My Chairman does not know what I am doing but thinks it is mathematics, The Math Department knows it is astronomy, I know astronomers who believe it is physics, Myself, I like what I do.
Joseph Ford (quoted in a paper by Robert H.G. Helleman)
For a realistic model describing the dynamics of NEOs, \({\color{blue}\dot{x}=f(x,t),\,x(0)=x_0}\), find the initial conditions \({\color{blue}x_* \in \mathbb{R}^d}\)\(({\color{blue}d= 6})\), and perhaps other relevant physical parameters, which better describe for the observations, in order to make precise predictions.
Raposo Pulido V,, PhD Thesis, U. Politécnica Madrid, (2019).
But:
No optical observations of \({\color{green}r}\) or \({\color{green}\dot{r}}\); they must be inferred;
Data is noisy: uncertainties
The problem
To estimate the collision probability, first,a nominal orbit is obtained, under the assumption that the uncertainties associated to the observations behave as iid variables.
The computation of the collision probability essentially is restricted to a 1d curve (direction of maximal instability), and through Monte Carlo calculations there.
Taylor theorem and Taylor models
Theorem (Taylor theorem): Let \({\color{green}f:\mathbb{D}\subset\mathbb{R}^d\to\mathbb{R}}\) is (n+1)-times partially differentiable in \({\color{green}\mathbb{D}=[\mathbf{a},\mathbf{b}]}\), and \({\color{green}\mathbf{x}_0\in \mathbb{D}}\). Then, for every \({\color{green}\mathbf{x}\in[\mathbf{a},\mathbf{b}]}\) there is a \({\color{purple}\xi\in [0,1]}\) such that
Definition (Taylor model with absolute remainder) Let \({\color{blue}f:\mathbb{D}\to\mathbb{R}}\) as before, where the \({\color{blue}\mathbb{D}}\) is the domain, \({\color{blue}\mathbf{x}_0\in\mathbb{D}}\) is an expansion point, \({\color{blue}P_{f,n}(\mathbf{x}-\mathbf{x}_0)}\) is a polynomial of degree (or order) \(n\), and an interval \({\color{blue}\Delta}\) such that \({\color{blue}0\in\Delta}\). We call \({\color{green}M=(P_{f,n}(\mathbf{x}-\mathbf{x}_0), \Delta, \mathbf{x}_0, \mathbb{D})}\) a Taylor model with absolute remainder of \({\color{blue}f}\) around \({\color{blue}\mathbf{x}_0}\) on \({\color{blue}\mathbb{D}}\) when for all \({\color{blue} x\in\mathbb{D}}\) we have \[\begin{equation}
{\color{blue}f(\mathbf{x})\in P_{f,n}(\mathbf{x}-\mathbf{x}_0) + \Delta,}
\end{equation}\] that is \[\begin{equation}
{\color{blue}\forall x\in\mathbb{D}, \exists\, \delta\in\Delta, f(\mathbf{x})-P_{f,n}(\mathbf{x}-\mathbf{x}_0) = \delta.}
\end{equation}\]
Taylor models with relative remainder
Definition (Taylor model with relative remainder): Let \({\color{blue}f:\mathbb{D}\to\mathbb{R}}\) as before, where the \({\color{blue}\mathbb{D}}\) is the domain, \({\color{blue}{x}_0\in\mathbb{D}}\) is an expansion point, \({\color{blue}P_{f,n}({x}-{x}_0)}\) is a polynomial of degree (or order) \(n\), and an interval \({\color{blue}\Delta}\) such that \({\color{blue}0\in\Delta}\). We call \({\color{green}M=(P_{f,n}({x}-{x}_0), \Delta, {x}_0, \mathbb{D})}\) a Taylor model with relative remainder of \({\color{blue}f}\) around \({\color{blue}\mathbf{x}_0}\) on \({\color{blue}\mathbb{D}}\) when for all \({\color{blue} x\in\mathbb{D}}\) we have \[\begin{equation}
{\color{blue}f({x})\in P_{f,n}({x}-{x}_0) + \Delta({x}-{x}_0)^{n+1},}
\end{equation}\] that is \[\begin{equation}
{\color{blue}\forall x\in\mathbb{D}, \exists\, \delta\in\Delta, f({x})-P_{f,n}({x}-{x}_0) = \delta({x}-{x}_0)^{n+1}.}
\end{equation}\]
Addition and multiplication
Let \({\color{blue}f, g:\mathbb{D}\to\mathbb{R}}\) have associated Taylor models around \({\color{blue}\mathbf{x}_0}\), i.e., for all \({\color{blue}\mathbf{x}\in\mathbb{D}}\)\[\begin{eqnarray*}
{\color{blue}f(x)} &{\color{blue}\in} & {\color{blue}P_{f,n}(\mathbf{x}-\mathbf{x}_0) + \Delta_f,}\\
{\color{blue}g(x)} &{\color{blue}\in} & {\color{blue}P_{g,n}(\mathbf{x}-\mathbf{x}_0) + \Delta_g,}
\end{eqnarray*}\] then \[\begin{eqnarray*}
{\color{blue}f(x) + g(x)} &{\color{blue}\in} & {\color{blue}(P_{f,n}(\mathbf{x}-\mathbf{x}_0) + P_{g,n}(\mathbf{x}-\mathbf{x}_0)) + (\Delta_f + \Delta_g),}\\
\\
{\color{blue}f(x) \cdot g(x)} &{\color{blue}\in} & {\color{blue}P_{f\cdot g,n}(\mathbf{x}-\mathbf{x}_0) + {\color{green}P_{f\cdot g,>n}(\mathbb{D}-\mathbf{x}_0)}\, + }\\
& & {\color{blue}P_{f,n}(\mathbb{D}-\mathbf{x}_0) \Delta_g + P_{g,n}(\mathbb{D}-\mathbf{x}_0) \Delta_f + \Delta_f \Delta_g.}
\end{eqnarray*}\]
Functions
Idea (Berz+Makino, Joldes): use Taylor theorem and (an explicit form of) Lagrange remainder (with some manipulations).
Joldes improves the Lagrange form of the remainder computing \({\color{blue}f^{[n+1]}(\mathbb{D})}\); if this quantity has a definite sign, the remainder is monotonic and its bound is improved.
TaylorModels.jl
In TaylorModels.jl we deal with the one-variable and the many-variable expansions, separately.
We exploit Julia’s type system and multiple dispatch thoroughly.
\[\begin{align*}
~\\[1.5cm]
\end{align*}\]
Code time
Summary
Some problems in dynamical astronomy deserve validated numerics!
TaylorModels.jl is on its way; still needs a lot to be done, specially the experience I do not have on validated techniques.
Collaborators:
L.E. Ramírez-Montoya (UNAM), J.A. Pérez-Hernández (Harvard Center for Astrophysics)
D.P. Sanders (USA), O. Hénot (Taiwan), B. Richard (Germany)
L. Ferranti (Vaasa), C. Schilling (Aalborg), M. Forrets (Uruguay), N. Revol (Lyon)
