扰动标量守恒定律的异常激波位移概率
Anomalous Shock Displacement Probabilities for a Perturbed Scalar Conservation Law
AI总结：
针对随机时空强迫的一维守恒律，利用大偏差理论计算异常激波轮廓位移的指数小概率，分析率函数的标度行为，并基于大偏差的重要性采样蒙特卡洛策略高效估计位移概率。
摘要：
我们考虑一个具有随机时空强迫的一维守恒律，并利用大偏差理论计算异常激波轮廓位移的指数小概率。在随机力的空间支撑和结构的适当假设下，我们分析了率函数的标度行为，即位移概率的指数衰减率。对于小位移，我们证明率函数的上界和下界由位移的平方除以时间给出。对于大位移，率函数的相应界与位移成正比。我们在不同条件下数值计算率函数，并表明标度行为的理论分析得到证实。我们还应用基于大偏差的重要性采样蒙特卡洛策略来估计位移概率。我们使用以强迫为中心的偏置分布，该强迫给出了异常激波轮廓的最可能转移路径，即率函数的最小化器。数值模拟表明，该策略比基本蒙特卡洛方法更有效和稳健。
英文摘要：
We consider an one-dimensional conservation law with random space-time forcing and calculate using large deviations the exponentially small probabilities of anomalous shock profile displacements. Under suitable hypotheses on the spatial support and structure of random forces, we analyze the scaling behavior of the rate function, which is the exponential decay rate of the displacement probabilities. For small displacements we show that the rate function is bounded above and below by the square of the displacement divided by time. For large displacements the corresponding bounds for the rate function are proportional to the displacement. We calculate numerically the rate function under different conditions and show that the theoretical analysis of scaling behavior is confirmed. We also apply a large-deviation-based importance sampling Monte Carlo strategy to estimate the displacement probabilities. We use a biased distribution centered on the forcing that gives the most probable transition path for the anomalous shock profile, which is the minimizer of the rate function. The numerical simulations indicate that this strategy is much more effective and robust than basic Monte Carlo.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
