Iterative method for solving linear operator equation of the first kind.

Area of Science:

• Numerical Analysis
• Inverse Problems
• Operator Theory

Background:

• Linear operator equations of the first kind are ill-posed, requiring regularization methods for stable solutions.
• Existing methods like modified Lavrentiev and Landweber iterative methods have limitations in accuracy and convergence.
• Inverse problems, such as the inverse heat equation, often involve solving these types of equations.

Purpose of the Study:

• To introduce and analyze a new iterative method for solving linear operator equations of the first kind.
• To enhance the modified Lavrentiev method by incorporating iterative performance.
• To compare the proposed method's efficiency and accuracy against established techniques like the standard modified Lavrentiev and Landweber methods.

Main Methods:

• A new iterative algorithm is developed based on the modified Lavrentiev method.
• The core linear operator is decomposed using polar decomposition to obtain a unitary operator.
• The iterative process leverages this unitary operator to potentially improve convergence.

Main Results:

• The proposed iterative method yields approximate solutions of higher quality compared to the standard modified Lavrentiev regularization method.
• Numerical experiments demonstrate the effectiveness of the new method in solving the inverse heat equation.
• The use of a unitary operator derived from polar decomposition enhances the convergence of the iteration.

Conclusions:

• The new iterative method is efficient and effective for solving linear operator equations of the first kind.
• The integration of polar decomposition and unitary operators offers a promising approach for improving iterative regularization techniques.
• The method shows particular utility in addressing complex inverse problems like the inverse heat equation.