Interpolation

Name
Instructor
CourseDate
Interpolation
SINC COLLOCATION METHOD
Sinc collocation method is used to solve both linear and nonlinear equations. However, unlike other methods like the finite-element, finite-difference, and boundary elements that are domain specific and not very accurate, SINC Collocation is a single method that can be used to solve the boundary value problems. Through trigonometric methods, use of the method helps one to get singularities near or at borders accurately. It is the most efficient method handling infinite and finite domains. It gains superiority to the multigrid method in that fewer nodal points are used, and exponential convergence can be achieved even when singularities are present (Narasimhan, Majdalani and Stenger 447-462).
Radial Basis Function RBF
RBFs are used to model function approximations of the form
yx=i=1nωi∅(x-x1)The function y(x) (approximating function) is represented as a sum of the radial basis functions (N), each having a different center (Xi) and a weighted coefficient (Wj). Since the approximating function is linear in value, it is easier to employ the matrix method to derive the weight coefficient. This type of the equation is applied where prediction of series is required as well as when controlling nonlinear systems that present unstable behaviors. Essentially, Radial Basis Functions are typically trained by a two-step algorithm.
Akima Interpolation
This is a continuously differentiable spline interpolation built from third order piecewise polynomials.

In Akima interpolation, the interpolation polynomials are only determined by the data from the next neighbor points. This makes the method computationally very efficient and hence there is no need to solve the large equation systems. The technique requires only a few functional points for the whole curve. As a result, the method does not cause wiggles in the curve. For a polynomial equation, the derivatives are generated from the ratios of j=i-n, i-n-1, i-n-2, and the weighting coefficients wi-(n-1) and wi-(n-2). For further derivatives, additional ratios must be used where the function is reduced to a lower polynomial at every stage.
Works Cited
Narasimhan, Susheela, Joseph Majdalani, and Frank Stenger. “A FIRST STEP IN APPLYING THE SINC COLLOCATION METHOD TO THE NONLINEAR NAVIER-STOKES EQUATIONS.” Numerical Heat Transfer, Part B: Fundamentals 41.5 (2002): 447-462. Web.