Local conditions for global convergence of gradient flows and proximal point sequences in metric spaces

article Local conditions for global convergence of gradient flows and proximal point sequences in metric spaces published yes Lorenzo Dello Schiavo author ECEBF480-9E4F-11EA-B557-B0823DDC885E0000-0002-9881-6870 Jan Maas author 4C5696CE-F248-11E8-B48F-1D18A9856A870000-0002-0845-1338 Francesco Pedrotti author d3ac8ac6-dc8d-11ea-abe3-e2a9628c4c3c JaMa department Optimal Transport and Stochastic Dynamics project Taming Complexity in Partial Differential Systems project Configuration Spaces over Non-Smooth Spaces project This paper deals with local criteria for the convergence to a global minimiser for gradient flow trajectories and their discretisations. To obtain quantitative estimates on the speed of convergence, we consider variations on the classical Kurdyka–Łojasiewicz inequality for a large class of parameter functions. Our assumptions are given in terms of the initial data, without any reference to an equilibrium point. The main results are convergence statements for gradient flow curves and proximal point sequences to a global minimiser, together with sharp quantitative estimates on the speed of convergence. These convergence results apply in the general setting of lower semicontinuous functionals on complete metric spaces, generalising recent results for smooth functionals on Rn. While the non-smooth setting covers very general spaces, it is also useful for (non)-smooth functionals on Rn. . American Mathematical Society2024 eng Transactions of the American Mathematical Society 0002-9947 1088-6850 2304.05239 00120327330000110.1090/tran/9156 37763779-3804 https://research-explorer.ista.ac.at/record/17336 Dello Schiavo L, Maas J, Pedrotti F. 2024. Local conditions for global convergence of gradient flows and proximal point sequences in metric spaces. Transactions of the American Mathematical Society. 377(6), 3779–3804. Dello Schiavo, L., Maas, J., & Pedrotti, F. (2024). Local conditions for global convergence of gradient flows and proximal point sequences in metric spaces. Transactions of the American Mathematical Society. American Mathematical Society. <a href="https://doi.org/10.1090/tran/9156">https://doi.org/10.1090/tran/9156</a> Dello Schiavo, Lorenzo, Jan Maas, and Francesco Pedrotti. “Local Conditions for Global Convergence of Gradient Flows and Proximal Point Sequences in Metric Spaces.” Transactions of the American Mathematical Society. American Mathematical Society, 2024. <a href="https://doi.org/10.1090/tran/9156">https://doi.org/10.1090/tran/9156</a>. Dello Schiavo, Lorenzo, et al. “Local Conditions for Global Convergence of Gradient Flows and Proximal Point Sequences in Metric Spaces.” Transactions of the American Mathematical Society, vol. 377, no. 6, American Mathematical Society, 2024, pp. 3779–804, doi:<a href="https://doi.org/10.1090/tran/9156">10.1090/tran/9156</a>. L. Dello Schiavo, J. Maas, and F. Pedrotti, “Local conditions for global convergence of gradient flows and proximal point sequences in metric spaces,” Transactions of the American Mathematical Society, vol. 377, no. 6. American Mathematical Society, pp. 3779–3804, 2024. Dello Schiavo L, Maas J, Pedrotti F. Local conditions for global convergence of gradient flows and proximal point sequences in metric spaces. Transactions of the American Mathematical Society. 2024;377(6):3779-3804. doi:<a href="https://doi.org/10.1090/tran/9156">10.1090/tran/9156</a> L. Dello Schiavo, J. Maas, F. Pedrotti, Transactions of the American Mathematical Society 377 (2024) 3779–3804. 171432024-06-16T22:01:06Z2026-04-07T13:00:02Z