Fei Liu

Fei Liu, Lexing Ying

This paper introduces the recursive sweeping preconditioner for the numerical solution of the Helmholtz equation in 3D. This is based on the earlier work of the sweeping preconditioner with the moving perfectly matched layers (PMLs). The key idea is to apply the sweeping preconditioner recursively to the quasi-2D auxiliary problems introduced in the 3D sweeping preconditioner. Compared to the non-recursive 3D sweeping preconditioner, the setup cost of this new approach drops from $O(N^{4/3})$ to $O(N)$, the application cost per iteration drops from $O(N\log N)$ to $O(N)$, and the iteration count only increases mildly when combined with the standard GMRES solver. Several numerical examples are tested and the results are compared with the non-recursive sweeping preconditioner to demonstrate the efficiency of the new approach.

Fei Liu, Qingfu Zhang, Jialong Shi et al.

Using Large Language Models (LLMs) in an evolutionary or other iterative search framework have demonstrated significant potential in automated algorithm design. However, the underlying fitness landscape, which is critical for understanding its search behavior, remains underexplored. In this paper, we illustrate and analyze the fitness landscape of LLM-assisted Algorithm Search (LAS) using a graph-based approach, where nodes represent algorithms and edges denote transitions between them. We conduct extensive evaluations across six algorithm design tasks and six commonly-used LLMs. Our findings reveal that LAS landscapes are highly multimodal and rugged, particularly in combinatorial optimization tasks, with distinct structural variations across tasks and LLMs. Moreover, we adopt four different methods for algorithm similarity measurement and study their correlations to algorithm performance and operator behaviour. These insights not only deepen our understanding of LAS landscapes but also provide practical insights for designing more effective LAS methods.

Fei Liu, Mingen Li, Jingpei Lu et al.

Simulation modeling of robots, objects, and environments is the backbone for all model-based control and learning. It is leveraged broadly across dynamic programming and model-predictive control, as well as data generation for imitation, transfer, and reinforcement learning. In addition to fidelity, key features of models in these control and learning contexts are speed, stability, and native differentiability. However, many popular simulation platforms for robotics today lack at least one of the features above. More recently, position-based dynamics (PBD) has become a very popular simulation tool for modeling complex scenes of rigid and non-rigid object interactions, due to its speed and stability, and is starting to gain significant interest in robotics for its potential use in model-based control and learning. Thus, in this paper, we present a mathematical formulation for coupling position-based dynamics (PBD) simulation and optimal robot design, model-based motion control and system identification. Our framework breaks down PBD definitions and derivations for various types of joint-based articulated rigid bodies. We present a back-propagation method with automatic differentiation, which can integrate both positional and angular geometric constraints. Our framework can critically provide the native gradient information and perform gradient-based optimization tasks. We also propose articulated joint model representations and simulation workflow for our differentiable framework. We demonstrate the capability of the framework in efficient optimal robot design, accurate trajectory torque estimation and supporting spring stiffness estimation, where we achieve minor errors. We also implement impedance control in real robots to demonstrate the potential of our differentiable framework in human-in-the-loop applications.