(µ) = f is well-posed, i.e.:
inf sup
u∈Xµ v ∈Yµ
bµ (u, v )
≥ β(µ),
kukXµ kv kYµ
sup sup
u∈Xµ v ∈Yµ
|bµ (u, v )|
≤ Cb (µ)
kukXµ kv kYµ
Re-define k · kXˆµ through (see also Nguyen/Patera/Rozza, Deparis, but 6=)
kukXˆµ := sup
v ∈Yµ
bµ (u, v )
= kBµ ukYµ0 = kRY−1
Bµ ukYµ ,
µ
kv kYµ
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
u ∈ Xµ , µ ∈ P
July 2, 2013
13 / 29
Finding Good Projections
Renormation
Renormation...Dahmen/Huang/Schwab/Welper, Demkowicz et al., Cai et al, Manteuffel et al.....
Suppose Bµ u(µ) = f is well-posed, i.e.:
inf sup
u∈Xµ v ∈Yµ
bµ (u, v )
≥ β(µ),
kukXµ kv kYµ
κXµ →Yµ0 (Bµ ) ≤ Cb (µ)/β(µ)
sup sup
u∈Xµ v ∈Yµ
|bµ (u, v )|
≤ Cb (µ)
kukXµ kv kYµ
Re-define k · kXˆµ through (see also Nguyen/Patera/Rozza, Deparis, but 6=)
kukXˆµ := sup
v ∈Yµ
bµ (u, v )
= kBµ ukYµ0 = kRY−1
Bµ ukYµ ,
µ
kv kYµ
u ∈ Xµ , µ ∈ P
Perfect Condition:
sup sup
v ∈Xµ y ∈Yµ
bµ (v , y)
bµ (v , y )
= inf sup
= 1,
v ∈Xµ y∈Yµ kykYµ kv kX
ky kYµ kv kXˆµ
ˆµ
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
⇒
κXˆµ →Y 0 (Bµ ) = 1
µ
July 2, 2013
13 / 29
Finding Good Projections
Renormation
Renormation...Dahmen/Huang/Schwab/Welper, Demkowicz et al., Cai et al, Manteuffel et al.....
Suppose Bµ u(µ) = f is well-posed, i.e.:
inf sup
u∈Xµ v ∈Yµ
bµ (u, v )
≥ β(µ),
kukXµ kv kYµ
κXµ →Yµ0 (Bµ ) ≤ Cb (µ)/β(µ)
sup sup
u∈Xµ v ∈Yµ
|bµ (u, v )|
≤ Cb (µ)
kukXµ kv kYµ
Re-define k · kXˆµ through (see also Nguyen/Patera/Rozza, Deparis, but 6=)
kukXˆµ := sup
v ∈Yµ
bµ (u, v )
= kBµ ukYµ0 = kRY−1
Bµ ukYµ ,
µ
kv kYµ
u ∈ Xµ , µ ∈ P
Perfect Condition:
sup sup
v ∈Xµ y ∈Yµ
bµ (v , y)
bµ (v , y )
= inf sup
= 1,
v ∈Xµ y∈Yµ kykYµ kv kX
ky kYµ kv kXˆµ
ˆµ
⇒
κXˆµ →Y 0 (Bµ ) = 1
µ
...How to use this numerically?
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
13 / 29
Finding Good Projections
A Saddle Point Formulation
Outline
1
Introduction
2
The Greedy Paradigm - Rate Optimality
3
Finding Good Projections - Stable Variational Formulations
Renormation
A Saddle Point Formulation
4
Double Greedy Scheme
5
Numerical Experiments
Convection-Diffusion Equations
Transport Equation
6
Summary and Outlook
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
14 / 29
Finding Good Projections
A Saddle Point Formulation
Main Result
un (µ) ∈ Xn ⊂ Xµ ,
bµ (un (µ), v ) = hf , v i, v ∈ Vn ,
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
15 / 29
Finding Good Projections
A Saddle Point Formulation
Main Result
un (µ) ∈ Xn ⊂ Xµ , yn (µ) ∈ Vn ⊂ Yµ
(yn (µ) ≈ f − Bµ un (µ)) solve:
(yn (µ), v )Yµ + bµ (un (µ), v ) = hf , v i, v ∈ Vn ,
bµ (w, yn (µ))
W. Dahmen (RWTH Aachen)
=
0,
How to Best Sample a Solution Manifold?
w ∈ Xn ,
July 2, 2013
15 / 29
Finding Good Projections
A Saddle Point Formulation
Main Result
un (µ) ∈ Xn ⊂ Xµ , yn (µ) ∈ Vn ⊂ Yµ
(yn (µ) ≈ f − Bµ un (µ)) solve:
(yn (µ), v )Yµ + bµ (un (µ), v ) = hf , v i, v ∈ Vn ,
bµ (w, yn (µ))
=
0,
w ∈ Xn ,
if and only if
bµ (un (µ), v ) = hf , v i,
W. Dahmen (RWTH Aachen)
∀ v ∈ Yn := PYµ ,Vn (RY−1
Bµ (Xn ))
µ
How to Best Sample a Solution Manifold?
July 2, 2013
15 / 29
Finding Good Projections
A Saddle Point Formulation
Main Result
un (µ) ∈ Xn ⊂ Xµ , yn (µ) ∈ Vn ⊂ Yµ
(yn (µ) ≈ f − Bµ un (µ)) solve:
(yn (µ), v )Yµ + bµ (un (µ), v ) = hf , v i, v ∈ Vn ,
bµ (w, yn (µ))
=
0,
w ∈ Xn ,
if and only if
bµ (un (µ), v ) = hf , v i,
Moreover
inf sup
w∈Xn v ∈Vn
W. Dahmen (RWTH Aachen)
∀ v ∈ Yn := PYµ ,Vn (RY−1
Bµ (Xn ))
µ
p
bµ (w, v )
≥ 1 − δ2
kwkXˆµ kv kYµ
How to Best Sample a Solution Manifold?
July 2, 2013
15 / 29
Finding Good Projections
A Saddle Point Formulation
Main Result
un (µ) ∈ Xn ⊂ Xµ , yn (µ) ∈ Vn ⊂ Yµ
(yn (µ) ≈ f − Bµ un (µ)) solve:
(yn (µ), v )Yµ + bµ (un (µ), v ) = hf , v i, v ∈ Vn ,
bµ (w, yn (µ))
=
0,
w ∈ Xn ,
if and only if
bµ (un (µ), v ) = hf , v i,
Moreover
inf sup
w∈Xn v ∈Vn
∀ v ∈ Yn := PYµ ,Vn (RY−1
Bµ (Xn ))
µ
p
bµ (w, v )
≥ 1 − δ2
kwkXˆµ kv kYµ
if and only if
sup inf kRY−1
Bµ w − φkYµ ≤ δkwkXˆµ
µ
w∈Xn φ∈Vn
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
15 / 29
Finding Good Projections
A Saddle Point Formulation
Main Result
un (µ) ∈ Xn ⊂ Xµ , yn (µ) ∈ Vn ⊂ Yµ
(yn (µ) ≈ f − Bµ un (µ)) solve:
(yn (µ), v )Yµ + bµ (un (µ), v ) = hf , v i, v ∈ Vn ,
bµ (w, yn (µ))
=
0,
w ∈ Xn ,
if and only if
bµ (un (µ), v ) = hf , v i,
Moreover
inf sup
w∈Xn v ∈Vn
∀ v ∈ Yn := PYµ ,Vn (RY−1
Bµ (Xn ))
µ
p
bµ (w, v )
≥ 1 − δ2
kwkXˆµ kv kYµ
if and only if
sup inf kRY−1
Bµ w − φkYµ ≤ δkwkXˆµ
µ
w∈Xn φ∈Vn
⇒ ku(µ) − un (µ)kXˆµ + kyn (µ)kYµ ≤
W. Dahmen (RWTH Aachen)
2
inf ku(µ) − wkXˆµ
1 − δ w∈Xn
How to Best Sample a Solution Manifold?
July 2, 2013
15 / 29
Double Greedy Scheme
Double Greedy Scheme: (Xn , Vn ) → (Xn+1 , Vn+1 )
stabilize: given Xn , generate inf-sup stable test space Vn - independent
of µ - through a greedy inner loop (see Gerner/Veroy for Stokes)

¯ = argminµ∈P,w∈X  sup
(µ,
¯ w)
n
v ∈Vn
W. Dahmen (RWTH Aachen)

bµ (w, v )
kv kYµ kwkXˆ

−1
¯ Vn → [Vn , v¯ ]
v¯ = RY Bµ
¯ w,
µ
¯
µ
How to Best Sample a Solution Manifold?
July 2, 2013
16 / 29
Double Greedy Scheme
Double Greedy Scheme: (Xn , Vn ) → (Xn+1 , Vn+1 )
stabilize: given Xn , generate inf-sup stable test space Vn - independent
of µ - through a greedy inner loop (see Gerner/Veroy for Stokes)

¯ = argminµ∈P,w∈X  sup
(µ,
¯ w)
n
v ∈Vn

bµ (w, v )
kv kYµ kwkXˆ

−1
¯ Vn → [Vn , v¯ ]
v¯ = RY Bµ
¯ w,
µ
¯
µ
each µ-query requires only solving a small n-dimensional
eigenvalue problem
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
16 / 29
Double Greedy Scheme
Double Greedy Scheme: (Xn , Vn ) → (Xn+1 , Vn+1 )
stabilize: given Xn , generate inf-sup stable test space Vn - independent
of µ - through a greedy inner loop (see Gerner/Veroy for Stokes)

¯ = argminµ∈P,w∈X  sup
(µ,
¯ w)
n
v ∈Vn

bµ (w, v )
kv kYµ kwkXˆ

−1
¯ Vn → [Vn , v¯ ]
v¯ = RY Bµ
¯ w,
µ
¯
µ
each µ-query requires only solving a small n-dimensional
eigenvalue problem
T
truth-independent termination even when Yµ 6= Y := µ∈P Yµ
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
16 / 29
Double Greedy Scheme
Double Greedy Scheme: (Xn , Vn ) → (Xn+1 , Vn+1 )
stabilize: given Xn , generate inf-sup stable test space Vn - independent
of µ - through a greedy inner loop (see Gerner/Veroy for Stokes)

¯ = argminµ∈P,w∈X  sup
(µ,
¯ w)
n
v ∈Vn

bµ (w, v )
kv kYµ kwkXˆ

−1
¯ Vn → [Vn , v¯ ]
v¯ = RY Bµ
¯ w,
µ
¯
µ
each µ-query requires only solving a small n-dimensional
eigenvalue problem
T
truth-independent termination even when Yµ 6= Y := µ∈P Yµ
stabilization
residual based surrogate Rn (µ) := kf − Bµ un (µ)kYN0
is tight with a small µ-independent condition
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
16 / 29
Double Greedy Scheme
Double Greedy Scheme: (Xn , Vn ) → (Xn+1 , Vn+1 )
stabilize: given Xn , generate inf-sup stable test space Vn - independent
of µ - through a greedy inner loop (see Gerner/Veroy for Stokes)

¯ = argminµ∈P,w∈X  sup
(µ,
¯ w)
n
v ∈Vn

bµ (w, v )
kv kYµ kwkXˆ

−1
¯ Vn → [Vn , v¯ ]
v¯ = RY Bµ
¯ w,
µ
¯
µ
each µ-query requires only solving a small n-dimensional
eigenvalue problem
T
truth-independent termination even when Yµ 6= Y := µ∈P Yµ
stabilization
residual based surrogate Rn (µ) := kf − Bµ un (µ)kYN0
is tight with a small µ-independent condition
SP-Galerkin projection ∼ best approximation
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
16 / 29
Double Greedy Scheme
Double Greedy Scheme: (Xn , Vn ) → (Xn+1 , Vn+1 )
stabilize: given Xn , generate inf-sup stable test space Vn - independent
of µ - through a greedy inner loop (see Gerner/Veroy for Stokes)

¯ = argminµ∈P,w∈X  sup
(µ,
¯ w)
n
v ∈Vn

bµ (w, v )
kv kYµ kwkXˆ

−1
¯ Vn → [Vn , v¯ ]
v¯ = RY Bµ
¯ w,
µ
¯
µ
each µ-query requires only solving a small n-dimensional
eigenvalue problem
T
truth-independent termination even when Yµ 6= Y := µ∈P Yµ
stabilization
residual based surrogate Rn (µ) := kf − Bµ un (µ)kYN0
is tight with a small µ-independent condition
SP-Galerkin projection ∼ best approximation
grow Xn → Xn+1 by outer greedy step
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
16 / 29
Double Greedy Scheme
Double Greedy Scheme: (Xn , Vn ) → (Xn+1 , Vn+1 )
stabilize: given Xn , generate inf-sup stable test space Vn - independent
of µ - through a greedy inner loop (see Gerner/Veroy for Stokes)

¯ = argminµ∈P,w∈X  sup
(µ,
¯ w)
n
v ∈Vn

bµ (w, v )
kv kYµ kwkXˆ

−1
¯ Vn → [Vn , v¯ ]
v¯ = RY Bµ
¯ w,
µ
¯
µ
each µ-query requires only solving a small n-dimensional
eigenvalue problem
T
truth-independent termination even when Yµ 6= Y := µ∈P Yµ
stabilization
residual based surrogate Rn (µ) := kf − Bµ un (µ)kYN0
is tight with a small µ-independent condition
SP-Galerkin projection ∼ best approximation
grow Xn → Xn+1 by outer greedy step
Theorem:
... The scheme is rate-optimal with surrogate condition close to one
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
16 / 29
Numerical Experiments
Convection-Diffusion Equations
Outline
1
Introduction
2
The Greedy Paradigm - Rate Optimality
3
Finding Good Projections - Stable Variational Formulations
Renormation
A Saddle Point Formulation
4
Double Greedy Scheme
5
Numerical Experiments
Convection-Diffusion Equations
Transport Equation
6
Summary and Outlook
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
17 / 29
Numerical Experiments
Convection-Diffusion Equations
Convection-Diffusion Equations


cos µ
 · ∇u + u = 1, in Ω = (0, 1)2 ,
−∆u + 
sin µ
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
u = 0, on ∂Ω
July 2, 2013
18 / 29
Numerical Experiments
Convection-Diffusion Equations
Convection-Diffusion Equations


cos µ
 · ∇u + u = 1, in Ω = (0, 1)2 ,
−∆u + 
sin µ
u = 0, on ∂Ω
Choice of the test norm:
1
hBµ u, v i + hBµ v , ui ,
2
1/2 2
1
v
:= sµ (v , v ) = |v |2H 1 (Ω) + c − div b(µ)
2
L2 (Ω)
sµ (u, v ) :=
kv k2Yµ
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
18 / 29
Numerical Experiments
Convection-Diffusion Equations
Convection-Diffusion Equations


cos µ
 · ∇u + u = 1, in Ω = (0, 1)2 ,
−∆u + 
sin µ
u = 0, on ∂Ω
Choice of the test norm:
1
hBµ u, v i + hBµ v , ui ,
2
1/2 2
1
v
:= sµ (v , v ) = |v |2H 1 (Ω) + c − div b(µ)
2
L2 (Ω)
sµ (u, v ) :=
kv k2Yµ
+ boundary penalization at outflow boundary:
(see Cohen/Dahmen/Welper, M2AN 2012)
¯ µ uk2¯ 0 = kB
¯ µ uk2 0 + λkuk2
kuk2X¯µ := kB
Yµ
Hb (µ)
Y
µ
1/2
¯µ := Yµ × Hb (µ)0
Hb (µ) = H00 (Γ+ (µ)), Y
W. Dahmen (RWTH Aachen)
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July 2, 2013
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Numerical Experiments
Convection-Diffusion Equations
Convection-Diffusion Equations


cos µ
 · ∇u + u = 1, in Ω = (0, 1)2 ,
−∆u + 
sin µ
u = 0, on ∂Ω
Choice of the test norm:
1
hBµ u, v i + hBµ v , ui ,
2
1/2 2
1
v
:= sµ (v , v ) = |v |2H 1 (Ω) + c − div b(µ)
2
L2 (Ω)
sµ (u, v ) :=
kv k2Yµ
+ boundary penalization at outflow boundary:
(see Cohen/Dahmen/Welper, M2AN 2012)
¯ µ uk2¯ 0 = kB
¯ µ uk2 0 + λkuk2
kuk2X¯µ := kB
Yµ
Hb (µ)
Y
µ
1/2
¯µ := Yµ × Hb (µ)0
Hb (µ) = H00 (Γ+ (µ)), Y
¯ µ w − f k2¯ 0 + λkγwk2
u(µ) = argminw∈X− kB
Hb (µ)
Y
µ
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July 2, 2013
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Numerical Experiments
Convection-Diffusion Equations
Convection-Diffusion Equations
(a) = 2−5
RB dim n = 6, m(n) = 13, angle µ = 0.885115,
(b) = 2−7
RB dim n = 7, m(n) = 20, angle µ = 0.257484,
(c) = 2−26
RB dim n = 20, m(n) = 57, angle µ = 0.587137
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July 2, 2013
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Numerical Experiments
Convection-Diffusion Equations
Convection-Diffusion Equations
= 2−5
= 2−7
= 2−26
0.12
6 · 10−2
0.15
8 · 10−2
4 · 10−2
6 · 10−2
0.1
4 · 10−2
2 · 10−2
0
0.2
0.1
5 · 10−2
2 · 10−2
2
3
4
5
reduced basis trial dimension
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6
0
2
3
4
5
6
7
reduced basis trial dimension
How to Best Sample a Solution Manifold?
0
5
10
15
20
reduced basis trial dimension
July 2, 2013
20 / 29
Numerical Experiments
Convection-Diffusion Equations
Convection-Diffusion Equations: = 2−5
a-post. error 0.0057
dim trial
dim test
δ
max surr
surr/a-post
2
3
2.51e-01
7.04e-02
1.24e+01
3
6
3.74e-01
3.08e-02
5.40e+00
4
7
3.74e-01
7.43e-03
1.30e+00
5
10
3.51e-01
5.81e-03
1.02e+00
6
13
1.86e-01
5.70e-03
1.00e+00
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July 2, 2013
21 / 29
Numerical Experiments
Convection-Diffusion Equations
Convection-Diffusion Equations: = 2−7
a-post. error 0.0197
dim trial
dim test
δ
max surr
surr/a-post
2
5
8.92e-03
1.25e-01
6.37e+00
3
8
1.22e-01
9.65e-02
4.90e+00
4
11
1.13e-02
3.21e-02
1.63e+00
5
14
1.27e-02
2.61e-02
1.32e+00
6
17
5.55e-03
2.21e-02
1.12e+00
7
20
4.82e-03
1.97e-02
1.00e+00
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July 2, 2013
22 / 29
Numerical Experiments
Convection-Diffusion Equations
Convection-Diffusion Equations: = 2−26
a-post. error 0.0011
trial
test
δ
surrogate
a-post
trial
test
δ
surrogate
surr/a-post
2
5
1.35e-03
2.11e-01
2.00e+02
12
33
3.47e-04
1.60e-02
1.52e+01
4
9
1.09e-02
7.58e-02
7.19e+01
14
39
1.10e-04
8.46e-03
8.02e+00
6
15
1.61e-03
5.02e-02
4.76e+01
16
45
9.39e-05
7.87e-03
7.46e+00
8
21
7.99e-04
2.39e-02
2.26e+01
18
51
6.11e-05
7.69e-03
7.29e+00
10
27
3.55e-04
2.10e-02
2.00e+01
20
57
5.28e-05
6.35e-03
6.02e+00
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July 2, 2013
23 / 29
Numerical Experiments
Transport Equation
Outline
1
Introduction
2
The Greedy Paradigm - Rate Optimality
3
Finding Good Projections - Stable Variational Formulations
Renormation
A Saddle Point Formulation
4
Double Greedy Scheme
5
Numerical Experiments
Convection-Diffusion Equations
Transport Equation
6
Summary and Outlook
W. Dahmen (RWTH Aachen)
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July 2, 2013
24 / 29
Numerical Experiments
Transport Equation
(Pure) Transport Equation
(1) Homogeneous B.C.s:


cos µ

 · ∇u + u = 1, in Ω = (0, 1)2 ,
sin µ
W. Dahmen (RWTH Aachen)
u = 0, on Γ− (µ)
How to Best Sample a Solution Manifold?
July 2, 2013
25 / 29
Numerical Experiments
Transport Equation
(Pure) Transport Equation
(1) Homogeneous B.C.s:


cos µ

 · ∇u + u = 1, in Ω = (0, 1)2 ,
sin µ
u = 0, on Γ− (µ)
(2) Discontinuous boundary data:



 0.5
cos µ

·∇u+u =
 1
sin µ
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
x <y 
,
x ≥y 

 1−y
u=
 0
How to Best Sample a Solution Manifold?

x ≤ 0.5 
,
x > 0.5 
on Γ− (µ)
July 2, 2013
25 / 29
Numerical Experiments
Transport Equation
(Pure) Transport Equation
(1) Homogeneous B.C.s:


cos µ

 · ∇u + u = 1, in Ω = (0, 1)2 ,
sin µ
u = 0, on Γ− (µ)
(2) Discontinuous boundary data:



 0.5
cos µ

·∇u+u =
 1
sin µ

x <y 
,
x ≥y 
kv kYµ = kBµ∗ v kL2 (Ω) ,

 1−y
u=
 0

x ≤ 0.5 
,
x > 0.5 
on Γ− (µ)
k · kXˆµ = k · kL2 (Ω) ,
i.e. RYµ = Bµ Bµ∗
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July 2, 2013
25 / 29
Numerical Experiments
Transport Equation
(Pure) Transport Equation
Figure: (1): reduced basis of
dimension n = 24, m(n) = 91, angle
µ = 0.244579
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Figure: (2): reduced basis of
dimension n = 24, m(n) = 96, angle
µ = 0.256311
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July 2, 2013
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Numerical Experiments
Transport Equation
(Pure) Transport Equation: Convergence
jump boundary
zero boundary
6 · 10−2
1.2 · 10−2
1 · 10−2
4 · 10−2
8 · 10−3
6 · 10−3
2 · 10−2
4 · 10−3
2 · 10−3
0
5
10
15
20
reduced basis trial dimension
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25
0
5
10
15
20
25
reduced basis trial dimension
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July 2, 2013
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Numerical Experiments
Transport Equation
(Pure) Transport Equation (2): truth L2 -error 0.0154
trial
test
δ
surr
rb truth
rb L2
surr/err
4
14
4.97e-01
5.91e-02
1.29e-01
1.30e-01
4.54e-01
8
31
4.29e-01
4.34e-02
7.78e-02
7.95e-02
5.46e-01
12
49
3.71e-01
3.48e-02
7.40e-02
7.53e-02
4.63e-01
16
64
3.74e-01
2.99e-02
6.20e-02
6.41e-02
4.67e-01
20
81
3.73e-01
2.71e-02
5.46e-02
5.62e-02
4.82e-01
24
96
3.91e-01
2.51e-02
4.51e-02
4.79e-02
5.25e-01
1 cycle of iterative tightening:
trial
test
δ
surr
rb truth
rb L2
surr/err
20
81
3.73e-01
2.71e-02
5.46e-02
5.62e-02
4.82e-01
10
87
3.51e-01
6.45e-02
7.40e-02
7.53e-02
8.57e-01
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July 2, 2013
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Summary
Summary and Outlook
Reference: W. Dahmen, C. Plesken, G. Welper, Double Greedy Algorithms: Reduced Basis Methods for Transport Dominated
Problems, ESAIM: Mathematical Modelling and Numerical Analysis, 48(3) (2014), 623–663.
W. Dahmen (RWTH Aachen)
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July 2, 2013
29 / 29
Summary
Summary and Outlook
Reference: W. Dahmen, C. Plesken, G. Welper, Double Greedy Algorithms: Reduced Basis Methods for Transport Dominated
Problems, ESAIM: Mathematical Modelling and Numerical Analysis, 48(3) (2014), 623–663.
works the same for space-time formulations
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
29 / 29
Summary
Summary and Outlook
Reference: W. Dahmen, C. Plesken, G. Welper, Double Greedy Algorithms: Reduced Basis Methods for Transport Dominated
Problems, ESAIM: Mathematical Modelling and Numerical Analysis, 48(3) (2014), 623–663.
works the same for space-time formulations
applies to “classical saddle point problems” as well, see Gerner/Veroy,
Rozza
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
29 / 29
Summary
Summary and Outlook
Reference: W. Dahmen, C. Plesken, G. Welper, Double Greedy Algorithms: Reduced Basis Methods for Transport Dominated
Problems, ESAIM: Mathematical Modelling and Numerical Analysis, 48(3) (2014), 623–663.
works the same for space-time formulations
applies to “classical saddle point problems” as well, see Gerner/Veroy,
Rozza
in principle, stability constants can be driven as close to one as one
wishes
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
29 / 29
Summary
Summary and Outlook
Reference: W. Dahmen, C. Plesken, G. Welper, Double Greedy Algorithms: Reduced Basis Methods for Transport Dominated
Problems, ESAIM: Mathematical Modelling and Numerical Analysis, 48(3) (2014), 623–663.
works the same for space-time formulations
applies to “classical saddle point problems” as well, see Gerner/Veroy,
Rozza
in principle, stability constants can be driven as close to one as one
wishes
iterative tightening restores efficient offline-online procedures for Y 6= Yµ
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
29 / 29
Summary
Summary and Outlook
Reference: W. Dahmen, C. Plesken, G. Welper, Double Greedy Algorithms: Reduced Basis Methods for Transport Dominated
Problems, ESAIM: Mathematical Modelling and Numerical Analysis, 48(3) (2014), 623–663.
works the same for space-time formulations
applies to “classical saddle point problems” as well, see Gerner/Veroy,
Rozza
in principle, stability constants can be driven as close to one as one
wishes
iterative tightening restores efficient offline-online procedures for Y 6= Yµ
low smoothness w.r.t. the parameters in the pure transport case ...
...limitation of RBMs...
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
29 / 29
Summary
Summary and Outlook
Reference: W. Dahmen, C. Plesken, G. Welper, Double Greedy Algorithms: Reduced Basis Methods for Transport Dominated
Problems, ESAIM: Mathematical Modelling and Numerical Analysis, 48(3) (2014), 623–663.
works the same for space-time formulations
applies to “classical saddle point problems” as well, see Gerner/Veroy,
Rozza
in principle, stability constants can be driven as close to one as one
wishes
iterative tightening restores efficient offline-online procedures for Y 6= Yµ
low smoothness w.r.t. the parameters in the pure transport case ...
...limitation of RBMs...
In progress:
High-dimensional (empirical) interpolation - compressed sensing
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
29 / 29
Summary
Summary and Outlook
Reference: W. Dahmen, C. Plesken, G. Welper, Double Greedy Algorithms: Reduced Basis Methods for Transport Dominated
Problems, ESAIM: Mathematical Modelling and Numerical Analysis, 48(3) (2014), 623–663.
works the same for space-time formulations
applies to “classical saddle point problems” as well, see Gerner/Veroy,
Rozza
in principle, stability constants can be driven as close to one as one
wishes
iterative tightening restores efficient offline-online procedures for Y 6= Yµ
low smoothness w.r.t. the parameters in the pure transport case ...
...limitation of RBMs...
In progress:
High-dimensional (empirical) interpolation - compressed sensing
reparametrizations
W. Dahmen (RWTH Aachen)
How to Best Sample a Solution Manifold?
July 2, 2013
29 / 29
Summary
Summary and Outlook
Reference: W. Dahmen, C. Plesken, G. Welper, Double Greedy Algorithms: Reduced Basis Methods for Transport Dominated
Problems, ESAIM: Mathematical Modelling and Numerical Analysis, 48(3) (2014), 623–663.
works the same for space-time formulations
applies to “classical saddle point problems” as well, see Gerner/Veroy,
Rozza
in principle, stability constants can be driven as close to one as one
wishes
iterative tightening restores efficient offline-online procedures for Y 6= Yµ
low smoothness w.r.t. the parameters in the pure transport case ...
...limitation of RBMs...
In progress:
High-dimensional (empirical) interpolation - compressed sensing
reparametrizations
Relaxation
W. Dahmen (RWTH Aachen)
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July 2, 2013
29 / 29