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Jennifer K. Ryan Heinrich Heine University, Dsseldorf, Germany - - PowerPoint PPT Presentation

Sep 05, 2022 292 likes •642 views

Utilizing Geometry of Smoothness-Increasing-Accuracy-Conserving (SIAC) filters for reduced errors Joint work with Julia Docampo Snchez (MIT) Jennifer K. Ryan Heinrich Heine University, Dsseldorf, Germany University of East Anglia,Norwich,

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SLIDE 1

Utilizing Geometry of Smoothness-Increasing-Accuracy-Conserving (SIAC) filters for reduced errors Jennifer K. Ryan Heinrich Heine University, Düsseldorf, Germany University of East Anglia,Norwich, United Kingdom

Advances in Applied Mathematics 18-20 December 2018

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Joint work with Julia Docampo Sánchez (MIT)

Portions of this work partial funded by Air Force Office of Scientific Research under grant number FA8655-09-1-3017

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SLIDE 2

SMOOTHNESS-INCREASING ACCURACY-CONSERVING (SIAC) FILTER

superconvergence extraction through siac filtering

Approximation order: p+1 SIAC DG order: 2p+1 SIAC filtering allows:

  • Extract global superconvergence
  • Create globally smooth approximations
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SLIDE 3

APPLICATIONS: FLOW VISUALIZATION

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Visualizing Vorticity (from NACA wing simulation)

Jallepelli, Docampo, Ryan, Haimes, and Kirby, IEEE TVCG, 2018

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SLIDE 4

DATA INFORMATION

➤ Re = 1.2x106, 12 degree angle of attack ➤ Results from Nektar++: ➤ continuous Galerkin (cG)

discretization

➤ p=5 polynomials ➤ 211180 tetrahedra ➤ 38680 prisms ➤ Box is sampled and vorticity is computed at a resolution of

90 x 90 x 90.

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SLIDE 5

2D SIAC FILTER: COMPUTATIONAL FOOTPRINT

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Tensor Product Filter 1D Line SIAC Filter

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SLIDE 6

OUTLINE

➤ Background ➤ Convergence properties of Discontinuous Galerkin method ➤ Smoothness-Increasing Accuracy-Conserving (SIAC) filter ➤ Divided Difference Estimates ➤ Line SIAC filter ➤ Numerical results ➤ Conclusion & Future Work

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SLIDE 7

DISCONTINUOUS GALERKIN: CONVERGENCE PROPERTIES

For linear hyperbolic equations over a regular grid:

➤ In L2: ➤ Outflow edge: ➤ Negative-Order norm:

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|(u − uh)(xj+1/2)| ≤ Ch2p+1

ku uhk0  Chp+1

k∂α

h (u uh)k−(p+1) = sup Φ∈C∞

(∂α

h (u uh), Φ)

kΦkp+1  Ch2p+1

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SLIDE 8

Extracting Superconvergence Smoothness-Increasing Accuracy-Conserving (SIAC) Filters

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SLIDE 9

SIAC FILTERED DG

SIAC filtered solution: SIAC filtered error:

u∗

h(x, t) = 1

H Z

R

K ✓x − y H ◆ uh(y, t) dy

(Uniform) k(u K(2p+1,p+1)

h

⇤ uh)(T)k0 Ch2p+1 (Non-uniform) k(u K(2p+1,p+1)

H

⇤ uh)(T)k0 Ch

2 3 (2p+1)

Mock & Lax (1978) Bramble & Schatz, Math. Comp (1977) Cockburn, Luskin, Shu, & Süli, Math. Comp (2003) (❩L-inf estimates)❪ Ji, Xu, Ryan, Math. Comp. (2012)

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SIAC KERNEL

➤ SIAC kernel: ➤ Linear combination of B-splines of order m+1. ➤ Filter width: (2r+m+1)H, where H is the scaling

(generally the mesh size).

➤ Alternatively: Can choose coefficients to satisfy data

requirements.

Chosen to maintain 2r moments B-spline chosen for desired smoothness

K(x) =

r

X

γ=−r

cγψ(m+1)(x − γ)

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SLIDE 11

SIAC KERNEL

➤ SIAC kernel: ➤ Linear combination of B-splines of order m+1. ➤ Filter width: (2r+m+1)H, where H is the scaling

(generally the mesh size).

➤ Alternatively: Can choose coefficients to satisfy data

requirements.

Chosen to maintain 2r moments B-spline chosen for desired smoothness

K(x) =

r

X

γ=−r

cγψ(m+1)(x − γ)

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SLIDE 12

SIAC KERNEL: FOURIER SPACE

➤ In physical space, the filter is ➤ In Fourier space this is:

ˆ K(k) = ✓sin(kπ) kπ ◆m+1 c0 + 2

r

X

γ=0

cγ cos(γkπ) ! K(x) =

r

X

γ=−r

cγψ(m+1)(x − γ)

Thomee, Math. Comp. (1977) Ji & Ryan, ICOSAHOM 2014 Proceedings

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SLIDE 13

SIAC KERNEL: FOURIER SPACE

Basic properties of dg and beyond

Plot of full kernel in Fourier space for preserving 2, 4 and 6 moments. SIAC filter: first-order B-spline (top hat function). Plot of full kernel in Fourier space for preserving 2, 4 and 6 moments. SIAC filter: fourth-order B-spline.

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SLIDE 14

TYPICAL PARAMETER CHOICE

p Number B-Splines B-Splne Order Number elements Possible accuracy 1 3 2 5 3 3 1 3 3 2 5 3 7 —> 9 5 5 1 7 5 3 1 4 —> 5 3 3 7 4 11 7 7 1 8 —> 9 7 5 1 6 —> 7 5

H < Dx Order p+1 H = Dx up to Order 2p+1

u∗

h(x, t) = 1

H Z

R

K ✓x − y H ◆ uh(y, t) dy

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SLIDE 15

SIAC KERNEL

superconvergence extraction through siac filtering

p=2 Kernel:

  • Preserves 4 moments
  • C1 continuity
  • Support of 7 elements

p=1 Kernel:

  • Preserves 2 moments
  • Continuous
  • Support of 5 elements
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SLIDE 16

SIAC KERNEL: MULTIPLE DIMENSIONS

➤ The post-processed solution:

where

➤ The post-processing kernel is the same in each dimension:

u∗

h(¯

x, t) = 1 Hd Z

Rd K

✓ ¯ x1 − x1 ∆x1 ◆ K ✓ ¯ x2 − x2 ∆x2 ◆ · · · K ✓ ¯ xd − xd ∆xd ◆ uh(x, t) dx

Hd = ∆x1∆x2 · · · ∆xd K(x) =

r

X

γ=−r

cγψ(m+1)(x − γ)

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SLIDE 17

SIAC FILTER: ERROR ESTIMATES

iltering

Through convolution, we can obtain a higher order accurate solution:

Determined by number of moments (2r) the filter preserves Ch2r+1 Determined by

  • Numerical scheme
  • Choice of kernel function
  • (Dual problem of PDE)
  • Chs

Error in filtered solution

ku u∗

hk0  ku u∗k0

| {z }

Filter Error

+ kKH ⇤ (u uh)k0 | {z }

Discretization Error

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SLIDE 18

SIAC FILTER: ERROR ESTIMATES

Second term where

Using properties of B-splines and convolution Properties of the scheme/Equation

kKH ⇤ (u uh)k0  X

|α|≤m+1

kDα(KH ⇤ (u uh))k−(m+1)  X

|α|≤m+1

kKk1k∂α

H(u uh)k−(m+1)

k∂α

h (u uh)k−(p+1)  Ch2p+1

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SLIDE 19

Question: How can we use a 1D filter for 2D data?

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SLIDE 20

LINE SIAC FILTER FOR MULTI-DIMENSIONAL FILTERING

➤ Cartesian-aligned filter vs. Rotated filter

h is the uniform DG element size, H is the kernel scaling

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SLIDE 21

SIAC FILTER: DIVIDED DIFFERENCE ESTIMATES

➤ We need to worry about: ➤ Requires: ➤ Relating coordinate-aligned derivatives with arc-length

derivatives

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k∂α

H(u uh)k−(m+1)

Dα e ψ(m+1)(x, y) = ∂α1 ∂xα1 ∂α2 ∂yα2 e ψ(m+1)(x, y) = cosα1 θ sinα2 θ d|α| dt|α| ψ(m+1)(t) = cosα1 θ sinα2 θ∂|α|

H ψ(m+1−α)

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SLIDE 22

SIAC FILTER: DIRECTIONAL DIVIDED DIFFERENCE ESTIMATES

➤ Need to relate directional divided differences to coordinate-

aligned divided differences.

➤ Direction vector: u=(ux,uy). ➤ Scaled directional divided difference with respect to u: ➤ a-th directional divided difference:

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∂u,Hf(t) = 1 H ✓ f ✓ x + H 2 ux, y + H 2 uy ◆ − f ✓ x − H 2 ux, y − H 2 uy ◆◆ =∂ux,Hf ✓ x, y + H 2 uy ◆ + ∂uy,Hf ✓ x − H 2 ux, y ◆ . ∂α

u,Hf(x, y) = ∂u,H

  • ∂α−1

u,H f(x, y)

  • ,

α > 1.

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SLIDE 23

SIAC FILTER: DIVIDED DIFFERENCE ESTIMATES

➤ We can relate the directional divided difference to the

coordinate aligned divided difference

➤ As long as q s 0, p/2, then we have superconvergence! ➤ Leads to a reduced error constant.

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D↵ψ(`)(t) = (cos θ)↵x(sin θ)↵y ∂↵

h ψ(`−↵)(x)

coordinate aligned

kKH ⇤ (u uh)k0  cosα1 θ sinα2 θ C X

|α|≤m+1

kKHk1 k∂α

H(u uh)k−(m+1)

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SLIDE 24

LINE SIAC FILTER FOR MULTI-DIMENSIONAL FILTERING

➤ Reducing the support to a line: axis-aligned vs. rotated

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Tensor Product SIAC filter Line SIAC filter

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SLIDE 25

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DG Errors Tensor Product Filter Line Filter p/4 rotation Line Filter 3p/4 rotation

LINE SIAC FILTER FOR MULTI-DIMENSIONAL FILTERING

SISC (2017)

ut + ux + uy = 0, u(x, y, 0) = sin(x + y)

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SLIDE 26

LINE SIAC FILTER FOR MULTI-DIMENSIONAL FILTERING

➤ Numerical test: 2D advection equation, u(x,u,0)=sin(x+y) ➤ Superconvergence and error reduction!

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Unfiltered Line Filtering 2D Filter θ = 3π/4 θ = π/4 θ = 0 N L2-error Order L2-error Order L2-error Order L2-error P1 20 9.7e-03

  • 1.5e-03
  • 2.7e-03
  • 1.6e-03

40 2.4e-03 2.02 1.9e-04 2.98 2.6e-04 3.33 2.0e-04 80 5.9e-04 2.01 2.4e-05 2.99 2.8e-05 3.21 P2 20 2.4e-04

  • 1.5e-06
  • 1.4e-04
  • 6.1e-06

40 2.9e-05 3.01 4.7e-08 4.99 2.3e-06 5.91 1.2e-07 80 3.6e-06 3.01 1.5e-09 5.00 3.7e-08 5.95

  • P3

20 4.5e-06

  • 7.7e-10
  • 1.6e-05
  • 1.4e-07

40 2.8e-07 4.01 6.9e-12 6.79 6.9e-08 7.87 5.6e-10

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SLIDE 27

LSIAC FILTER: SMOOTHNESS

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SLIDE 28

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LINE SIAC FILTER FOR MULTI-DIMENSIONAL FILTERING

ut + ux + uy = 0, u(x, y, 0) = sin(x) cos(y)

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SLIDE 29

LINE SIAC FILTER FOR MULTI-DIMENSIONAL FILTERING

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ut + ux + uy = 0, u(x, y, 0) = sin(x) cos(y)

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SLIDE 30

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LINE SIAC FILTER FOR MULTI-DIMENSIONAL FILTERING

ut + ✓u2 2 ◆

x

+ ✓u2 2 ◆

y

= 0, u(x, y, 0) = 2 + 1 2 sin(x + y).

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SLIDE 31

LINE SIAC FILTER FOR MULTI-DIMENSIONAL FILTERING

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ut + ✓u2 2 ◆

x

+ ✓u2 2 ◆

y

= 0, u(x, y, 0) = 2 + 1 2 sin(x + y).

Nonlinear!

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SLIDE 32

LSIAC FILTER: COMPUTATIONAL COST

➤ Total operations per point. ➤ Elapsed time per point.

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Filter Type Intersection Scans Integrals Quadrature Sums Line Filter 4 10 10 2D Rotated Filter 64 93 8649 2D No Rotation 64 63 3969

  • No. of Splines and degree

Line Filter 2D Rotated Filter 2D No Rotation 3, 1 0.09 0.87 0.68 5, 2 0.35 3.49 2.60 7, 3 0.41 10.42 6.75

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SLIDE 33

SUMMARY

➤ A Line SIAC filter can be applied for multi-dimensional data ➤ Reduces error ➤ Increases smoothness in all directions ➤ Reduced computational cost ➤ Improves the convergence rate from p+1 to 2p+1 ➤ Requires choosing the rotation wisely. ➤ Essential to have the appropriate divided difference estimates. ➤ Can generalise to higher dimensions given the appropriate

parameterisation.

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SLIDE 34

REFERENCES

Basic properties of dg and beyond

  • 1. J.H. Bramble and A.H. Schatz, ” Higher order local accuracy by averaging in the finite element

method”, Mathematics of Computation, 31 (1977), pp.94–111.

  • 2. B. Cockburn, M. Luskin, C.-W. Shu, and E. Süli, ” Enhanced accuracy by post-processing for finite

element methods for hyperbolic equations”, Mathematics of Computation, 72 (2003), pp.577–606.

  • 3. J. Docampo Sánchez, J.K. Ryan, M. Mirzargar, and R.M. Kirby, "Multi-dimensional Filtering: Reducing

the dimension through rotation." SIAM Journal on Scientific Computing, awaiting publication.

  • 4. J. King, H. Mirzaee, J.K. Ryan, and R.M. Kirby, ”Smoothness-Increasing Accuracy-Conserving (SIAC)

Filtering for discontinuous Galerkin Solutions: Improved Errors Versus Higher-Order Accuracy”, Journal

  • f Scientific Computing, 53 (2012), 129–149.
  • 5. H. Mirzaee, J.K. Ryan, and R.M. Kirby, “Efficient Implementation of Smoothness-Increasing Accuracy-

Conserving (SIAC) Filters for Discontinuous Galerkin Solutions”, Journal of Scientific Computing, vol. 52 (2012), pp. 85–112.

  • 6. M. Mirzargar, J.K. Ryan and R.M. Kirby, "Smoothness-Increasing Accuracy-Conserving (SIAC) Filtering

and Quasi-Interpolation: A Unified View." Journal of Scientific Computing, 67 (2016) pp 237--261.

  • 7. M.S. Mock and P

.D. Lax, ”The computation of discontinuous solutions of linear hyperbolic equations”, Communications on Pure and Applied Mathematics, 31 (1978), pp.423–430.

  • 8. J.K. Ryan, "Exploiting Superconvergence through Smoothness-Increasing Accuracy-Conserving (SIAC)

Filtering”, Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2014, Salt Lake City, Utah. Lecture Notes in Computational Science and Engineering, Springer, 106 (2015), pp 87–102.

  • 9. V. Thomee, ” High order local approximations to derivatives in the finite element method”,

Mathematics of Computation, 31 (1977), pp. 652–660.