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1. Tikhonov regularization of a second order dynamical system with Hessian driven damping

Author

Szilárd László, Radu Ioan Boţ, and Ernö Robert Csetnek

Subjects
Hessian matrix, General Mathematics, 0211 other engineering and technologies, Dynamical Systems (math.DS), 02 engineering and technology, Dynamical system, 01 natural sciences, Hessian-driven damping, 90C26, Tikhonov regularization, symbols.namesake, 34G25, 47J25, 47H05, 90C26, 90C30, 65K10, Convergence (routing), FOS: Mathematics, Applied mathematics, 0101 mathematics, Mathematics - Dynamical Systems, Mathematics - Optimization and Control, Mathematics, 65K10, 021103 operations research, Full Length Paper, 47J25, 47H05, 010102 general mathematics, Hilbert space, 90C30, Function (mathematics), Convex optimization, Optimization and Control (math.OC), Second order dynamical system, 34G25, symbols, Fast convergence methods, Convex function, Software
Abstract

We investigate the asymptotic properties of the trajectories generated by a second-order dynamical system with Hessian driven damping and a Tikhonov regularization term in connection with the minimization of a smooth convex function in Hilbert spaces. We obtain fast convergence results for the function values along the trajectories. The Tikhonov regularization term enables the derivation of strong convergence results of the trajectory to the minimizer of the objective function of minimum norm.

Published
2020

2. Randomness and Nonlinear Evolution Equations

Author

Gigliola Staffilani, Andrea R. Nahmod, and Massachusetts Institute of Technology. Department of Mathematics

Subjects
010101 applied mathematics, Set (abstract data type), Applied Mathematics, General Mathematics, 010102 general mathematics, Applied mathematics, Uniqueness, 0101 mathematics, Nonlinear evolution, 01 natural sciences, Randomness, Term (time), Mathematics
Abstract

In this paper we survey some results on existence, and when possible also uniqueness, of solutions to certain evolution equations obtained by injecting randomness either on the set of initial data or as a perturbative term., National Science Foundation (U.S.) (Grant DMS 1362509), National Science Foundation (U.S.) (Grant DMS 1462401)

Published
2018