Radial Basis Function Collocation Method

Alexander ChengUniversity of Mississippi

Keynote presented at Methods of Fundamental SolutionsAyia Napa, Cyprus, June 11-13, 2007

Collaborators: Zi-Cai LeeJason HuangEdward KansaJaime Cabral

An Engineer’s Apology

• No proof (only numerical experiments)• Sales pitch (like a used car salesman)• This is not a MFS presentation

Why Collocation Method?

• Accuracy• Simplicity• Solve ill-posed BVP directly• Meshless

Accuracy

Accuracy

• Its accuracy is impossible to match by FEM or FDM.

• In an example solving Poisson equation, an accuracy of the order 10-16 is reached using a 20x20 grid.

To Make It Dramatic

• Assume that in an initial mesh, FEM/FDM can solve to an accuracy of 1%.

• Using a quadratic element or central difference, the error estimate is h2

• To reach an accuracy of 10-16, h needs to be refined 107

fold• In a 3D problem, this means 1021 fold more degrees of

freedom• The full matrix is of the size 1042

• The effort of solution could be 1063 fold• If the original CPU is 0.01 sec, this requires 1054 years• The age of universe is 2 x 1010 years

Why Is Collocation Method So Accurate?

• First, It is not supposed to be the most accurate method, Galerkin method is better.

• It beats FEM/FDM because it uses global, rather than local, interpolation.

• Let’s look at an example.

Method of Weighted Residuals

• Governing equation

• Essential and natural boundary conditions

Minimizing Weighted Residual• Approximation

• Satisfying governing equation

• Satisfying boundary conditions

Galerkin Method

• Weight: Wi = Ni

• Satisfying governing equation

Collocation Method

• Weight

• Collocate for governing equation

• Collocate for boundary condition

Other Methods

• Subdomain method• Moving least square method• Trefftz method• Method of fundamental solutions• Etc.

A Simple Example

Solution Strategy

• Approximate solution

• Solved by collocation, subdomain, and Galerkin method

• Integration performed exactly

Solution Error

Comments

• Galerkin method is the most accurate, collocation the least.

• Integration tends to distribute the error and point collocation concentrates the error.

• So why collocation method is the best?

Errors

• Approximation (truncation) error• Round off error • Integration error

In Multi-Dimensional Geometry

• Integration requires the division of domain into cells

• Geometric approximation error O(hn)• Quadrature error O(hn)• When truncation error decreases

(exponentially for collocation method), the above errors dominate.

• Hence point collocation is the most accurate!

FEM Mistake

• FEM made a mistake by switching from global interpolation to local interpolation.

• It uses low degree polynomials within each element.

• Linear interpolation needs weak formulation• Higher degree interpolation still violates the

derivative requirement on edges of elements• It can never do better than hn error convergence,

while collocation method can have exponential error convergence.

Example — Poisson Equation

Exact SolutionHbL

00.25

0.50.75

1x0

0.25

0.5

0.75

1

y

-0.2

0

0.2

u

00.25

0.50.75

1x

MQ and FD Solutions

Simplicity

BVP (Well and Ill-Posed)

• Governing equation

• Boundary condition

• Interior condition

Well- and Ill-Posed BVP

• Well-posed problem

and

• Ill-posed problem

or

• Assume the approximate solution is given by

( )φ x( )φ xChoices of Basis Functions

• Monomial• Chebyshev polynomial• Fourier series• Wavelet• Fundamental solutions• Non-singular solution (Trefftz)• Radial basis function

Point Collocation

Examples of Ill-Posed Problems

• Harbor wave field

Examples

• Groundwater field

Examples

• Geoprospecting

Examples

• Non-Destructive Testing

Problem 1: Onishi (1995) FEM

• Governing equation

• BC

• Internal values: u(1,1) = 0, u(2,1) = 3, u(1,2) = -3, u(2,2) = 0

Problem 1

• Onishi (1995), FEM– Governing equation

– BC

– Internal values: u(1,1) = 0, u(2,1) = 3, u(1,2) = -3, u(2,2) = 0

Collocation Nodes

Result

Problem 2: Lesnic (1998) BEM

Lesnic Result

RBF Result

Problem 3: “Groundwater”Exact potential9 data given

Kriging

Solving Ill-Posed Problem

Error Estimate(Experimental)

Test Problem

Solution method

• Approximation by inverse multiquadric

Watch out for the “c”

What Is the Role of c?

• People observe that as c increases, error decreases

• It is generally believed that as c →∞, ε →0 • If this is true, we have a dream method:

higher and higher precision without paying a price

• However, matrix ill-condition gets in the way; the dream cannot be fulfilled.

What Is the Role of c?

• What if we can compute with infinite precision?

• Then, is it true that as c →∞, ε →0?• (Or, is it true that for MFS, as R →∞, ε→0?)

Error Estimate

• 0 < λ < 1, a > 0• What is our clue?

MQ Interpolation Error Estimate

• for interpolation of smooth function: – Madych & Nelson [1992]: O(λ(1/h)), 0<λ<1– Madych [1992]: O(eac λ(c/h)), 0<λ<1– Wendland [2001]: O(λ◊(1/h)), 0<λ<1

• for solving PDE– O(???)

• Use 6x6 mesh (h = 0.2, 4x4 interior collocation)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1collocation nodes (x-governing eqn, o-bc)

Result: h = 1/5

Result: h = 1/10

Find Error Estimate Constants by Data Fitting

Error Estimate Fit

Optimal c

• If the error estimate

is true, then there exists an optimal cwhere error is minimum

Revised Error Estimate

• If we can always use optimal c with a given mesh, what is the new error estimate?

Effective Error Estimate If cmax Is Used

• h = 1/5, ε ~ 10-2

• h = 1/10, ε ~ 10-6

• h = 1/20, ε ~ 10-15

But Wait!

• How can I find optimal c, given a realproblem and a mesh?

• I need the two constants λ and a.• I cannot get them by fitting the error data

because I do not know the error!• What is the remedy?• Use residual error to estimate!

Residual Error

• Residual error is not solution error!• It is however proportional to solution error

by a (big and unknown) constant

Solution Strategy

• Select a mesh, find solution with a given c.• Calculate residual on a sparse monitoring

grid.• Fix the mesh, solve the problem again with

two more c values and calculate residual.• a and ln λ can be determined from the

following formula with 3 data points

Result

Comment

• Residual error and solution error have the same trend.

• Three point fit produces a reasonable prediction of residual error.

• It captures the optimal c value.• That c will be used in the computation to

produce the best result with that mesh.• The actual error with that computation is

still not known!

Conclusion

• For a continuous function, global basis function is much, much better than local, piecewise continuity interpolation.

• Point collocation is better than weighted integration.

• Some basis functions, such as multiquadric and Gaussian (fundamental solution too), have exponential convergence property.

• For basis functions that have a parameter to adjust its shape, the flatter the basis function is, the more accurate the solution.

• Varying the parameter, instead of refining the mesh, is the better strategy to achieve better accuracy, as it is without cost.

• The error does not goes to zero by pushing the parameter value, as there is an error bound for a given mesh.

• Once that limit is reached, the mesh needs to be refined, then the parameter value can be pushed further.

• The combined error convergence rate is AMAZING!

• Practical way that utilizes the residual error can be devised to predict the optimal parameter value, thus achieving optima accuracy.

Final Comment

• Is infinite (high) precision computation possible?

• Mathematica, Fortran, and C code exist.