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Generation and Coherent Control of Pulsed Quantum Frequency Combs

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Published on: June 8, 2018

Localized bases of eigensubspaces and operator compression.

Weinan E¹, Tiejun Li, Jianfeng Lu

¹Department of Mathematics and Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, USA.

Proceedings of the National Academy of Sciences of the United States of America

|January 19, 2010

Summary

This study introduces methods for finding localized bases to create sparse, finite-dimensional approximations of complex operators. This approach enhances the efficiency of analyzing large-scale systems like Markov chains and differential operators.

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Area of Science:

Numerical Analysis
Linear Algebra
Scientific Computing

Background:

Complex local operators, such as Markov chain generators or discretized differential operators, require efficient finite-dimensional approximations.
Standard eigenvalue-based projections may not yield unique or optimal representations for these operators.

Purpose of the Study:

To develop methods for finding the best finite-dimensional approximation of complex local operators.
To identify localized bases for eigensubspaces that preserve sparsity in the reduced operator.

Main Methods:

Investigating various techniques for obtaining localized bases for operator eigensubspaces.
Characterizing the decay rates of these localized basis functions.
Developing efficient numerical algorithms for basis and reduced operator computation.

Main Results:

Localized bases can be derived for operator eigensubspaces, leading to sparse reduced operators.
Explicit characterization of the decay rates of these basis functions is provided.
Efficient numerical algorithms are presented for practical implementation.

Conclusions:

Localized bases offer a powerful tool for creating sparse, finite-dimensional approximations of complex operators.
The developed methods and algorithms facilitate efficient analysis and computation for large-scale systems.