张量列车加速求解非光滑刚体动力学
Tensor Train accelerated solvers for nonsmooth rigid body dynamics
AI总结：
本文提出利用低秩压缩快速直接求解器加速第二阶方法中的牛顿步线性系统，通过量化张量列车分解高效表示系统矩阵及其逆矩阵，实现非光滑刚体动力学问题的快速求解。
摘要：
在过去二十年里，对颗粒介质中时间演化和力传播的高保真模拟需求推动了离散元（DEM）摩擦接触建模的再关注。力惩罚方法虽然经济且易于使用，但引入了人工刚度，需要小时间步以保持数值稳定性。基于优化的方法通过互补约束几何地强制接触，允许使用更大的时间步，但每次时间步都要解决一个非线性互补问题（NCP）。我们回顾了最新努力，聚焦于将NCP松弛为锥互补问题（CCP）并通过等价的二次优化问题求解。我们区分了线性收敛的一阶方法和二阶方法，后者通过二次收敛和更稳健的性能，以牺牲求解大规模稀疏线性系统为代价。我们提出了一种新颖的加速方法，利用低秩压缩快速直接求解器加速第二阶方法中的牛顿步线性系统，使用量化张量列车（QTT）分解高效近似表示系统矩阵及其逆矩阵。这为在直接或预条件迭代方法中加速其求解提供了稳健的框架。在多个数值测试中，我们证明这种方法表现出预计算成本的亚线性缩放，可以高效更新跨牛顿迭代和时间步，并导致牛顿步的快速最优复杂度求解。这使我们的方法在中等至大规模系统中比最先进的预条件技术快一个数量级，缓解了二阶方法的计算瓶颈。
英文摘要：
In the last two decades, increased need for high-fidelity simulations of the time evolution and propagation of forces in granular media has spurred renewed interest in discrete element method (DEM) modeling of frictional contact. Force penalty methods, while economic and accessible, introduce artificial stiffness, requiring small time steps to retain numerical stability. Optimization-based methods, which enforce contacts geometrically through complementarity constraints, allow the use of larger time steps at the expense of solving a nonlinear complementarity problem (NCP) each time step. We review the latest efforts to produce solvers for this NCP, focusing on its relaxation to a cone complementarity problem (CCP) and solution via an equivalent quadratic optimization problem with conic constraints. We distinguish between linearly convergent first order methods and second order methods, which gain quadratic convergence and more robust performance at the expense of the solution of large sparse linear systems. We propose a novel acceleration for the solution of Newton step linear systems in second order methods using low-rank compression based fast direct solvers. We use the Quantized Tensor Train (QTT) decomposition to produce efficient approximate representations of the system matrix and its inverse. This provides a robust framework to accelerate its solution in a direct or a preconditioned iterative method. In a number of numerical tests, we demonstrate that this approach displays sublinear scaling of precomputation costs, may be efficiently updated across Newton iterations as well as across time steps, and leads to a fast, optimal complexity solution of the Newton step. This allows our method to gain an order of magnitude speedups over state-of-the-art preconditioning techniques for moderate to large-scale systems, mitigating the computational bottleneck of second order methods.
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