Summary of Methods and Applications
2.4 Applications to PDEs
Perhaps the most important concrete example of applications is the use of ra- dial basis functions for solving partial differential equations. These methods
30 2. Summary of methods
are particularly interesting when nonlinear partial differential equations are solved and/or nongrid approaches are used, e.g. because of nonsmooth do- main boundaries, where nonuniform knot placement is important to mod- elling the solution to good accuracy. This is no contradiction to our analysis above, where equal spacing was chosen merely for the purpose of theoret- ical analysis. As we shall see in the next chapter, there are, for more than two or three-dimensions, not many alternative methods that allow nongrid approaches.
Two ways to approach the numerical solution of elliptic boundary value prob- lems are by collocation and by the dual reciprocity method. We begin with a description of the collocation approach. This involves an important decision whether to use the well-known, standard globally supported radial basis func- tions such as multiquadrics or the new compactly supported ones which are described in Chapter 6 of this book. Since the approximation properties of the latter are not as good as the former ones, unless multilevel methods (Floater and Iske, 1996) are used, we have a trade-off between accuracy on one hand and sparsity of the collocation matrix on the other hand. Compactly supported ones give, if scaled suitably, banded collocation matrices while the globally supported ones give dense matrices. When we use the compactly supported radial basis functions we have, in fact, another trade-off, because even their scaling pits accuracy against population of the matrix. We will come back to those important questions later in the book.
One typical partial differential equation problem suitable for collocation techniques reads
Lu( x ) = f ( x ), x ∈ ⊂ Rn,
Bu|∂ = q,
where is a domain with suitably smooth, e.g. Lipschitz-continuous, boundary
∂ and f ,q are prescribed functions. The L is a linear differential operator and
B a boundary operator acting on functions defined on∂. Often, B is just point evaluation (this gives rise to the so-called Dirichlet problem) on the boundary or taking normal derivatives (for Neumann problems). We will come to nonlinear examples soon in the context of boundary element techniques.
For centres that are partitioned into two disjoint sets 1 and 2, the former from the domain, the latter from its boundary, the usual approach to collocation is to solve the so-called Hermite interpolation system
ξ uh = f (ξ ), ξ ∈ 1,
2.4 Applications 31
which involves both derivatives of different degrees and function evaluations. The approximants uh are defined by the sums
uh( x ) =
ξ ∈1cξ ξ φ( x − ξ ) +
ζ ∈2d ζ ζ φ( x − ζ ).
The ξ and ζ are suitable functionals to describe our operators L and B on the discrete set of centres. This is usually done by discretisation, i.e. replacing derivatives by differences.
Thus we end up with a square symmetric system of linear equations whose collocation matrix is nonsingular if, for instance, the radial basis function is positive definite and the aforementioned linear functionals are linearly inde- pendent functionals in the dual space of the native space of the radial basis functions (see Chapter 5 for the details about ‘native spaces’ which is another name, commonly used in the literature, for the reproducing kernel semi-Hilbert spaces treated there).
An error estimate is given in Franke and Schaback (1998). For those error estimates, it has been noted that more smoothness of the radial basis func- tion is required than for a comparable finite element setting, but clearly, the radial basis function setting has the distinct advantage of availability in any dimension and the absence of grids or triangulations which take much time to compute.
If a compactly supported radial basis function is used, it is possible to scale so that the matrix is a multiple of the identity matrix, but then the approximation quality will necessarily be bad. In fact, the conditioning of the collocation matrix is also affected which becomes worse the smaller the scaling η is with
φ(·/η) being used as scaled radial basis function. A Jacobi preconditioning by the diagonal values helps here, so the matrix A is replaced by P−1AP−1 where
P =
diag(A) (Fasshauer, 1999).We now outline the second method, that is a boundary element method (BEM). The dual reciprocity method as in Pollandt (1997) uses the second Green formula and a fundamental solution φ( · ) of the Laplace operator
= ∂ 2 ∂ x 12 + · · · + ∂2 ∂ x 2 n
to reformulate a boundary value problem as a boundary integral problem over a space of one dimension lower. No sparsity occurs in the linear systems that are solved when BEM are used, but this we are used to when applying noncompactly supported radial basis functions (see Chapter 7).
32 2. Summary of methods
The radial basis function that occurs in that context is this fundamental solu- tion, and, naturally, it is highly relevant in this case that the Laplace operator is rotationally invariant. We wish to give a very concrete practical example from Pollandt (1997), namely, for a nonlinear problem on a domain ⊂ Rn with
Dirichlet boundary conditions such as
u( x ) = u2( x ), x ∈ ⊂ Rn,
u|∂ = q,
one gets after two applications of Green’s formula (Forster, 1984) the equation on the boundary (where g will be defined below)
(2.4) 1 2
u( x ) − g( x )
+
∂
φ( x − y) ∂ ∂n y
u( y) − g( y)
−
u( y) − g( y)
∂ ∂n y φ( x − y)
d y = 0, x ∈ ∂, where ∂n∂y is the normal derivative with respect to y on y = = ∂. We will
later use (2.4) to approximate the boundary part of the solution, that is the part of the numerical solution which satisfies the boundary conditions. In order to define the function g which appears in (2.4), we have to assume that there are real coefficients λξ such that the – usually infinite – expansion (which will be
approximated by a finite series in an implementation) (2.5) u2( y) =
ξ ∈
λξ
φ ( y − ξ ), y ∈ ,holds, and set
g( y) =
ξ ∈
λξ
( y − ξ ), y ∈ ,so that g = u2 everywhere with no boundary conditions. Here
φ,
are suitable radial basis functions with the property that
( · ) =
φ( · ) and the centres ξ are from .The next goal is to approximate the solution u of the PDE on the domain by g which is expanded in radial basis functions plus a boundary term ˜r that satisfies r ˜ ≡ 0 on . To this end, we require that (2.4) holds at finitely many boundary points x = ζ j ∈ ∂, j = 1,2, . . . , t , only. Then we solve for the
coefficients λξ by requiring that (2.5) holds for all y ∈ . The points in ⊂
must be chosen so that the interpolation problem is solvable.
Therefore we have fixed theλξ by interpolation (collocation in the language of
differential equations), whereas (2.4) determines the normal derivative ∂n∂
2.4 Applications 33
on , where we are replacing ∂n∂
y u( y) by another approximant, a spline, say, as
in Chapter 3, call it τ ( y). Thus the spline is found by requiring (2.4) for all x =
ζ j ∈ , j = 1,2, . . . , t , and choosing a suitable t . Finally, an approximation
u( x ) to u( x ) is determined on by the identity
u( x ) := g( x ) +
(q( y) − g( y)) ∂ ∂n y φ( x − y)d y −
φ( x − y)
τ ( y) − ∂g( y)∂n y
d y, x ∈ ,where ˜r corresponds to the second and third terms on the right-hand side of the display (Pollandt, 1997).
Now, all expressions on the right-hand side are known. This is an outline of the approach but we have skipped several important details. Nonetheless, one can clearly see how radial basis functions appear in this algorithm; indeed, it is most natural to use them here, since many of them are fundamental so- lutions of the rotationally invariant Laplace operators in certain dimensions. In the above example and n = 2, φ(r ) = 2π1 logr ,
φ(r ) = r 2 logr (thin- plate splines) and
(r ) = 161 r 4 logr − 321 r 4 are the correct choices. An undesirable feature of those functions for this application, however, is their unbounded support because it makes it harder to solve the linear systems for the λξ etc., especially since in the approximative solution of partial differen- tial equations usually very many collocation points are used to get sufficient accuracy.One suitable approach to such problems that uses radial basis functions with compact support is with the ‘piecewise thin-plate spline’ that we shall describe now. With it, the general form of the thin-plate spline is retained as well as the nonsingularity of the interpolation matrix for nonuniform data. In fact, the interpolation matrix turns out to be positive definite. To describe our new radial basis functions, let φ be the radial basis function
(2.6) φ(r ) =
∞
0
(1 − r 2/β)λ+ (1 − βµ)ν+d β, r ≥ 0.
Here (·)t + is the so-called truncated power function which is zero for negative argument and (·)t for positive argument. From this we see immediately that supp φ = [0,1]; it can be scaled for other support sizes. An example with
µ = 12, ν = λ = 1 is (2.7) φ(r ) =
2r 2 logr − 4 3 r 3 + r 2 + 1 3, if r ≤ 1, 0 otherwise.34 2. Summary of methods
which explains why we have called φ a ‘piecewise thin-plate spline’. The pos- itive definiteness of the interpolation matrix follows from a theorem which is stated and established in full generality in Chapter 6.
We now state a few more mathematical applications explicitly where the methods turned out to be good. Casdagli (1989) for instance used them to interpolate componentwise functions F : Rn → Rn that have to be iterated to
simulate what is called a discrete dynamical system. In such experiments we especially seek the attractor of the discrete dynamical system that maps F :
R2 → R2. An example is the H´enon map
F ( x , y) = ( y,1 + bx − ay2)
(a and b being suitable parameters). Note that often in such mathematical applications, thedimension is much larger than two, so that radialbasis functions are very suitable.
Since F often is far too expensive to be evaluated more than a few times, the idea is to interpolate F by s and then iterate with s instead. For instance, if F
can reasonably be evaluated m times, beginning from a starting value ξ 0 ∈ Rn,
interpolation points
ξ 1 = F (ξ 0), ξ 2 = F (ξ 1), . . . , ξ m = F (ξ m−1)
are generated, and we let = {ξ j}m j =0. Then we wish to interpolate F by s on the basis of that set . We note that thus the points incan be highly irregularly distributed, and at any rate their positions are not foreseeable. Moreover it is usual in this kind of application that n is large. Therefore both spline and poly- nomial interpolation are immediately ruled out, whereas Casdagli notes that, e.g., interpolation by multiquadrics is very suitable and gives good approxima- tions to the short term and long term asymptotic behaviour of the dynamical system.
Hence radial basis functions are useful for such applications where inter- polation is required to arbitrarily distributed data sites. There is, so far, no comprehensive theoretical explanation of this particular successful application, but the numerical results are striking as documented in Casdagli (1989).
In summary, this chapter has presented several concepts fundamental to ra- dial basis functions and highly relevant to Chapters 4–10, namely complete monotonicity, positive definiteness, quasi- and cardinal interpolation, polyno- mial reproduction and convergence orders, localness of cardinal functions, ra- dial basis functions of compact support. Three of the principal tools that we use here, namely the Bernstein representation theorem, Wiener’s lemma, the
2.4 Applications 35
Poisson summation formula, are so central to our work that they will come up frequently in the later chapters as well.
In the following chapter we will show several other approaches to approxi- mation and interpolation of functions with many variables. The main purpose of that chapter is to enable the reader to contrast our approach with other possible methods.