radial basis functions: developments and applications to...
TRANSCRIPT
Radial Basis Functions:
Developments and applications toplanetary scale flows
Natasha Flyer NCAR, IMAGeInstitute for Mathematics Applied to the Geosciences
in collaboration with, and presented by
Bengt Fornberg University of Colorado Boulder, Department of Applied Mathematics
Slide 1 of 27
Presentation outline:
Slides Topic
3 Finite Difference (FD) approximations
4 - 5 Pseudospectral (PS) approximations in 1-D and 2-D
6 - 8 RBF introduction
9 - 10 Flat RBFs - why are they interesting?
11 - 12 Numerically stable algorithms: Contour-Padé and RBF -QR
13 - 14 PS vs. RBF derivative approximations
15 - 17 RBF interpolation and PDE solution on a sphere
18 - 19 Vortex roll-up test case
20 - 22 Shallow water equations on a sphere
23 - 26 RBFs for computing mantle convection
27 Conclusions
Slide 2 of 27
Finite difference methods: Limits of increasing o rders exist
Approximate derivatives by local difference formulas.
Examples of some explicit FD formulas on an equispa ced grid:
[1]ØuØx = [ − 1
2 0 12 ] u
h + O(h2)= [ 1
12 − 23 0 2
3 − 112 ] u
h + O(h4)= [ − 1
603
20 − 34 0 3
4 − 320
160 ] u
h + O(h6)o o o o o o o
[¢ − 13
12 − 1 0 1 − 1
213 ¢] u
h PS limit
Slide 3 of 27
Pseudospectral (PS) methods can be seen as limits of increasing order FD methods
Periodic problem - Fourier-PS Method:
Use equispaced grid: We get exactly the same derivative approximations at nodes if we:
i. Extend periodically, and then apply limiting FD method,ii. Find interpolating trig polynomial by FFT, take its analytic derivative. O(N log N) op. for N points
Non-Periodic Problem - Chebyshev-PS Method:
Runge Phenomenon Does not arise if the nodes are suitably clustered near the boundaries.Example: f(x) = 1/(1+16x2)
Equispaced nodes Chebyshev-spaced nodes
i. Create global FD formulas separately for each node point, O(N 2) operations for N pointsii. Get Chebyshev polynomial expansion (by FFT), take its analytic derivative; O(N log N) operations.
Slide 4 of 27
How do PS methods work in more than 1-D ?
2-D:
They generalize to Tensor product type grids
BUT
NO set of basis functions can guarantee a nonsingular system in the case of scattered nodes
Scattered nodes Interpolant s(x) = �k=0
N
ck Tk(x)System that determines the expansion coefficients:
T0(x0) T1(x
0) £ TN(x
0)
T0(x1) T1(x
1) £ TN(x
1)
§ § • §
T0(xN
) T1(xN) £ TN(x
N)
c0
c1
§
cN
=
f0
f1
§
fN
Move two nodes so they exchange locations:
Two rows become interchanged, determinant changes sign
fl determinant is zero somewhere along the way.
Slide 5 of 27
RBF idea, In pictures:
1970 Invention of RBFs (for application in cartography)
Some other key dates:
1940 Unconditional non-singularity for many types of radial functions1984 Unconditional non-singularity for multiquadrics ( )�(r) = 1 + (� r )2
1990 First application to numerical solutions of PDEs2002 Flat RBF limit exists - generalizes all 'classical' pseudospectral methods 2004 First numerically stable algorithm in flat basis function limit2007 First application of RBFs to large-scale geophysical test problems2009 A new physical phenomenon (a certain mantle convection instability) initially discovered
in an RBF calculation
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RBF idea, In formulas:
Given scattered data (xk , fk), k = 1, 2, ... , N, in d-D, the coefficients in the interpolant�k
s(x) = �k=1
N
�k �(|| x − xk||)
are found by collocation: , k = 1, 2, ... , N :s(xk) = fk
�(||x1
− x1||) �(||x
1− x
2||) £ �(||x1 − x
N||)
�(||x2
− x1||) �(||x
2− x
2||) £ �(||x
2− x
N||)
§ § •§
�(||xN
− x1||) �(||x
N− x
2||)£ �(||x
N− x
N||)
�1
§
§
�N
=
f1
§
§
fN
Two key theorems:
- For most radial functions , this system can never be singular�(r)
- The interpolation is spectrally accurate for smooth radial functions
Slide 7 of 27
Many types of RBF s:
Piecewise smooth φ(r) Infinitely smooth φ(r)
Cubics TP splines Multiquadric Gaussian Inverse quadratic
r3 r2 log r 1 + (� r)2 e−(� r)2 11 + (� r)2
Interpolation guaranteed nonsingular forall commonly used radial functions.
- Piecewise smooth RBFs:- algebraic accuracy (cf. cubic splines)- minimize 'wiggles'- basis function can have compact support
- Infinitely smooth RBFs:- spectral accuracy (if no Runge phenomenon)
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Numerical conditioning, and the flat RBF limit (ε →ε →ε →ε → 0)
Classical basis functions are usually RBFs are translates of onehighly oscillatory single function - here �(r) = e−(� r)2
In case of 41 scattered nodes in 1-D: cond(A) = , det(A) = .O(�−80) O(�1640)
2-D: cond(A) = , det(A) = .O(�−16) O(�416)
Exact formulas available for any number of nodes in any number of dimensions(Fornberg and Zuev, 2007)
Extreme ill-conditioning typical as ε → 0.
Slide 9 of 27
Why are flat (or near-flat) RBFs interesting ?
- Intriguing error trends as ε → 0'Toy-problem' example: 41 node MQ interpolation of f(x1,x2) = 59
67 + (x1 + 17 )2 + (x2 − 1
11 )
- RBF interpolant in 1-D reduces to Lagrange's interpolation polynomial (Driscoll and Fornberg, 2002)
- The ε → 0 limit reduces to 'classical' PS methods if used on tensor type grids.
- The RBF approach generalize PS methods in many ways:- Guaranteed nonsingular also for scattered nodes on irregular geometries- Allow spectral accuracy to be combined with mesh refinement- Best accuracy often obtained for non-zero ε.
Solving followed by evaluating A� = f s(x, �) =�k=1N�k �(||x − x
k||)
is merely an unstable algorithm for a stable problem
Slide 10 of 27
Numerical computations for small values of εεεε (near-flat RBFs)
It is possible to create algorithms that completely bypass ill-conditioning all the wayinto ε→ 0 limit, while using only standard precision arithmetic:
Concept: Find a computational path from f to s(x,ε) that does not go via theill-conditioned λ.
- Contour-Padé algorithm First algorithm of its kind; established that the concept is possible
Limited to relatively small N-values (Fornberg and Wright, 2004)
Improved versions under development (Fornberg, Wright, and Trefethen).
- RBF-QR method Initially developed for nodes scattered over the surface of a sphereNo limit on N; cost about five times that of RBF-Direct
Original version for nodes on sphere (Fornberg and Piret, 2007).Version for 2-D developed (Fornberg, Larsson, and Flyer, 2009).Version for 3-D under development (Fornberg and Larsson).
Probably many more completely stable algorithms to come
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The concept for the RBF-QR method Recognize that the existence of an ill-conditioned basis does not imply that the spanned space is bad.
Ex. 1: 3-D space Ex. 2: Polynomials of degree ≤≤≤≤ 100
Bad Basis Good Basis
xn , n = 0, 1,¢, 100 Tn(x) , n = 0, 1,¢, 100Bad Basis Good Basis
Ex. 3: Space spanned by RBFs in their flat limit� d 0- The spanned space turns out to be excellent
for computational work - just the basis that isbad.
- Is there any Good Basis in exactly the same space?
- RBF-QR finds such a basis through someanalytical expansions, leading to numerical steps that all remain completely stable evenin the flat basis function limit.
Bad Basis
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-1-0.5
00.5
10
2
4
10
20
50
100
-1
0
1
n
x
y
-1-0.5
0
0.5
10
2
4
10
20
50
100
0
0.5
1
n
x
y
-1-0.5
0
0.51
0
2
4
10
20
50
100
0
0.5
1
n
x
y
PS vs. RBF derivative approximations (Fornberg, Flyer, Russell, 2008)
There is something strange about how FD Where should pick up its data from?Ø
Øxand PS methods approximate Answer: Very heuristic analysis suggests1
2Ø
Øx + ØØy
something reminiscent of the
function x8 0F1(3,− 14 (x2 + y2))
Even on a lattice, RBFs typically pick up
information for also from beside theØ
Øxx-axis.
Slide 13 of 27
→x
↑ y
The main strength of RBFs is their geometric flexib ility. However, even in a pure 2-Dperiodic geometry, they can approximate just as wel l or even better than Fourier-PS
Example: Interpolation errors in a 2-D periodic test case
Cartesian : Fourier - PS
Other three cases : RBFs on different node layouts
Spectral convergence in all cases. Slide 14 of 27
Interpolation on a sphere - RBF-Direct vs. RBF-QR (Fornberg and Piret, 2007)
Test function: 1849 minimal energy nodes Errors as function of εεεε
f(x) = e−7(x+ 12 )2−8(y+ 1
2 )2−9(z− 12
)2
RBF-Direct: with the obtained by solving .s(x) = �k=1
N
�k �(|| x − xk||) �k A� = f
cond(A) = O(ε -84)
To lower the critical ε by a factor of 100 by means of high precision arithmetic requiresthe numerical precision to be raised from standard 16 digits to close to 1000 digits.
RBF-QR: With the new basis functions in exactly the same approximation space,cond(A') = O(1).
Slide 15 of 27
Solving PDEs on a sphere
Governing PDE:
⇒ØuØt + (cos � − tan � sin� sin �) Øu
Ø�− cos� sin � Øu
�= 0
RBF collocation of PDE in spherical coordinates eliminatesall coordinate-based singularities
Spectrally accurate numerical methods: DF: Double Fourier series
RBF: Radial Basis Functions Below: 1849 minimum energy nodes
SE: Spectral ElementsImplemented by means of cubed sphere
SPH: Spherical HarmonicsCollocation with orthogonal set of trig-like basis functions, which lead to entirely uniformresolution over surface of sphere
Slide 16 of 27
Convective flow over a sphere - Comparison between methods (Flyer and Wright, 2007) (10- 3)
Snapshot of 4096 node RBF calculation after 12 days Error (parts per thousand)(1 full revolution) of a cosine bell
yes> 10006 minutes7,7760.005Spectral elementsSEno> 10090 seconds32,7680.005Double FourierDFno> 50090 seconds32,7680.005Spherical harmonicsSPHyes< 401/2 hour4,0960.006Radial basis functionsRBF
Local refine-ment feasible
Code length(lines)
Time step(RK4)
Number of nodepoints/
free parameters
l2 errorMethod
(Fornberg and Piret, 2008): Time stepping thousands of full revolutions and using RBF-QR - still nosignificant error growth or trailing dispersive waves.
Slide 17 of 27
Moving Vortex Roll-Up on A Sphere: Local Node Refin ement (Flyer and Lehto, 2009)
Initial condition Solution after 12 daysLinear convectionwith a vortex-likeflow field
Numerical implementation Minimal energy (ME) nodes Refined nodes
IMQ RBFs: �(r) = 11 + �2r2
N = 3136 nodes (1849 shown in figures to the right)
Method of lines (MOL) time stepping with standardRunge-Kutta, 4th order
Slide 18 of 27
Numerical RBF solution at 12 days Error at 12 days (10- 4)
2 ⋅ 10 -31-35 o - 0.625 o- Finite Volume (3 levels. lat-long)FV8 ⋅ 10 -520-3,136Radial basis functionsRBF
With local refinement7 ⋅ 10 -36-9,600Discontinuous GalerkinDG2 ⋅ 10 -330-38,400Finite Volume (cubed sphere)FV2 ⋅ 10 -3100.625 o165,888Finite Volume (lat-long grid)FV4 ⋅ 10 -3606.4 o3,136Radial basis functionsRBF
Without local refinementl2 errorMinutesTypical angularN (total)ErrorTime stepResolutionMethod
Slide 19 of 27
Nonlinear Unsteady Shallow Water Equations (Flyer and Wright, 2009)
Forcing terms added to the shallow water equations to generate a flow that mimics a shortwave trough embedded in a westerly jet.
Initial Velocity Field Initial geopotential height field(equivalent to pressure field)
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Errors after wave trough traveled one revolution
Total physical run time 5 days
Increase in error with time for different N nodes, N = 3136Numerical time step �t = 10 minutes,Error in max norm, white < 10−5
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Comparison with Commonly Used Methods and Benchmark s
Accuracy obtained with different methods, and numeri cal time step needed for stability
4.0 ⋅ 10 -545 seconds24,5766.5 ⋅ 10 -390 seconds6,144Spectral Element2.0 ⋅ 10 -33 minutes8,192 (1,849)Spherical Harmonic4.0 ⋅ 10 -490 seconds32,7688.2 ⋅ 10 -43 minutes8,1923.9 ⋅ 10 -16 minutes2,048Double Fourier1.0 ⋅ 10 -86 minutes5,0412.5 ⋅ 10 -7 8 minutes4,0968.8 ⋅ 10 -615 minutes3,1363.5 ⋅ 10 -324 minutes1,8494.8 ⋅ 10 -140 minutes784RBF
Relative l2 errorTime stepNMethod
Computational times for the RBF method , in Matlab on 2.66 GHz PC
24 minutes 0.605,041 6 minutes0.414,096 2 minutes0.253,13633 seconds0.111,849
5 seconds 0.03784Total RuntimeRuntime per time step (s)N
Slide 22 of 27
Thermal Convection in a 3-D Spherical Shell (Flyer and Wright, 2009)
Equations : (Ra = Rayleigh number; related to ratio of heat convection to heat conduction)
= $u = 0<u −=p + RaTe
r= 0
ØTØt + u $ =T = <T
Node Layout for hybrid RBF-Chebyshev discretization :
Slide 23 of 27
Ra = 7,000 test case
Initial condition N =1849 nodes on each spherical shellY40 + 5
7Y44
M = 31 shells, Perturbation temperature shownBlue - down flow, Yellow - up flow, Red - core
Comparisons against main previous results from the literature
0.2157831.08233.6096 57,319RBF-ChebyshevRBF-CH0.2157831.08213.6096 552,960Spherical harmonics -FDSH-FD0.2159431.02263.5983 663,552Finite volumeFV0.2159732.63083.4945 12,582,912Finite differencesFD0.2176 31.09 3.6254 393,216Finite elementsFE
T<VRMS >NuouterNo of nodesMethod
Slide 24 of 27
Example of computed solution for Ra = 500,000
Isosurfaces of perturbedtemperature:
Calculation on PC system;
Regime not reached withany alternative numericalmethod.
At somewhat lower Ra numbers, a similar RBF calculation revealed anunexpected physical instability, afterwards confirmed on the Japanese EarthSimulator.
Slide 25 of 27
What is next in modeling geophysical flows with RBF s?
Some things currently on the radar:
1) Dynamically adaptive node refinement
2) RBF-FD, implemented on scattered nodes (create FD formulas are exact forRBFs rather than for polynomials)
3) Carry out RBF-FD analysis, and devise hyperviscosity-type local smoothingoperators.
4) Implement RBF codes on parallel Graphical Processing Units (GPUs), and lateron other parallel architectures.
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Conclusions
Established:
- RBFs can be seen as a generalization of PS methods to arbitrarily shaped domains.
- RBFs combine spectral accuracy with flexible opportunities for local node refinement.
- RBFs can offer excellent accuracy also over very long integration times.
- The near-flat basis function regime (ε small) has been found to be of particular interest,and genuinely stable numerical algorithms have been developed.
- For certain classes of convective flow problems, RBF-based solutions are more accurateand more cost-effective than any other numerical method.
Current research issues:
- Compare RBFs against alternative methods in more major applications.
- Explore further the combination of spectral accuracy with local node refinement.
- Find RBF algorithms that combine high speed with numerical stability (for small ε).
- Develop further the concept of RBF-FD (RBF-generated FD formulas for scattered nodecases ).
- Develop effective implementations first on GPUs, and later on massively parallel(peta-scale) computer hardware.
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