6. Duchon splines

This procedure becomes very useful in the geometrical reconstruction of oil reservoirs (RS). Indeed, we reconstructed the geometry of RS in [12] by using B-spline surfaces. The technique exploits input mesh's simplicity to build a robust piecewise continuous geometrical representation using Bèzier bicubic patches. We manage the reservoir's topology with interpolation

Figure 18. An aircraft wing by translating an NACA profile accordingly.

Figure 19. A translational surface.

18 Topics in Splines and Applications

Figure 20. An interpolation surface.

In the context of applications in statistical analysis involving very high dimensional data sets, response surfaces are growing popularity. By running the simulations at a set of points (e.g., experimental design) and fitting response surfaces, i.e., splines, for instance, to the resulting input-output data that is characterized by sparsity, we can obtain fast surrogates for the objective function for optimization purposes [14, 15]. The appeal of the latter approach goes beyond reducing runtime. Since the method begins with experimental design, statistical analyses can be done to identify which input variables are the most important, and thus we can create "main effect plots" to visualize input-output relationships [14]. We must recognize interpolation methods in which the basis functions are fixed and those in which they have parameters that are tuned (e.g., kriging, which has a statistical interpretation that allows one to construct an estimate of the potential error in the interpolator). We refer the reader to [14, 15] for further reading.

There are different ways to approximate a function of several variables: multivariate piecewise polynomials, splines, and tensor product methods, among others. All these approaches have advantages and drawbacks, but if the rank of the linear system to solve may become large, a natural choice is radial basis functions, which are also useful in lower dimensional problems [14, 16, 17]. This may be particularly true if the input data is scattered, which excludes tensor product methods at first glance. Duchon splines are a class of positive definite and compactly supported radial functions, which consist of univariate polynomial within their support. It can be proven that they are of minimal degree and unique up to a constant factor, for given smoothness and space dimension [18]. They are particularly suitable to compute interpolants for very large scatter datasets [17].

Duchon splines, denoted herein as s, are defined by [17, 18]

$$s(\underline{x}) = \sum\_{j} \lambda\_{j} \cdot \varphi \left(\rho\_{j}\right) + p\_{n}(\underline{x}) \quad \text{;} \quad n = 2, 3$$

$$\rho\_{\dot{\gamma}} = ||\underline{x} - \underline{x}^{\prime}||$$

$$\varphi(\rho) = \rho^{2} \ln \rho,$$

where pnð Þx is a linear polynomial in two or three dimensions:

$$\begin{aligned} p\_2(\underline{\mathbf{x}}) &= a\mathbf{x} + by + c \\ p\_3(\underline{\mathbf{x}}) &= d\mathbf{x} + ey + f\underline{\mathbf{z}} + \underline{\mathbf{g}} \ ; \quad \lambda\_{\dot{\boldsymbol{\mu}}}, a \dots, \boldsymbol{\mathcal{g}} \in \mathbb{R} \end{aligned} \tag{37}$$

7. Concluding remarks

Acknowledgements

Author details

References

Horacio Florez\* and Belsay Borges

\*Address all correspondence to: florezg@gmail.com

4th ed. San Diego: Academic Press; 1993

Pittsburgh, Pennsylvania. 2001

versity of Texas at Austin. 2012

We presented a concise introduction to scalar and parametric spline interpolants. We introduced cubic and tension splines for scalar functions, and then we generalized them for the parametric case via Bèzier, B-spline, and NURBS curves. These latter entities are of the particular interest for applications in CAGD. We thus elaborated on topics such as inverse design and interpolation. We extended the treatment also to cover interpolation and translational surfaces with examples in mechanical and petroleum engineering. We wrapped up with the topic of interpolating sparse very high dimensional data sets via Duchon splines which are a kind of response surfaces suitable for applications in statistical analysis and optimization.

Scalar and Parametric Spline Curves and Surfaces http://dx.doi.org/10.5772/intechopen.74929 21

We acknowledge the financial support of the project "Reduced-Order Parameter Estimation

Computer Science Department, The University of Texas at El Paso, El Paso, USA

[1] Curtis F, Wheatley P. Applied Numerical Analysis. Pearson: USA; 2004

[2] Kincaid D, Cheney W. Numerical Analysis. USA: Brooks/Cole Publishing Company; 1991 [3] Atkinson K. An Introduction to Numerical Analysis. New York: John Wiley & Sons; 1978 [4] Thompson J, Weatherill N. Handbook of Grid Generation. CRC Press: Boca Raton; 1999 [5] Farin G. Curves and Surfaces for Computer-aided Geometric Design A Practical Guide.

[6] Florez H. A new method for building B-spline curves and its application to geometry design and structured grid generation. In: ASME International 2001 DETC Conference;

[7] Florez H. Domain decomposition methods for geomechanics [Ph.D. thesis]. In: The Uni-

for Underbody Blasts" funded by Army Research Laboratory.

Notice that λ<sup>j</sup> and the polynomial coefficients are all scalar quantities. In order to guarantee existence and uniqueness for these splines, an orthogonality condition with respect to linear polynomials is enforced, for instance, in two dimensions this yields to

$$
\sum\_{j} \lambda\_{j} = \sum\_{j} \lambda\_{j} \mathbf{x}\_{j} = \sum\_{j} \lambda\_{j} y\_{j} = 0. \tag{38}
$$

By considering this result, the interpolation problem becomes

$$\mathbf{s}(\underline{\mathbf{x}}^i) = \sum\_j \lambda\_j \cdot \boldsymbol{\varphi}(\rho\_j^i) + p\_n(\underline{\mathbf{x}}^i) = F^i\_\prime \tag{39}$$

which implies <sup>m</sup> points plus <sup>n</sup> <sup>þ</sup> 1 orthogonality conditions; here, Fi are the nodal values to be interpolated. The resultant linear system to solve for is of ð Þ m þ n þ 1 rank.

Duchon splines are certainly suitable to interpolate scattered data sets that we cannot tackle with the tensor product surfaces that we discussed before. Indeed, Figure 21 depicts such an application, in optimization, where an objective function that we wish to minimize was sampled randomly by Monte-Carlo (MC) realizations. To compute a minimum, we interpolate the black dots, and then we minimize the resulting spline with standard Newton stochastic techniques [15]. It is true that Duchon splines are a valid choice for "surrogate" models for such applications.

Figure 21. Discrete MC data with Duchon splines.
