Radial Basis Functions mappings

We now turn to the analysis of the RBF mappings, introduced in Sect. 2.7. In particular, we state some conditions ensuring the well-posedness of the interpolation problem (2.65), and an a

with respect to the parameter µ and the set of interpolation points.

Since we do not use RBF mappings in shape optimization problems, we do not derive any condition ensuring that the norm of the displacement field obtained through a RBF mapping is bounded by a suitable norm of the parameter vector µ as in (4.24). Recalling from Sect. 2.7, a RBF mapping is given by

T (x; µ) = c(µ) +A(µ)x + W(µ)Ts(x), (4.25) being c(µ) ∈ R2, A(µ) ∈ R2×2, W(µ) ∈ Rk×2 the coefficients of the mapping. Recall that

s(x) ∈Rk is given by s(x) = (Φ(kx − P1k), . . . , Φ(kx − Pkk))T.

Vector and matrices components cm(µ), (A(µ))mn = Amn(µ) and (W(µ))im = wim(µ), for

1 ≤ m, n ≤ 2, 1 ≤ i ≤ k are obtained by solving the following linear systems:

 M P PT ₀   Wm(µ) γ_m(µ)  =  Pm_{+ µ}m 0  , m = 1, 2 (4.26)

where (for sake of analysis, notation slightly differs from (2.71)):

• _Wm(µ) = (w1m, (µ) . . . , wkm(µ))T and γm(µ) = (cm(µ), A1m(µ), A2m(µ))T are the coeffi-

cients related to radial and polynomial components to deformation, for the m-th coordinate;

• Pm_{= (P}

1m, . . . , Pkm)T the vector of the m-th coordinates of the k control points in the

undeformed, reference configuration; in the same way, Pi= (Pi1, Pi2)T for any 1 ≤ i ≤ k;

• µm_{= (µ}

1m, µkm) the displacement of the set of k control points in the m-th direction;

• M ∈ Rk×k_{the interpolation matrix, of components M}

ij = Φ(kPi−Pjk), for any 1 ≤ i, j ≤ k;

• _{P ∈ R}k×3 _{the matrix arising from the side constraints and given byP = [ 1 | P}1 _{| P}2_].

It is possible to show (see e.g. [50], Proposition 2.1) that the interpolation matrixM is in general symmetric and positive semi-definite; it is positive definite if Φ(r) = φ(r2), where φ :R+ →R

is a continuous completely monotonic function, i.e. φ : R+ → R, φ ∈ C∞(0, ∞) is such that

(−1)l_φ(l)_{(t) ≥ 0 for l = 0, 1, 2, . . . and for all positive t. In this case, the RBF interpolation}

problem admits a unique solution even without adding a polynomial function to the RBF map, provided that {Pi}ki=1⊂R2 is a set of k > 2 distinct, non-collinear control points. Then, if we

use e.g. Gaussian or Multiquadric RBFs (see table 2.1), the interpolation problem is well posed.

On the other hand, we can recover the possibility to solve the RBF interpolation problem even for function which are not completely monotonic, by relaxing the requirement on Φ(·) and the definition of positive definiteness. In fact, we can always interpolate uniquely with (strictly)

conditionally positive definite functions5_{if we add polynomials to the interpolant, and if the only}

polynomial that vanishes on the set of control points {Pi}ki=1 ⊂R

2 _{sites is zero.}

5_{A function F : R}d _{→ R is conditionally positive definite (of order k) if for all finite subsets Ξ ⊂ R}d_{, the} quadratic formP_ξ∈ΞP_ζ∈ΞλξλζF (ξ − ζ) is nonnegative for all λ = {λξ}ξ∈Ξwhich satisfiesP_ξ∈Ξλξq(ξ) = 0 for any polynomial functions q ∈ Πk−1_d of total degree at most k − 1 in d unknowns. In our case, F (x) = Φ(kxk) and it can be proved that Φ(kxk) is strictly conditionally positive definite (of order k) if (−1)k_dk_/dtk_φ(√_{t), t > 0} is completely monotonic (see e.g. [50], Theorem 5.1). Common choices of RBFs satisfy this requirement wit k = 2, so that adding a polynomial of degree k − 1 = 1 is sufficient.

The side constraints in (4.26) are used to take up the extra degrees of freedom introduced through the use of the polynomial π(x) ∈ Π1

2. Moreover, we can prove the following result:

Proposition 4.15. The RBF mapping (4.25) satisfies the following properties:

1. T (x; µ) = x + θ(x; µ), where θ(x; µ) = 0 if and only if µ = 0; 2. the following a priori estimate holds:

1 κ2(K) kPm_k 2 kµm_k 2 ≤ (kγ_m(µ)k2+ kWm(µ)k2)1/2≤ κ2(K) kµm_k 2 kPm_k 2 , (4.27) where κ2(K) = maxiσi(K) miniσi(K)

is the spectral condition number ofK =

 M P PT ₀



.

Proof. First of all, we can rewrite expression (4.25) as follows: T (x; µ) = x + c(µ) + (A(µ) − I)x + W(µ)Ts(x).

It is easy to check that, since P = [ 1 | P1_{| P}2_{], the solution of the linear systems (4.26) with}

µm_{= 0 is given by W}

m(0) = 0 for m = 1, 2 and γ1(0) = [0, 1, 0]T, γ2(0) = [0, 0, 1]T, i.e. we

haveA(0) = I, c(0) = 0, W(0) = 0, provided that K is nonsingular. To prove this, let us remark that:

• if_{M is symmetric positive definite, K has a saddle-point structure and is nonsingular if and} only ifP has full column rank, i.e. if and only if rank(P) = 3. In the present case, this condition is verified if and only if the k > 2 control points are non-collinear. In fact,K is congruent to the block diagonal diag(M, S), where S = −PT_{MP is the Schur complement,}

so thatK is nonsingular if and only if S is nonsingular, provided that M is nonsingular. • if_{M is symmetric but positive semi-definite, we can rely on theorem 5.1 in [50].}

To prove the a priori estimate (4.27), we simply apply a basic result in numerical linear algebra (see e.g. [258], Theorem 3.2) for the stability analysis of a linear systemKx = b, b 6= 0:

K(x + δx) = b + δb ⇒ _κ(1_K)kδbk_kbk ≤kδxk kxk ≤ κ(K) kδbk kbk , by taking, for m = 1, 2: b =  Pm 0  , δb =  µm 0  , x =  Wm(0) γ_m(0)  , δx =  Wm(µ) −Wm(0) γ_m(µ) − γ_m(0)  .

In this way, by taking the Euclidean norm k · k2 vectors (and the induced matrix norm to get the

spectral condition number κ2(K), the estimate directly follows.

We point out that some relationships between the spectral condition numbers of the interpolation matrix κ2(M) and of the global matrix κ2(K) can also be derived, by exploiting the theory of

saddle-point problems (see e.g. [29], Theorem 3.5). In this way, we can characterize the stability

estimate (4.27) by means of the spectral condition number κ2(M) of M.

problem, related to location/number of control points and radial basis. In particular, upper bounds under the following form can be established:

κ2(M) ≤ k

maxi6=j|Φ(kPi− Pj))|

G(q) , q =

1

2mini6=j kPi− Pjk2,

where q is the separation radius of the control points positions and G(·) a real function depending on the radial basis; e.g., for the Gaussian RBF Φ(h) = e−h2 we have G(h) = h−de−γ/h2, for the spline-type Φ(h) = hλ_{we have G(h) = h}λ_{. See e.g. [50], Chapter 5 or [291] and references therein}

for further details and a general setting.

For a small number of control points, as in our approach, linear systems (4.26) can be efficiently solved by a suitable direct method (matrix factorization is not depending on the parameters). When using a large number of control points – as for example in fluid-structure interaction coupled problems or, more generally, when dealing with mesh motion through RBF – the matrix appearing in (4.26) may be badly conditioned and non-sparse (because of the global character of RBF), and some difficulties may arise. In these cases, suitable scaling or preconditioning strategies may help, as discussed for example in [50]. See also [155] for theoretical construction of shape deformations based on RBFs for mesh deformations and the works by Schaback and Wendland (e.g., [321, 293]) for computational aspects related to RBFs interpolation.