What is the Tikhonov regularization method

In theory, an indefinite problem is usually defined by a set of linear algebraic equations, and this set of equations is usually the indefinite inverse problem with a large number of conditions. A large number of conditions means that rounding errors or other errors can seriously affect the outcome of the problem.

An explanatory definition is also given: for linear equations.

ax=b, when the solution x does not exist or the solution is not unique, it is called an ill-posed problemBut in many cases, we need to solve a pathological problem, so how do we do that?

In the case where the solution does not exist, the solution is to add some conditions to find an approximate solution; In cases where the solution is not unique, the solution is to add some constraints to narrow the scope of the solution. This method of solving a pathological problem by adding conditions or constraints is the regularization method.

The English word for regularization is regularization, that is, regularization, adjustment. Through some adjustments or other methods, the problem of pathology can also be solved in a single way. In this process of adjustment, the technique used is the regularization technique, and the method used is the regularization method.

The standard method for solving linear equations is least squares.

That is, solve the min<>

For pathological linear equations, Tikhonov proposed the use of <>

method, <>

It's called the Tikhonov matrix.

regularization regularizing operator regularization operator.

Dekhonov's regularization is the introduction of regular operators to transform the indefinite problem into a definite problem. There are three conditions for judging suitability.

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Regularization is a method of using regular expressions to quickly find matches in a large amount of data.

A text pattern consisting of normal characters, such as characters A through Z, as well as special characters, called metacharacters. This pattern describes one or more strings to be matched when looking for a body of text. A regular expression acts as a template to match a character pattern to the searched string.