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Numerical operator calculus in higher dimensions.

Gregory Beylkin1, Martin J Mohlenkamp

  • 1Applied Mathematics, University of Colorado, Boulder, CO 80309, USA. beylkin@colorado.edu

Proceedings of the National Academy of Sciences of the United States of America
|July 26, 2002
PubMed
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We introduce a new method to overcome the curse of dimensionality in computational problems. This approach uses a novel representation for vectors and matrices, enabling efficient one-dimensional operations for high-dimensional analysis.

Area of Science:

  • Computational Science
  • Numerical Analysis
  • Quantum Mechanics

Background:

  • The curse of dimensionality significantly increases computational cost with increasing dimensions.
  • Existing methods struggle with high-dimensional problems due to exponential scaling.
  • Solving complex problems in physics and engineering is hindered by computational limitations.

Purpose of the Study:

  • To propose a novel representation for vectors and matrices to mitigate the curse of dimensionality.
  • To enable efficient numerical analysis in high dimensions by reducing computational complexity.
  • To demonstrate the applicability of this representation to key scientific problems.

Main Methods:

  • Generalizing the separation of variables technique for high-dimensional data.

Related Experiment Videos

  • Developing a representation for vectors and matrices that allows controlled accuracy.
  • Performing basic linear algebra operations using one-dimensional computations.
  • Applying the representation to the multiparticle Schrödinger operator and inverse Laplacian.

    • Main Results:

    • The proposed representation bypasses exponential scaling, enabling efficient one-dimensional operations.
    • The multiparticle Schrödinger operator and inverse Laplacian can be efficiently represented.
    • Numerical evidence suggests eigenfunctions inherit the efficient representation property.
    • The ground-state eigenfunction for a 30-particle Schrödinger operator was computed efficiently.

    Conclusions:

    • The novel representation effectively addresses the curse of dimensionality for specific operators.
    • Numerical operator calculus in higher dimensions becomes feasible with this approach.
    • Further research can explore the compatibility of various operators and algorithms with this representation.