Diffusion-based inverse algorithms have shown remarkable performance across various inverse problems, yet their reliance on numerous denoising steps incurs high computational costs. While recent developments of fast diffusion ODE solvers offer effective acceleration for diffusion sampling without observations, their application in inverse problems remains limited due to the heterogeneous formulations of inverse algorithms and their prevalent use of approximations and heuristics, which often introduce significant errors that undermine the reliability of analytical solvers. In this work, we begin with an analysis of ODE solvers for inverse problems that reveals a linear combination structure of approximations for the inverse trajectory. Building on this insight, we propose a canonical form that unifies a broad class of diffusion-based inverse algorithms and facilitates the design of more generalizable solvers. Inspired by the linear subspace search strategy, we propose Learnable Linear Extrapolation (LLE), a lightweight approach that universally enhances the performance of any diffusion-based inverse algorithm conforming to our canonical form. LLE optimizes the combination coefficients to refine current predictions using previous estimates, alleviating the sensitivity of analytical solvers for inverse algorithms. Extensive experiments demonstrate consistent improvements of the proposed LLE method across multiple algorithms and tasks, indicating its potential for more efficient solutions and boosted performance of diffusion-based inverse algorithms with limited steps. Codes for reproducing our experiments are available at https://github.com/weigerzan/LLE_inverse_problem.

The success of the model predictive path integral control (MPPI) approach depends on the appropriate selection of the input distribution used for sampling. However, it can be challenging to select inputs that satisfy output constraints in dynamic environments. The main contribution of this paper is to propose an output-sampling-based MPPI (o-MPPI), which improves the ability of samples to satisfy output constraints and thereby increases MPPI efficiency. Comparative simulations and experiments of dynamic autonomous driving of bots around a track are provided to show that the proposed o-MPPI is more efficient and requires substantially (20-times) less number of rollouts and (4-times) smaller prediction horizon when compared with the standard MPPI for similar success rates.