A Variational Surface-Evolution Approach to Optimal Transport over Transitioning Compact Supports with Domain Constraints
Abstract
:1. Introduction
Novelties and Contributions Presented in This Work
2. Spatiotemporal Hypersurface
2.1. Assumptions
2.1.1. Compact Support
2.1.2. Balanced Density
2.1.3. Smoothness
2.1.4. Piece-Wise Smooth Boundary
2.1.5. Smooth Extension
2.2. Local Geometry
2.2.1. Parameterization
2.2.2. Unit Normal
2.2.3. Metric Tensor
2.2.4. Area Element
2.2.5. Normal Variations
3. Spatiotemporal Formulation of Optimal Mass Transport
3.1. Spatiotemporal Advection Field
3.2. Solenoidal Vector Field
3.3. Extended Velocity
3.3.1. Local Kinetic Energy
3.3.2. Extended Velocity
3.3.3. Generalized Momentum
3.4. Variational Formulation
3.4.1. Action Integral
3.4.2. First Variation
3.4.3. Optimality Condition for
3.4.4. Partial Gradient with Respect to (Fixed Support)
3.4.5. Shape Optimality Condition for
3.4.6. Conjectures on the Existence of Optimal Transport with Intermediate Support Constraints
4. Shape Optimization Strategy with Added Density Prior
4.1. Density Prior
4.2. Optimality Conditions and Partial Gradients
4.3. Optimization in (Fixed Support)
4.3.1. Computing an Initial Solenoidal Field
4.3.2. Iterated Gradient Descent for the Solenoidal Field
4.4. Total Shape Gradient
5. Results
5.1. Interpolation between Two Different Non-Convex Supports
5.2. Optimal Transport with Support Constraints
5.2.1. Known Densities with Intermediate Support Constraints
5.2.2. Unknown Final Density but with Known Support
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. Detailed First Variation and Gradient Calculation
Appendix B. Coupled Boundary and Flux Perturbations
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Yezzi, A. A Variational Surface-Evolution Approach to Optimal Transport over Transitioning Compact Supports with Domain Constraints. Fluids 2024, 9, 118. https://doi.org/10.3390/fluids9050118
Yezzi A. A Variational Surface-Evolution Approach to Optimal Transport over Transitioning Compact Supports with Domain Constraints. Fluids. 2024; 9(5):118. https://doi.org/10.3390/fluids9050118
Chicago/Turabian StyleYezzi, Anthony. 2024. "A Variational Surface-Evolution Approach to Optimal Transport over Transitioning Compact Supports with Domain Constraints" Fluids 9, no. 5: 118. https://doi.org/10.3390/fluids9050118
APA StyleYezzi, A. (2024). A Variational Surface-Evolution Approach to Optimal Transport over Transitioning Compact Supports with Domain Constraints. Fluids, 9(5), 118. https://doi.org/10.3390/fluids9050118