-
Proceedings of the National Academy of... Apr 2023First-order optimization algorithms are widely used today. Two standard building blocks in these algorithms are proximal operators (proximals) and gradients. Although...
First-order optimization algorithms are widely used today. Two standard building blocks in these algorithms are proximal operators (proximals) and gradients. Although gradients can be computed for a wide array of functions, explicit proximal formulas are known for only limited classes of functions. We provide an algorithm, HJ-Prox, for accurately approximating such proximals. This is derived from a collection of relations between proximals, Moreau envelopes, Hamilton-Jacobi (HJ) equations, heat equations, and Monte Carlo sampling. In particular, HJ-Prox smoothly approximates the Moreau envelope and its gradient. The smoothness can be adjusted to act as a denoiser. Our approach applies even when functions are accessible only by (possibly noisy) black box samples. We show that HJ-Prox is effective numerically via several examples.
PubMed: 36989305
DOI: 10.1073/pnas.2220469120 -
Plastic and Reconstructive Surgery.... Jul 2014There is a dearth of detailed published work on the anatomy of ulnar artery perforators. The objective of this study was to fully document the vascular basis of the free...
BACKGROUND
There is a dearth of detailed published work on the anatomy of ulnar artery perforators. The objective of this study was to fully document the vascular basis of the free proximal ulnar artery perforator flap and report its use in reconstruction of the hand.
METHODS
(1) The ulnar artery perforators were studied in 25 fresh cadavers and 10 cast preparations. Cadavers were injected with lead oxide for 3-dimensional reconstruction. The origin, course, and distribution of the ulnar artery perforators were comprehensively documented. (2) Between August 2011 and January 2013, 29 free proximal ulnar artery perforator flaps were utilized for reconstruction of soft-tissue defects of the hand in 25 patients. Flap size varied from 3.5 × 2.0 cm to 24.0 × 4.0 cm, with a consistent thickness of approximately 3 mm.
RESULTS
(1) There were 7 ± 2.0 ulnar artery perforators. The average external diameter was 0.6 ± 0.2 mm. Each perforator supplied an average area of 26 ± 7.0 cm(2). Extensive anastomoses were found between the ulnar artery perforators and multiple adjacent source arteries. (2) All flaps survived. The clinical results were satisfactory after 10.2 ± 5.3 months of follow-up. The flaps were considered cosmetically acceptable by both patients and doctors.
CONCLUSIONS
The main advantage of the proximal ulnar artery perforator flap is that it is a thin flap that is ideal for upper extremity reconstruction, either as proximally or distally based local perforator flap or as a free flap. The donor site is excellent, and the vascular anatomy is very consistent.
PubMed: 25426362
DOI: 10.1097/GOX.0000000000000113