Interpolation method: basic types and computational algorithms

A significant number of mathematical problems is connected with the finding of a distributed unevenly distributed information space. We are talking about information systems geographically, because it is in them that it is possible to measure the necessary quantities at certain points. To solve these problems, one or another method of interpolation is often used.

Definition

Interpolation is a way of calculating the intermediate values of quantities over the available discrete set of values. The most common methods of interpolation are: the method of inverse weighted distances, the surface of the trend and kriging.

Basic methods of interpolation

So, let's take a closer look at the first method, its essence lies in the influence of points closer to the estimated ones in comparison with the one located further. When used, this method of interpolation involves choosing from a certain topography in a certain neighborhood a specific point that has the greatest impact on it. So choose the maximum search radius or the number of points that are located close to a certain point. The weight is then given the height at each specific point, calculated depending on the distance from the given point. Only in this way can the greater contribution of the nearest points to the interpolated height be achieved by comparison with points farther from a given distance.

The second method of interpolation is used when researchers have an interest in general surface trends. Similarly to the first method for a trend, points that are within a given surface can be used. Many best approximations are constructed here, based on mathematical equations (splines or polynomials). In general, the least squares method is used, based on equations with nonlinear dependencies. The method is based on the replacement of curves and other forms of sequences of numerical type by simple ones. For the purpose of constructing a trend, each value on a given surface must be substituted into the equation. The result is the only value assigned to the interpolated solution (point). For all other points, the process continues.

Another method of interpolation, kriging, mentioned above, provides for the optimization of the interpolation procedure, taking as a basis the statistical nature of the surface.

Use of quadratic interpolation

There is another tool for determining specific points - the method of quadratic interpolation, the essence of which is the replacement of a function on a certain interval by a quadratic parabola. Her extremum is calculated analytically. After its approximate finding (minimum or maximum), it is necessary to specify a certain range of values, after which the search for the solution to continue. Repeating this procedure, it is possible, using an iterative procedure, to refine the value of this equation to the result with the accuracy specified in the statement of the problem.