Parallel multigrid method for solving inverse problems

Area of Science:

• Numerical Analysis
• Computational Mathematics
• Scientific Computing

Background:

• Iterative methods like Landweber's are used for linear operator equations but often exhibit slow convergence.
• The Multigrid method is a powerful technique for accelerating convergence and solving large linear systems.
• Inverse ill-posed problems require efficient and accurate solution methods.

Purpose of the Study:

• To develop a novel parallel algorithm for the Multigrid method.
• To enhance the computational efficiency and accuracy of iterative solvers for inverse ill-posed problems.
• To accelerate the solution process for linear operator equations.

Main Methods:

• A new parallel V-cycle algorithm was developed, combining V-cycle and two-grid methods for parallel computation at each level.
• The algorithm incorporates coarse grid operators and residual right-hand side vectors.
• The Landweber iterative method was employed as the base iterative solver.

Main Results:

• The developed parallel Multigrid algorithm significantly accelerates computation compared to sequential methods.
• Numerical experiments, including solving the heat equation's initial value problem, demonstrated superior efficiency.
• The parallel V-cycle algorithm provides a highly accurate and fast approximation solution.

Conclusions:

• The new parallel V-cycle algorithm is highly efficient for solving inverse ill-posed problems.
• This method accelerates the convergence of iterative solvers, offering a significant improvement over sequential approaches.
• The parallel algorithm enhances the speed and accuracy of finding approximate solutions for linear algebra equations.