Many imaging problems can be formulated as inverse problems expressed as finite-dimensional optimization problems. These optimization problems generally consist of minimizing the sum of a data fidelity and regularization terms. In Darbon (SIAM J. Imag. Sci. 8:2268–2293, 2015), Darbon and Meng, (On decomposition models in imaging sciences and multi-time Hamilton-Jacobi partial differential equations, arXiv preprint arXiv:1906.09502, 2019), connections between these optimization problems and (multi-time) Hamilton-Jacobi partial differential equations have been proposed under the convexity assumptions of both the data fidelity and regularization terms. In particular, under these convexity assumptions, some representation formulas for a minimizer can be obtained. From amore »
A min-plus fundamental solution semigroup for a class of approximate infinite dimensional optimal control problems
By exploiting min-plus linearity, semiconcavity,
and semigroup properties of dynamic programming, a fundamental
solution semigroup for a class of approximate finite
horizon linear infinite dimensional optimal control problems is
constructed. Elements of this fundamental solution semigroup
are parameterized by the time horizon, and can be used to
approximate the solution of the corresponding finite horizon
optimal control problem for any terminal cost. They can also
be composed to compute approximations on longer horizons.
The value function approximation provided takes the form of
a min-plus convolution of a kernel with the terminal cost. A
general construction for this kernel is provided, along with a
spectral representation for a restricted class of sub-problems.
- Award ID(s):
- 1908918
- Publication Date:
- NSF-PAR ID:
- 10170566
- Journal Name:
- Proceedings of the American Control Conference
- Page Range or eLocation-ID:
- 794-799
- ISSN:
- 0743-1619
- Sponsoring Org:
- National Science Foundation
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