Interpolation method: basic types and computational algorithms

A considerable number of mathematical problemsis connected with the finding of a distributed non-uniformly in the space of information. 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 method of calculatingintermediate values ​​of values ​​by 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 essenceis due to the influence of points closer to the estimated compared to the one located further. When used, this method of interpolation involves choosing from a certain topography in a certain neighborhood a particular point that has the greatest impact on it. So you select the maximum search radius or the number of points that are located close to a certain point. Then the weight is 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 point.

The second interpolation method is used when yresearchers 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. Here a lot of best approximation is constructed, 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 determiningspecific points is the method of quadratic interpolation, the essence of which is the replacement of a function on a certain interval by a quadratic parabola. At the same time, its extremum is estimated 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.