Differential Inversion of the Implicit Euler Method: Symbolic Analysis

Uwe Naumann
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Abstract

The implicit Euler method integrates systems of ordinary differential equations $$\frac{d x}{d t}=G(t,x(t))$$ with differentiable right-hand side $G : R \times R^n \rightarrow R^n$ from an initial state $x=x(0) \in R^n$ to a target time $t \in R$ as $x(t)=E(t,m,x)$ using an equidistant discretization of the time interval $[0,t]$ yielding $m>0$ time steps. We aim to compute the product of its inverse Jacobian $$ (E')^{-1} \equiv \left (\frac{d E}{d x}\right )^{-1} \in R^{n \times n} $$ with a given vector efficiently. We show that the differential inverse $(E')^{-1} \cdot v$ can be evaluated for given $v \in R^n$ with a computational cost of $\mathcal{O}(m \cdot n^2)$ as opposed to the standard $\mathcal{O}(m \cdot n^3)$ or, naively, even $\mathcal{O}(m \cdot n^4).$ The theoretical results are supported by actual run times. A reference implementation is provided.
隐式欧拉法的微分反演:符号分析
隐式欧拉法积分常微分方程系统 $$\frac{d x}{d t}=G(t,x(t))$$ 具有可微分右边$G:R \times R^n \rightarrow R^n$ 从初始状态 $x=x(0) \in R^n$ 到目标时间 $t \in R$ 为 $x(t)=E(t,m,x)$,使用时间区间 $[0,t]$ 的等距离散化,产生 $m>0$ 的时间步长。我们的目标是在 R^{n \times n} $$中用给定矢量高效地计算其逆雅各布值 $ (E')^{-1} \equiv \left (\frac{d E}{d x}\right )^{-1} \$ 的乘积。我们证明,对于 R^n$ 中的给定 $v ,可以用 $\mathcal{O}(m\cdot n^2)$ 的计算成本求出微分逆 $(E')^{-1} \cdot v$,而不是标准的 $\mathcal{O}(m\cdot n^3)$ ,甚至不是 $\mathcal{O}(m\cdot n^4) 。我们提供了一个参考实现。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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