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Mapping Cortical Dynamics Using Simultaneous MEG/EEG and Anatomically-constrained Minimum-norm Estimates: an Auditory Attention Example

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Monge-Ampere grids and the multidimensional mapped Fourier method.

¹Mathematics Institute, University of Bergen, Johannes Brunsgate 12, Bergen 5008, Norway. ilan.degani@mi.uib.no

The Journal of Chemical Physics

|May 2, 2008

Summary

Solving the Monge-Ampere equation enables better coordinate transformations for numerical methods. This improves accuracy in multidimensional problems like Schrodinger's equation, benefiting computational quantum mechanics.

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Area of Science:

Computational physics
Numerical analysis
Quantum mechanics

Background:

Mapped Fourier methods enhance numerical efficiency through tailored coordinate transformations.
A key challenge in multidimensional applications is finding effective coordinate transformations.
Standard Fourier methods face accuracy limitations in complex systems.

Purpose of the Study:

To investigate the use of the Monge-Ampere equation for generating coordinate transformations.
To apply these transformations within the mapped Fourier method for multidimensional problems.
To assess the impact on the accuracy of solving Schrodinger's equation.

Main Methods:

Coordinate transformations were derived by solving the Monge-Ampere equation.
The derived transformations were integrated into the mapped Fourier method.
The combined approach was applied to eigenvalue calculations for multidimensional Schrodinger's equation.

Main Results:

Significant improvements in accuracy were achieved compared to the standard Fourier method.
The method demonstrated enhanced performance in eigenvalue calculations for two-dimensional systems.
The Monge-Ampere equation proved effective in constructing efficient representations.

Conclusions:

The Monge-Ampere equation is a valuable tool for creating optimal coordinate transformations.
This approach offers a pathway to more accurate and efficient multidimensional simulations.
The findings have implications for computational quantum mechanics and related scientific fields.