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A novel restart technique improves iterative projection methods for nonlinear eigenvalue problems. This method uses local eigenvalue enumeration to prevent subspace growth, reducing computational costs for large-scale problems.

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Area of Science:

  • Numerical Analysis
  • Computational Mathematics
  • Scientific Computing

Background:

  • Iterative projection methods are crucial for solving large-scale eigenvalue problems.
  • Nonlinear eigenvalue problems (NEPs) present unique computational challenges.
  • Existing methods often suffer from increasing computational costs due to subspace growth.

Purpose of the Study:

  • Introduce a new restart technique for iterative projection methods applied to NEPs.
  • Address the issue of computational cost escalation in existing methods.
  • Enhance the efficiency of computing multiple or interior eigenvalues of NEPs.

Main Methods:

  • Develop a restart technique based on local enumeration of eigenvalues in inner iterations.
  • Integrate the technique into Nonlinear Arnoldi and Jacobi-Davidson methods.
  • Extend the method to NEPs lacking the minmax property but possessing dominant real or imaginary parts.

Main Results:

  • The proposed local numbering avoids the need for all previous eigenvectors in the search subspace.
  • This effectively eliminates subspace growth and associated super-linear cost increases.
  • Demonstrated efficiency on quadratic, rational, exponential, and a real-life gyroscopic eigenvalue problem.

Conclusions:

  • The new restart technique significantly enhances the efficiency of iterative projection methods for NEPs.
  • It provides a computationally advantageous approach for problems requiring many eigenvalues or interior spectrum computation.
  • The method shows promise for both theoretical and industrial applications in eigenvalue analysis.