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Motivated by gradient methods in optimization theory, we give methods based onψ‐fractional derivatives of orderαin order to solve unconstrained optimization problems. The convergence of these methods is analyzed in detail. This paper also presents an Adams–Bashforth–Moulton (ABM) method for the estimation of solutions to equations involvingψ‐fractional derivatives. Numerical examples using the ABM method show that the fractional orderαand weightψare tunable parameters, which can be helpful for improving the performance of gradient descent methods.
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This content will become publicly available on August 1, 2026
EDGE: Epsilon-Difference Gradient Evolution for Buffer-Free Flow Maps
We propose the Epsilon Difference Gradient Evolution (EDGE) method for accurate flow-map calculation on grids via Hermite interpolation without using velocity buffers. Our key idea is to integrate Gradient Evolution for accurate first-order derivatives and a tetrahedron-based Epsilon Difference scheme to compute higher-order derivatives with reduced memory consumption. EDGE achievesO(1) memory usage, independent of flow map length, while maintaining vorticity preservation comparable to buffer-based methods. We validate our methods across diverse vortical flow scenarios, demonstrating up to 90% backward map memory reduction and significant computational efficiency, broadening the applicability of flow-map methods to large-scale and complex fluid simulations.
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- Award ID(s):
- 1919647
- PAR ID:
- 10648629
- Publisher / Repository:
- ACM
- Date Published:
- Journal Name:
- ACM Transactions on Graphics
- Volume:
- 44
- Issue:
- 4
- ISSN:
- 0730-0301
- Page Range / eLocation ID:
- 1 to 11
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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