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Professor Luis Nunes Vicente will talk about 'Reducing Sample Complexity in Stochastic Derivative-Free Optimization via Tail Bounds and Hypothesis Testing';

We introduce and analyze new probabilistic strategies for enforcing sufficient decrease conditions in stochastic derivative-free optimization, with the goal of reducing sample complexity and simplifying convergence analysis. First, we develop a new tail bound condition imposed on the estimated reduction in function value, which permits flexible selection of the power used in the sufficient decrease test, q in (1,2]. This approach allows us to reduce the number of samples per iteration from the standard O(delta^{−4}) to O(delta^{-2q}), assuming that the noise moment of order q/(q-1) is bounded. Second, we formulate the sufficient decrease condition as a sequential hypothesis testing problem, in which the algorithm adaptively collects samples until the evidence suffices to accept or reject a candidate step. This test provides statistical guarantees on decision errors and can further reduce the required sample size, particularly in the Gaussian noise setting, where it can approach O(delta^{−2-r}) when the decrease is of the order of delta^r. We incorporate both techniques into stochastic direct-search and trust-region methods for potentially non-smooth, noisy objective functions, and establish their global convergence rates and properties. 

This is joint work with Anjie Ding, Francesco Rinaldi, and Damiano Zeffiro.

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In this talk, we focus on the existence of time-periodic leapfrogging vortex rings for the three-dimensional incompressible Euler equations, thereby providing a rigorous realization of a phenomenon first conjectured by Helmholtz (1858). In the leapfrogging motion, two coaxial vortex rings periodically exchange positions, a striking behavior repeatedly observed in experiments and numerical simulations, yet lacking complete mathematical justification. Our construction relies on a desingularization of two interacting vortex filaments within the contour dynamics formulation, which yields a Hamiltonian description of nearly concentric vortex rings. The main difficulty stems from a singular small-divisor problem arising in the linearized transport dynamics, where the effective time scale degenerates with the ring thickness parameter. To overcome this obstruction, we develop a degenerate KAM-type analysis combined with pseudo-differential operator techniques to control the linearized dynamics around symmetric configurations. Combining these tools with a Nash-Moser iteration scheme, we construct families of nontrivial time-periodic solutions in an almost uniformly translating frame. This establishes the first rigorous construction of classical leapfrogging motion for axisymmetric Euler flows without swirl, with no restriction on the time interval of existence. 

This is a joint work with Zineb Hassainia and Taoufik Hmidi.

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Dr Lindon Roberts will talk about: 'Optimization Algorithms for Bilevel Learning with Applications to Imaging'

Many imaging problems, such as denoising or inpainting, can be expressed as variational regularization problems. These are optimization problems for which many suitable algorithms exist. We consider the problem of learning suitable regularizers for imaging problems from example (training) data, which can be formulated as a large-scale bilevel optimization problem. 

In this talk, I will introduce new deterministic and stochastic algorithms for bilevel optimization, which require no or minimal hyperparameter tuning while retaining convergence guarantees. 

This is joint work with Mohammad Sadegh Salehi and Matthias Ehrhardt (University of Bath), and Subhadip Mukherjee (IIT Kharagpur).

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This is a joint OxPDE and Numerical Analysis seminar. 

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