利用半定规划确定多项式扩散平稳平均值的界
Bounding stationary averages of polynomial diffusions via semidefinite programming
摘要：
我们介绍了一个算法的基础上半定规划，产量增加（分别）。降序（递减）具有多项式漂移向量和扩散系数的多项式平稳平均的上界。通过优化的目标，确定的固定平均的利益，在一组真正的向量定义的某些线性等式和半定不等式，满足的时刻的任何固定措施的扩散。我们使用的方法，通过几个应用程序：贝叶斯推理问题;计算的线性常微分方程的李雅普诺夫指数受乘性白噪声;和可靠性问题，从结构力学。此外，我们证明了边界收敛到下确界和上确界的一组平稳平均值与计算的李雅普诺夫指数，我们提供了数值证据的收敛在更一般的设置。
英文摘要：
We introduce an algorithm based on semidefinite programming that yields increasing (resp. decreasing) sequences of lower (resp. upper) bounds on polynomial stationary averages of diffusions with polynomial drift vector and diffusion coefficients. The bounds are obtained by optimising an objective, determined by the stationary average of interest, over the set of real vectors defined by certain linear equalities and semidefinite inequalities which are satisfied by the moments of any stationary measure of the diffusion. We exemplify the use of the approach through several applications: a Bayesian inference problem; the computation of Lyapunov exponents of linear ordinary differential equations perturbed by multiplicative white noise; and a reliability problem from structural mechanics. Additionally, we prove that the bounds converge to the infimum and supremum of the set of stationary averages for certain SDEs associated with the computation of the Lyapunov exponents, and we provide numerical evidence of convergence in more general settings.
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