Monte Carlo determination of multiple extremal eigenpairs

Area of Science:

• Computational physics
• Numerical analysis
• Matrix computations

Background:

• Determining extremal eigenpairs of large matrices is computationally intensive.
• Existing methods often require significant memory and computational resources.
• The power method is a foundational iterative technique for eigenvalue problems.

Purpose of the Study:

• To develop a memory-efficient Monte Carlo algorithm for finding extremal eigenpairs of large matrices.
• To extend the capabilities of the power method using probabilistic techniques.
• To demonstrate the algorithm's efficacy on a relevant physics problem.

Main Methods:

• Monte Carlo algorithm based on an extension of the power method.
• Utilized the 'comb' splitting and termination technique.
• Incorporated the weight cancellation method and the 'sewing' sampling method.
• Applied to calculate eigenvalues of transfer matrices for 2D Ising models.

Main Results:

• Successfully determined the two largest eigenvalues for various sizes of 2D Ising models.
• Algorithm demonstrated effectiveness without requiring inner products or full vector storage.
• The method proved to be a memory-efficient alternative to deterministic approaches.

Conclusions:

• The presented Monte Carlo algorithm offers an efficient and general approach for eigenvalue problems.
• Applicable to large matrices where traditional methods are infeasible.
• Potential for wide application in physics and other fields requiring dominant eigenvalue determination.