Auxiliary variables for numerically solving nonlinear equations with softly broken symmetries
1Institute of Cosmology, Department of Physics and Astronomy, Tufts University, Medford, Massachusetts 02155, USA.
Physical Review. E
|July 16, 2017
Summary
This study introduces a faster method for solving nonlinear equations with narrow error valleys, common in physics. By adding symmetry generators as variables, it accelerates convergence to solutions, particularly for problems like false vacuum decay.
Area of Science:
- Applied Mathematics
- Theoretical Physics
- Computational Science
Background:
- Standard methods for solving simultaneous nonlinear equations struggle with error functions exhibiting narrow, curved valleys.
- Slow convergence to solutions occurs once numerical methods reach these 'valleys', hindering efficiency.
Purpose of the Study:
- To develop a more efficient method for solving nonlinear equations in cases with softly broken symmetries.
- To accelerate the convergence of numerical solvers when encountering narrow error valleys.
Main Methods:
- Introducing generators of softly broken symmetries as auxiliary variables.
- Modifying numerical methods to handle cases with more variables than equations.
- Implementing Powell's hybrid method with this generalization.
Main Results:
- Demonstrated significantly faster convergence to solutions in the presence of softly broken symmetries.
- Successfully applied the method to benchmark problems, including false vacuum decay.
- Developed a Mathematica package for practical implementation.
Conclusions:
- The proposed method offers a substantial improvement for solving specific classes of nonlinear equations.
- This approach is particularly valuable for complex physical problems involving symmetry breaking.
- The generalized Powell's hybrid method provides a practical tool for researchers.
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