The Hierarchical Poincar -Steklov schemefastdirect/notes/CBMS... · 2014. 6. 25. · CBMS-NSF...

Motivation Outline of scheme Leaf computation Merging two DtNs Review Numerical Results Free-space scattering Summary

The Hierarchical Poincare-Steklov scheme

Adrianna Gillman

Dartmouth College

Collaborators:

Alex Barnett Per-Gunnar MartinssonDartmouth College University of Colorado, Boulder

Additional people working on these methods: T. Babb, S. Hao and T. Haut.

CBMS-NSF Conference: Fast Direct Solvers for Elliptic PDEsJune 25, 2014


Recall: Model problem from Lecture 6

Let Ω = [0, 1]2. We introduce a square n × n grid on with nodes xjNj=1 where

N = n2. Let u = [u(j)]Nj=1 denote a potential vector, and let f = [f(j)]Nj=1

denote a given load vector. The equilibrium equation we seek to solve is

Au = f

where A is a finite difference operator discretizing an elliptic boundary valueproblem on Ω. (Laplace, Helmholtz, convection-diffusion, etc).


5-point stencil

The archetypical example of an elliptic finite difference equation is the 5-pointstencil

[Au](k) =1

h2(4u(k)− u(ks)− u(ke)− u(kn)− u(kw))

where h is the grid spacing (h = 1/(n − 1)), where k is a node in the mesh,and where ks , ke , kn, kw are the nodes that are immediate neighbors of k tothe south, east, north, and west of k , respectively.

kkw ke

ks

kn

ThenAu = f

is a discrete analog of the Poisson equation (with ∆ = ∂2

∂x21+ ∂2

∂x22) and

−∆u = f .


Nested dissection

Ω2Ω1

Ω3

The non-zero blocks of the coefficient matrix are:

A =

A11 A13

A22 A23

A31 A32 A33

=

I

I

A31A−111 A32A

−122 I

A11 A13

A22 A23

S

where the Schur complement S33 is given by

S33 = A33 − A31A−111 A13 − A32A

−122 A23.

Note: A31, A13, A23, and A32 are diagonal.


Scaling of the nested dissection method

We know the scaling of the method is O(N1.5).

What impact does the order of the discretization have on the computationalcost?


Scaling of the nested dissection method

We know the scaling of the method is O(N1.5).

What impact does the order of the discretization have on the computationalcost?

104

105

106

107

10−2

10−1

100

101

102

103

104

N

Tim

ein

seconds

5-pt stencil 9-pt stencil 13-pt stencil

Performance of UMFPACK


Nested dissection and the 9-pt stencil

The naive fourth-order finite difference scheme of an elliptic finite differenceequation is the following 9-point stencil

[Au](k) =1

h2

(

5u(k)− 1

12u(kss)−

4

3u(ks)−

1

12u(kee)−

4

3u(ke)−

1

12u(knn)− 4

3u(kn)− 1

12u(kww)− 4

3u(kw)

)

where h is the grid spacing (h = 1/(n − 1)), where k is a node in the mesh,and where ks , kss , ke , kee , knn, kn, kw , kww are the nodes that are immediatelysouth, east, north and west of k , as pictured below.

kkww kw keeke

kss

ks

knn

kn



Ω2Ω1

Ω3

The coefficient matrix is given by:

A =

A11 A12 A13

A21 A22 A23

A31 A32 A33

Changes in A:

• A12 and A21 have gone from zero matrices to diagonal.

• A31, A13, A23, and A32 are banded diagonal.



Ω2Ω1

Ω3

The coefficient matrix is given by:

A =

A11 A13

A22 A23

A31 A32 A33

Changes in A:

• A31, A13, A23, and A32 are banded diagonal.

• A33 is also banded diagonal.


Accuracy for Helmholtz problems

Let Ω = [0, 1]2. Consider ∆u + κ2u = 0 with 3 choices of κ.

Matrix N Epot Epot Epot

Helmholtz I Helmholtz II Helmholtz IIIΩ ∼ 12× 12λs Ω ∼ 102 × 102λs 12 pts/λ

5-point stencil 40000 2.7e0 1.2e0 3.1e0O(h2) 160000 2.0e1 2.5e1 1.9e1

640000 3.1e-1 6.7e1 1.4e12560000 6.1e-2 8.8e1 3.7e1

10240000 1.5e-2 1.6e1 3.5e19-point stencil 40000 ≥5.5e-2 ≥3.8e0 ≥1.3e0O(h4) 160000 ≥8.0e-3 ≥1.8e1 ≥3.2e-1

640000 ≥1.4e-4 ≥1.4e1 ≥9.6e-12560000 ≥1.0e-5 ≥6.1e-1 ≥2.4e0

13-point stencil 40000 ≥4.7e-4 ≥8.7e1 ≥8.6e-3O(h6) 160000 ≥1.7e-5 ≥1.1e1 ≥1.3e-1

640000 ≥5.9e-7 ≥4.9e-1 ≥2.9e-12560000 ≥4.1e-9 ≥8.3e-2 ≥1.4e-1

Epot =max

∣

∣uapp(xk)− uexact(xk)∣

∣

N

k=1

max∣

∣uexact(xk)∣

∣

N

k=1

,


General idea

Goal: Design a high order accurate discretization technique which has a solverthat has

• O(N1.5) dense linear algebra computational cost,

• O(N) fast linear algebra computational cost, and

• small constant.


General idea

Goal: Design a high order accurate discretization technique which has a solverthat has

• O(N1.5) dense linear algebra computational cost,

• O(N) fast linear algebra computational cost, and

• small constant.

Idea:

1. Partition the geometry into a collection of small regions.

2. Discretize each small region with a spectral method.

3. “Glue” regions together with approximate Dirichlet-to-Neumann (DtN)operators.


Definition of the DtN

Consider the following model problem on Ω = [0, 1]2

−∆u(x) + b(x)u(x) = 0 x ∈ Ωu(x) = f (x) x ∈ Γ,

where Γ = ∂Ω, f (x) and b(x) are given. b(x) is smooth.


Definition of the DtN

Consider the following model problem on Ω = [0, 1]2

−∆u(x) + b(x)u(x) = 0 x ∈ Ωu(x) = f (x) x ∈ Γ,

where Γ = ∂Ω, f (x) and b(x) are given. b(x) is smooth.

For x ∈ Γ, let g(x) =∂u

∂ν(x) denote the normal derivative in the outwards

direction of u at x.

The Dirichlet-to-Neumann (DtN) operator Λ maps f (x) to g(x). In otherwords,

g = Λf for x ∈ Γ.


Precomputation

Let Ω = [0, 1]2 and Γ = ∂Ω. Consider the boundary value problem

−∆u(x) + b(x)u(x) = 0 x ∈ Ωu(x) = f (x) x ∈ Γ.

Partition the domain into a collection of 4L equisized small boxes. The size ofthe boxes is chosen so that the local problem can be solved to high accuracyvia a spectral method.These small boxes are called leaf boxes.

∂Ω Ω

Here L = 2.


Precomputation


−∆u(x) + b(x)u(x) = 0 x ∈ Ωu(x) = f (x) x ∈ Γ.

Discretize each small box via a spectral method.

∂Ω Ω


Precomputation


−∆u(x) + b(x)u(x) = 0 x ∈ Ωu(x) = f (x) x ∈ Γ.

Eliminate interior nodes to construct construct an operator that maps thesolution on the boundary to the solution the interior.

⇒


Precomputation


−∆u(x) + b(x)u(x) = 0 x ∈ Ωu(x) = f (x) x ∈ Γ.

Retabulate from Chebyshev nodes to Legendre nodes on the boundary of eachleaf.

⇒

Results in a collection of DtN maps Λ and solution operators Ψ for each box.


Precomputation


−∆u(x) + b(x)u(x) = 0 x ∈ Ωu(x) = f (x) x ∈ Γ.

Merge boxes by eliminating a series of horizontal interior connections creatingrectangular boxes.

⇒

Results in a collection of DtN maps Λ and solution operators Ψ for eachrectangular box.


Precomputation


−∆u(x) + b(x)u(x) = 0 x ∈ Ωu(x) = f (x) x ∈ Γ.


⇒



Precomputation


−∆u(x) + b(x)u(x) = 0 x ∈ Ωu(x) = f (x) x ∈ Γ.


⇒



Precomputation


−∆u(x) + b(x)u(x) = 0 x ∈ Ωu(x) = f (x) x ∈ Γ.


⇒



Solve


−∆u(x) + b(x)u(x) = 0 x ∈ Ωu(x) = f (x) x ∈ Γ.

We know the solution on the boundary. Now we can now apply Ψ to find thesolution on the interior points.

⇒


Solve


−∆u(x) + b(x)u(x) = 0 x ∈ Ωu(x) = f (x) x ∈ Γ.

With the solution on the boundary of the red boxes, we apply Ψ to find thesolution on the interior points.

⇒


Solve


−∆u(x) + b(x)u(x) = 0 x ∈ Ωu(x) = f (x) x ∈ Γ.


⇒


Solve


−∆u(x) + b(x)u(x) = 0 x ∈ Ωu(x) = f (x) x ∈ Γ.


⇒


Solve


−∆u(x) + b(x)u(x) = 0 x ∈ Ωu(x) = f (x) x ∈ Γ.


⇒


Leaf computation

On each leaf box τ , we require thesolution u to satisfy[−∆+ b(x)] u = 0 in Ωτ

and u = f for x ∈ Γτ .Note: f is used as a place holder.

ui

ub


Leaf computation



ui

ub

Use spectral discretization with a p × p product Chebychev grid of nodes xj

Order geometry by interior and boundary; i.e. u =

[ub

ui

]

.


Leaf computation



ui

ub

Use spectral discretization with a p × p product Chebychev grid of nodes xj

Order geometry by interior and boundary; i.e. u =

[ub

ui

]

.

Let Dx and Dy denote p2 × p2 spectral differentiation matrices.Then the discrete approximation of the differential operator is given by

A = −D2x − D

2y + diag b(xj )

and the discretized differential equation is given by

Bu :=

[I 0

Ai,i Ai,e

] [ub

ui

]

=

[f

0

]

.


Leaf computation

The discretized differential equationis given by

[I 0

Ai,i Ai,e

] [ub

ui

]

=

[f

0

]

ui

ub


Leaf computation

The discretized differential equationis given by

[I 0

Ai,i Ai,e

] [ub

ui

]

=

[f

0

]

ui

ub

Rewriting the bottom row equation results in the following a map that takesthe solution on the boundary to the solution on the interior.

ui = −A−1i,i Ai,eub

We are now in a position to build our approximate DtN operator defined onGaussian points on the boundary of the box.


Construction of the approximate DtN

Step 1: Re-tabulate from Gaussian nodes to Chebyshev nodesFor each edge, build interpolation operators that maps a function from theGuassian points to the Chebychev points.Define this operator L1.

Note: ui = −A−1i,i Ai,eL1u

gb where u

gb is the solution at the Gaussian points on

the boundary.

We define Ψ = −A−1i,i Ai,eL1 to be the solution operator.





gb where u


the boundary.


Step 2: Spectral differentiationLeft multiply Ψ by the corresponding rows of Dx and Dy so that the normalderivative is taken. Let L2 denote this operator.(This operator is defined on the Chebychev points.)





gb where u


the boundary.



Step 3: Re-tabulate from Chebychev nodes to Gaussian nodesFor each edge, build interpolation operators that maps a function from theChebychev points to the Gaussian points. Let L3 denote this operator.





gb where u


the boundary.



Step 3: Re-tabulate from Chebychev nodes to Gaussian nodesFor each edge, build interpolation operators that maps a function from theChebychev points to the Gaussian points. Let L3 denote this operator.

Then the approximate DtN is given by

Λ = L3L2Ψ.


Merge two DtN maps

Ωα ΩβI1 I2I3

[v1v3

]

=

[Λα

1,1 Λα1,3

Λα3,1 Λα

3,3

] [u1

u3

] [v2v3

]

=

[Λ

β2,2 Λ

β2,3

Λβ3,2 Λ

β3,3

] [u2

u3

]

The flux is equal across the shared edge.

Λα3,1u1 + Λ

α3,3u3 = v3 = Λ

β3,2u2 + Λ

β3,3u3.


Merge two DtN maps

Ωα ΩβI1 I2I3

Λα3,1u1 + Λ

α3,3u3 = v3 = Λ

β3,2u2 + Λ

β3,3u3

Solving for u3, we find

u3 =(Λ

α3,3 − Λ

β3,3

)−1(

Λβ3,2u2 − Λ

α3,1u1

)=

(Λ

α3,3 − Λ

β3,3

)−1[−Λ

α3,1

∣∣ Λ

β3,2]

︸︷︷︸

=Ψτ

[u1

u2

]


Merge two DtN maps

I1 I2Ωτ

Now u3 can be eliminated by plugging

u3 =(Λ

α3,3 − Λ

β3,3

)−1(

Λβ3,2u2 − Λ

α3,1u1

).

into

[v1v3

]

=

[Λα

1,1 Λα1,3

Λα3,1 Λα

3,3

] [u1

u3

]

,

[v2v3

]

=

[Λ

β2,2 Λ

β2,3

Λβ3,2 Λ

β3,3

] [u2

u3

]

.


Merge two DtN maps

I1 I2Ωτ

We get

v1 =Λα1,1u1 + Λ

α1,3u3 = Λ

α1,1u1 + Λ

α1,3

(Λ

α3,3 − Λ

β3,3

)−1(

Λβ3,2u2 − Λ

α3,1u1

),

v2 =Λβ2,2u2 + Λ

β2,3u3 = Λ

β2,2u2 + Λ

β2,3

(Λ

α3,3 − Λ

β3,3

)−1(

Λβ3,2u2 − Λ

α3,1u1

).


Merge two DtN maps

I1 I2Ωτ

Rewriting in factored form gives the expression for the DtN map Λτ ,

[v1v2

]

=

([Λα

1,1 0

0 Λβ2,2

]

+

[Λα

1,3

Λβ2,3

](Λ

α3,3 − Λ

β3,3

)−1[−Λ

α3,1

∣∣ Λ

β3,2]

)

︸︷︷︸

=Λτ

[u1

u2

]

.


Review of algorithm

Pre-computation

Step 1 Partition the domain into a collection of small squares. Define a binary treeconsisting of the boxes formed by eliminating shared edges by neighbors.

Step 2 For each leaf box τ , compute an approximate DtN operator Λτ .

Step 3 Move up the tree. For each parent box τ with children α, and β, computeΛτ and Ψτ .


Review of algorithm

Pre-computation




Solve

Step 1 Moving from the top of the tree down, apply the matrix Ψτ to computethe solution on the interior of the box.


Review of algorithm

Pre-computation




Solve

Step 1 Moving from the top of the tree down, apply the matrix Ψτ to computethe solution on the interior of the box.

Post Processing

The solution can be obtained at x in Ω via high order interpolation.


Computational cost

Dense linear algebra

Let N denote the total number of discretization points.

Apply Λ: O(N)

Construct Λ: O(N1.5) Solve: O(N logN)1

2 3

4 5 6 7

8 9 10 11 12 13 14 15

......

......

......

......

......

......

......

......


Computational cost



Apply Λ: O(N)


2 3

4 5 6 7

8 9 10 11 12 13 14 15

......

......

......

......

......

......

......

......


Computational cost



Apply Λ: O(N)


2 3

4 5 6 7

8 9 10 11 12 13 14 15

......

......

......

......

......

......

......

......


Improving the scaling of the merge procedure

I1 I2Ωτ

Recall the merge formula

[v1v2

]

=

([Λα

1,1 0

0 Λβ2,2

]

+

[Λα

1,3

Λβ2,3

](Λ

α3,3 − Λ

β3,3

)−1[−Λ

α3,1

∣∣ Λ

β3,2]

) [u1

u2

]

.

Things that are exploited to improve asymptotic complexity.

• Λβ3,2, Λ

β3,1, Λ

β1,3, and Λ

β2,3 are low-rank.

•(Λα

3,3 − Λβ3,3

)is an HBS matrix.

• Λα1,1 and Λ

β2,2 are HBS matrices.


Computational cost

Fast linear algebra


Apply Λ: O(√N)

Construct Λ: O(N) Solve: O(N)

1

2 3

4 5 6 7

8 9 10 11 12 13 14 15

......

......

......

......

......

......

......

......


Numerical results

Experiments were carried out on a workstation equipped with two quad-coreIntel Xeon E5-2643 processors and 128 GB of RAM.

Most of the code is written in Matlab. (There is some Fortran.)


Scaling

N Tbuild Tsolve Tapply R(seconds) (seconds) (seconds) (MB)

Laplace

1815681 91.68 0.34 0.035 1611.197252225 371.15 1.803 0.104 6557.2728987905 1661.97 6.97 0.168 26503.29115909633 6894.31 30.67 0.367 106731.61

Helmholtz I1815681 62.07 0.202 0.027 1611.417252225 363.19 1.755 0.084 6557.12

Ω ∼ 12 × 12λs28987905 1677.92 6.92 0.186 26503.41115909633 7584.65 31.85 0.435 106738.85

Helmholtz II1815681 93.96 0.29 0.039 1827.727252225 525.92 2.13 0.074 7151.60

Ω ∼ 102 × 102λs 28987905 2033.91 8.59 0.175 27985.41

Helmholtz III1815681 93.68 0.29 0.038 1839.717252225 624.24 1.67 0.086 7865.13

12 pts per λ 28987905 4174.91 10.28 0.206 33366.45

N Total number of discretization points.

Tbuild Time for building the solution operator.

Tsolve Time to solve for interior nodes (edge nodes only).

Tapply Time to apply the approximate Dirichlet-to-Neumann operator Λ.

R Amount of memory required to store the solution operator.

λ wavelength ( 2πκ).

User prescribed tolerance was set to ǫ = 10−7.


High order discretization and the nested dissection method

Performance comparison with UMFPACK

104

105

106

107

108

10−2

10−1

100

101

102

103

104

N

Tim

ein

seconds

5-pt stencil 9-pt stencil 13-pt stencil – Laplace


Accuracy of method

Fix N = 7, 252, 225 and vary ǫ.

ǫ = 10−7 ǫ = 10−10 ǫ = 10−12

Epot Egrad Epot Egrad Epot Egrad

Laplace 3.57e-04 1.35e-02 1.59e-07 6.92e-06 7.32e-10 1.01e-07Helmholtz I 1.19e-04 1.31e-04 7.99e-08 9.72e-08 2.06e-09 1.71e-09Helmholtz II 2.90e-05 2.19e-05 5.72e-08 5.02e-08 6.21e-09 4.64e-09Helmholtz III 6.76e-05 5.05e-05 1.21e-07 1.31e-07 1.05e-07 1.10e-07

where

Epot =max

∣

∣uapp(xk)− uexact(xk)∣

∣

N

k=1

max∣

∣uexact(xk)∣

∣

N

k=1

,

and

Egrad =

maxk : xk∈Γ

∣

∣un,app(xk)− un,exact(xk)∣

∣

maxk : xk∈Γ

∣

∣un,exact(xk)∣

∣

.


Variable-medium freq-domain scattering problem

Given a frequency κ and incident wave ui

(satisfying ∆ui + κ2ui = 0 in R2),

consider the free space scattering problem

∆u + κ2 (1− b(x)) u = 0 in R2

where u = ui + us, and the unknownscattered field us satisfies a radiativecondition.


Variable-medium freq-domain scattering problem

Given a frequency κ and incident wave ui

(satisfying ∆ui + κ2ui = 0 in R2),

consider the free space scattering problem

∆u + κ2 (1− b(x)) u = 0 in R2

where u = ui + us, and the unknownscattered field us satisfies a radiativecondition.

Applications: underwater acoustics, seismic imaging, ultrasound, optics ingraded index, metamaterials.

We want a solver for smooth b, large κ and multiple incident angles.


Solution technique

Introduce an artificial domain Ω such thatthe support of b ⊂ Ω. We solve for thetotal field in Ω via composite spectral methodand couple it us in R

2/Ω viaDirichlet-to-Neumann maps.

Ω

∂Ω


Solution technique



Λ+

Λ+: the exterior Dirichlet-to-Neumann map

Λ+us|∂Ω =

∂us

∂ν:= u

sν

us is radiative and satisfies (∆ + κ2)us = 0 in R2/Ω.


Solution technique



Λ+

Λ−

Λ+: the exterior Dirichlet-to-Neumann map

Λ+us|∂Ω =

∂us

∂ν:= u

sν

us is radiative and satisfies (∆ + κ2)us = 0 in R2/Ω.

Λ−: the interior Dirichlet-to-Neumann map

Λ−u|∂Ω =

∂u

∂ν:= uν

where[∆+ κ2 (1− b(x))

]u = 0 in Ω.

Λ− exists for κ 6= κj , Dirichlet resonances of Ω


Solution technique

Using the definition of the DtN for thetotal field on the boundary, we find

Λ+

Λ−

Λ−(ui + us)|∂Ω = uiν + us

ν = uiν + Λ+us|∂Ω


Solution technique

Using the definition of the DtN for thetotal field on the boundary, we find

Λ+

Λ−

Λ−(ui + us)|∂Ω = uiν + us

ν = uiν + Λ+us|∂Ω

⇒(Λ− − Λ+

)us |∂Ω = ui

ν − Λ−ui |∂Ω

(Kirsch-Monk ’94, FEM coupling)

This system is ill-conditioned. Thus inversion leads to a loss in accuracy.


2nd kind integral equation

Using the definition of the exterior Dirichlet-to-Neumann operatorΛ+ = S−1

(D − 1

2

), the previous equation can regularized resulting in the

following boundary integral equation

(1

2− D + SΛ−)us|∂Ω = S(ui

n − Λ−ui|∂Ω)

where S and D are the single and double layer boundary integral operators,respectively.

(Sφ)(x) :=

∫

∂Ω

i

4H

(1)0 (κ|x− y|)φ(y)ds(y)

(Dφ)(x) :=

∫

∂Ω

i

4

∂

∂νyH

(1)0 (κ|x− y|)φ(y)ds(y).


2nd kind integral equation

Using the definition of the exterior Dirichlet-to-Neumann operatorΛ+ = S−1

(D − 1

2

), the previous equation can regularized resulting in the

following boundary integral equation

(1

2− D + SΛ−)us|∂Ω = S(ui

n − Λ−ui|∂Ω)

where S and D are the single and double layer boundary integral operators,respectively.

(Sφ)(x) :=

∫

∂Ω

i

4H

(1)0 (κ|x− y|)φ(y)ds(y)

(Dφ)(x) :=

∫

∂Ω

i

4

∂

∂νyH

(1)0 (κ|x− y|)φ(y)ds(y).

Theorem: Let ∂Ω be Lipschitz, b(x) bounded, and 0 < κ 6= κj .Then A := 1

2− D + SΛ− = I + compact and thus the boundary integral

equation is 2nd kind.


Numerical results

Experiments were carried out on a workstation equipped with two quad-coreIntel Xeon E5-2643 processors and 128 GB of RAM.

Most of the code is written in Matlab. (There is some Fortran.)

Everything is done with dense linear algebra.


Graded-index lens at high frequencyui(x) = e iκd·x d = (cos θ, sin θ) κ = 300 100 shortest-λ’s

N ppw error tpre tnew ext tnew int

231361 5 3e-3 17 s 0.013 s 0.16 s923521 10 2e-7 78 s 0.05 s 0.7 s3690241 20 7e-10 6 m 0.16 s 2.7 s

movie 1000 angles θ, 3 s to process each FMM evaluated solution at240,000 exterior points


“Bathroom glass” at high frequency

b(x) = random smooth, rolled-off to zero

80λ on a side

Timings and accuracy are similar to previous example.

movie


Photonic crystal waveguide

lattice of 400 bumps, each a resonator: choose κ in bandgap

Again, 9 digits N = 3.7e6


Summary

• The method presented get high order accuracy and comes convenientlywith its own direct solver.

• For non-oscillatory problems, the solver can be accelerated to have linearcomplexity.

• While scaling for highly oscillatory problems is not optimal, the O(N1.5)dense direct solver has small constants.

• The Dirichlet-to-Neumann operator is used in this presentation, butanalogous schemes can be constructed for other Poincare-Steklovoperators.

• The scheme works conceptually the same in 3D.

• The scheme can be used to rapidly construct and apply the time evolutionoperator associated with parabolic and hyperbolic PDEs.

Date post:	23-Aug-2020
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The Hierarchical Poincar -Steklov schemefastdirect/notes/CBMS... · 2014. 6. 25. · CBMS-NSF...

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