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Title: Fast structured Jacobi-Jacobi transforms
Award ID(s):
1720440
PAR ID:
10167328
Author(s) / Creator(s):
; ;
Date Published:
Journal Name:
Mathematics of Computation
Volume:
88
Issue:
318
ISSN:
0025-5718
Page Range / eLocation ID:
1743 to 1772
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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  1. 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. 
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