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Compressed modes for variational problems in mathematics and physics.

Vidvuds Ozolins1, Rongjie Lai, Russel Caflisch

  • 1Departments of Materials Science and Engineering, and Mathematics, University of California, Los Angeles, CA 90095-1555.

Proceedings of the National Academy of Sciences of the United States of America
|October 31, 2013
PubMed
Summary
This summary is machine-generated.

This study introduces a method for finding localized solutions in mathematical physics problems, like quantum mechanics. This approach uses regularization to create "compressed modes" for efficient computation.

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Area of Science:

  • Mathematical Physics
  • Quantum Mechanics
  • Computational Science

Background:

  • Variational optimization problems are common in mathematical physics.
  • Solving Schrödinger's equation requires efficient methods for eigenvalue spectrum and eigenfunctions.

Purpose of the Study:

  • To develop a general formalism for obtaining spatially localized solutions.
  • To introduce a method for generating "compressed modes" with compact support.

Main Methods:

  • Recasting problems as variational optimization.
  • Incorporating a regularization term into the variational principle.
  • Utilizing linear combinations of compressed modes.

Main Results:

  • Achieved spatially localized ("sparse") solutions.
  • Generated "compressed modes" with compact support.
  • Demonstrated systematic improvement in approximating eigenvalue spectrum and eigenfunctions.

Conclusions:

  • The developed formalism provides a general approach for sparse solutions.
  • Compressed modes offer an efficient alternative for numerical algorithms.
  • Linear scaling algorithms can be effectively employed with these localized modes.