Laurent Mertz

Alain Bensoussan, Hector Jasso Fuentes, Laurent Mertz

In a previous work by the first author with J. Turi (AMO, 08), a stochastic variational inequality has been introduced to model an elasto-plastic oscillator with noise. A major advantage of the stochastic variational inequality is to overcome the need to describe the trajectory by phases (elastic or plastic). This is useful, since the sequence of phases cannot be characterized easily. In particular, there are numerous small elastic phases which may appear as an artefact of the Wiener process. However, it remains important to have informations on these phases. In order to reconcile these contradictory issues, we introduce an approximation of stochastic variational inequalities by imposing artificial small jumps between phases allowing a clear separation of the phases. In this work, we prove that the approximate solution converges on any finite time interval, when the size of jumps tends to 0.

Yumeng Ren, Yaofang Liu, Aitor Artola et al.

Diffusion denoising models have become a popular approach for image generation, but they often suffer from slow convergence during training. In this paper, we identify that this slow convergence is partly due to the complexity of the Brownian motion driving the forward-time process. To address this, we represent the Brownian motion using the Karhunen-Loève expansion, truncating it to a limited number of eigenfunctions. We propose a novel ordinary differential equation with augmented random initials, termed KL diffusion, as a new forward-time process for training and sampling. By developing an appropriate denoising loss function, we facilitate the integration of our KL-diffusion into existing denoising-based models. Using the widely adopted DDIM framework as our baseline ensures a fair comparison, as our modifications focus solely on the forward process and loss function, leaving the network architecture and sampling methods unchanged. Our method significantly outperforms baseline diffusion models, achieving convergence speeds that are twice faster to reach the best FID score of the baseline and ultimately yielding much lower FID scores. Notably, our approach allows for highly parallelized computation, requires no additional learnable parameters, and can be flexibly integrated into existing diffusion methods. The code will be made publicly available.