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Alekseev–Gröbner formula

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Formula for expressing the global error of a perturbation

The Alekseev–Gröbner formula, or nonlinear variation-of-constants formula, is a generalization of the linear variation of constants formula which was proven independently by Wolfgang Gröbner in 1960 and Vladimir Mikhailovich Alekseev in 1961. It expresses the global error of a perturbation in terms of the local error and has many applications for studying perturbations of ordinary differential equations.

Formulation

Let d ∈ N {\displaystyle d\in \mathbb {N} } be a natural number, let T ∈ ( 0 , ∞ ) {\displaystyle T\in (0,\infty )} be a positive real number, and let μ : [ 0 , T ] × R d → R d ∈ C 0 , 1 ( [ 0 , T ] × R d ) {\displaystyle \mu \colon \times \mathbb {R} ^{d}\to \mathbb {R} ^{d}\in C^{0,1}(\times \mathbb {R} ^{d})} be a function which is continuous on the time interval [ 0 , T ] {\displaystyle } and continuously differentiable on the d {\displaystyle d} -dimensional space R d {\displaystyle \mathbb {R} ^{d}} . Let X : [ 0 , T ] 2 × R d → R d {\displaystyle X\colon ^{2}\times \mathbb {R} ^{d}\to \mathbb {R} ^{d}} , ( s , t , x ) ↦ X s , t x {\displaystyle (s,t,x)\mapsto X_{s,t}^{x}} be a continuous solution of the integral equation X s , t x = x + ∫ s t μ ( r , X s , r x ) d r . {\displaystyle X_{s,t}^{x}=x+\int _{s}^{t}\mu (r,X_{s,r}^{x})dr.} Furthermore, let Y ∈ C 1 ( [ 0 , T ] , R d ) {\displaystyle Y\in C^{1}(,\mathbb {R} ^{d})} be continuously differentiable. We view Y {\displaystyle Y} as the unperturbed function, and X {\displaystyle X} as the perturbed function. Then it holds that X 0 , T Y 0 − Y T = ∫ 0 T ( ∂ ∂ x X r , T Y s ) ( μ ( r , Y r ) − d d r Y r ) d r . {\displaystyle X_{0,T}^{Y_{0}}-Y_{T}=\int _{0}^{T}\left({\frac {\partial }{\partial x}}X_{r,T}^{Y_{s}}\right)\left(\mu (r,Y_{r})-{\frac {d}{dr}}Y_{r}\right)dr.} The Alekseev–Gröbner formula allows to express the global error X 0 , T Y 0 − Y T {\displaystyle X_{0,T}^{Y_{0}}-Y_{T}} in terms of the local error ( μ ( r , Y r ) − d d r Y r ) {\displaystyle (\mu (r,Y_{r})-{\tfrac {d}{dr}}Y_{r})} .

The Itô–Alekseev–Gröbner formula

The Itô–Alekseev–Gröbner formula is a generalization of the Alekseev–Gröbner formula which states in the deterministic case, that for a continuously differentiable function f ∈ C 1 ( R k , R d ) {\displaystyle f\in C^{1}(\mathbb {R} ^{k},\mathbb {R} ^{d})} it holds that f ( X 0 , T Y 0 ) − f ( Y T ) = ∫ 0 T f ′ ( ∂ ∂ x X r , T Y s ) ∂ ∂ x X s , T Y s ( μ ( r , Y r ) − d d r Y r ) d r . {\displaystyle f(X_{0,T}^{Y_{0}})-f(Y_{T})=\int _{0}^{T}f'\left({\frac {\partial }{\partial x}}X_{r,T}^{Y_{s}}\right){\frac {\partial }{\partial x}}X_{s,T}^{Y_{s}}\left(\mu (r,Y_{r})-{\frac {d}{dr}}Y_{r}\right)dr.}

References

  1. Gröbner, Wolfgang (1960). Die Lie-Reihen und Ihre Anwendungen. Berlin: VEB Deutscher Verlag der Wissenschaften.
  2. Alekseev, V. "An estimate for the perturbations of the solution of ordinary differential equations (Russian)". Vestn. Mosk. Univ., Ser. I, Math. Meh. 2, 1961.
  3. Iserles, A. (2009). A first course in the numerical analysis of differential equations (second ed.). Cambridge: Cambridge Texts in Applied Mathematics, Cambridge University Press.
  4. Hudde, A.; Hutzenthaler, M.; Jentzen, A.; Mazzonetto, S. (2018). "On the Itô-Alekseev-Gröbner formula for stochastic differential equations". arXiv:1812.09857 .
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