On the use of hybrid coarse-level models in multilevel minimization methods

Solving large-scale nonlinear minimization problems is computationally demanding. Nonlinear multilevel minimization (NMM) methods explore the structure of the underlying minimization problem to solve such problems in a computationally efficient and scalable manner. The efficiency of the NMM methods relies on the quality of the coarse-level models. Traditionally, coarse-level models are constructed using the additive approach, where the so-called $\tau$-correction enforces a local coherence between the fine-level and coarse-level objective functions. In this work, we extend this methodology and discuss how to enforce local coherence between the objective functions using a multiplicative approach. Moreover, we also present a hybrid approach, which takes advantage of both, additive and multiplicative, approaches. Using numerical experiments from the field of deep learning, we show that employing a hybrid approach can greatly improve the convergence speed of NMM methods and therefore it provides an attractive alternative to the almost universally used additive approach.