Variational approach to coarse-graining of generalized gradient flows.
Manh Hong Duong1, Agnes Lamacz2, Mark A Peletier3
11Mathematics Institute, University of Warwick, Coventry, United Kingdom.
Calculus of Variations and Partial Differential Equations
|July 2, 2019
Summary
We developed a unified variational technique for coarse-graining and limit analysis, applicable to systems with non-dissipative effects and approximate solutions. This method addresses complex problems like the Vlasov-Fokker-Planck equation limits.
Area of Science:
- Mathematical Physics
- Computational Mathematics
Background:
- Variational techniques are crucial for analyzing complex systems.
- Coarse-graining and limit behavior are often studied separately.
- Duality structures arise naturally in many physical systems.
Purpose of the Study:
- To present a unified variational technique for coarse-graining and limit analysis.
- To demonstrate the technique's applicability to systems with non-dissipative effects.
- To handle approximate solutions in variational evolutions.
Main Methods:
- A novel variational technique based on a duality structure.
- Unified approach to coarse-graining and passing to a limit.
- Application to systems derived from large-deviations principles.
Main Results:
- The technique naturally integrates duality structures with coarse-graining.
- Successfully applied to systems with non-dissipative effects.
- Demonstrated effectiveness on approximate solutions with errors.
Conclusions:
- The presented variational technique offers a unified framework for complex systems.
- It provides a robust method for analyzing limits of PDEs and stochastic systems.
- The approach is versatile, handling non-dissipative effects and approximate solutions effectively.
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