Local linearization method

Local Linearization Method

In numerical analysis, the Local Linearization (LL) method is a general strategy for designing numerical inte-

grators for differential equations based on a local (piecewise) linearization of the given equation on consecutive

time intervals. The numerical integrators are then iteratively defined as the solution of the resulting piecewise

linear equation at the end of each consecutive interval. The LL method has been development for a variety of

equations such that the ordinary, delayed, random and stochastic differential equations. The LL integrators

are key component in the implementation of inference methods for the estimation of unknown parameters

and unobserved variables of differential equations given time series of (potentially noisy) observations. The

LL schemes are ideals to deals with complex models in a variety of fields as neuroscience, finance, forestry

management, control engineering, mathematical statistics, etc.

High Order Local Linearization Method

High Order Local Linearization (HOLL) method is a generalization of the Local Linearization method oriented

to obtain high order integrators for differential equations that preserve the stability and dynamics of the linear

equations. The integrators are obtained by splitting, on consecutive time intervals, the solution x of the original

equation in two parts: the solution z of the locally linearized equation plus an high order approximation of the

residual ${\displaystyle \mathbf {r} =\mathbf {x} -\mathbf {z} }$ .

Local Linearization scheme

A Local Linearization (LL) scheme is the final recursive algorithm that allows the numerical implementation

of a discretization derived from the LL or HOLL method for a class of differential equations.

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LL methods for ODEs

Consider the d-dimensional Ordinary Differential Equation (ODE).

${\displaystyle {\frac {d\mathbf {x} \left(t\right)}{dt}}=\mathbf {f} \left(t,\mathbf {x} \left(t\right)\right),\qquad t\in \left[t_{0},T\right],\qquad \qquad \qquad \qquad (1).}$ 

with initial condition ${\displaystyle \mathbf {x} (t_{0})=\mathbf {x} _{0}}$ , where ${\displaystyle \mathbf {f} }$  is a differentiable function.

Let ${\displaystyle \left(t\right)_{h}=\{t_{n}:n=0,..,N\}}$  be a time discretization of the time interval ${\displaystyle [t_{0},T]}$  with maximum stepsize h such that ${\displaystyle t_{n}<t_{n+1}\quad and\quad h_{n}=t_{n+1}-t_{n}\leq h}$ . After the local linearization of the equation (1) at the time step ${\displaystyle t_{n}}$  the variation of constants formula yields

${\displaystyle \mathbf {x} (t_{n}+h)=\mathbf {x} (t_{n})+\mathbf {\phi } (t_{n},\mathbf {x} (t_{n});h)+\mathbf {r} (t_{n},\mathbf {x} (t_{n});h),}$ 

where

${\displaystyle \mathbf {\phi } (t_{n},\mathbf {z} _{n};h)=\int \limits _{0}^{h}e^{\mathbf {f} _{\mathbf {x} }\left(t_{n},\mathbf {z} _{n}\right)(h-s)}(\mathbf {f} \left(t_{n},\mathbf {z} _{n}\right)+\mathbf {f} _{t}\left(t_{n},\mathbf {z} _{n}\right)s)ds\qquad }$ 

results from the linear approximation, and

${\displaystyle \mathbf {r} (t_{n},\mathbf {z} _{n};h)=\int \limits _{0}^{h}e^{\mathbf {f} _{\mathbf {\%x} }\left(t_{n},\mathbf {z} _{n}\right)(h-s)}\mathbf {g} _{n}(s,\mathbf {x} \%(t_{n}+s))ds,\qquad \qquad \qquad (2).}$ 

is the residual of the linear approximation. Here, ${\displaystyle \mathbf {f} _{\mathbf {x} }}$ and ${\displaystyle \mathbf {f} _{t}}$  denote the partial derivatives of f with respect to the variables x and t, respectively, and ${\displaystyle \mathbf {g} _{n}(s,\mathbf {u} )=\mathbf {f} (s,\mathbf {u} )-\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {z} _{n})\mathbf {u} -\mathbf {f} _{t}\left(t_{n},\mathbf {z} _{n}\right)(s-t_{n})-\mathbf {f} \left(t_{n},\mathbf {z} _{n}\right)+\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {z} _{n})\mathbf {z} _{n}}$ 

Local Linear discretization

For a time discretization ${\displaystyle \left(t\right)_{h}}$ , the Local Linear discretization of the ODE (1) at each point ${\displaystyle t_{n+1}\in \left(t\right)_{h}}$  is deffined by the recursive expression

${\displaystyle \mathbf {z} _{n+1}=\mathbf {z} _{n}+\mathbf {\phi } (t_{n},\mathbf {z} _{n};h_{n}),\qquad with\quad \mathbf {z} _{0}=\mathbf {x} _{0}{\text{.}}\qquad \qquad \qquad \qquad (3)}$ 

The Local Linear discretization (3) converges with order 2 to the solution of nonlinear ODEs, but it match the

solution of the linear ODEs. The recursion (3) is also known as Exponential Euler discretization.

High Order Local Linear discretizations

For a time discretization ${\displaystyle \left(t\right)_{h},}$  a High Order Local Linear (HOLL) discretization of the ODE (1) at each point ${\displaystyle t_{n+1}\in \left(t\right)_{h}}$  is deffined by the recursive expression

${\displaystyle \mathbf {z} _{n+1}=\mathbf {z} _{n}+\mathbf {\phi } (t_{n},\mathbf {z} _{n};h_{n})+{\widetilde {\mathbf {r} }}(t_{n},\mathbf {z} _{n};h_{n}),\qquad with\quad \mathbf {z} _{0}=\mathbf {x} _{0},\qquad \qquad \qquad (4)}$ 

where ${\displaystyle {\tilde {r}}}$  is an approximation to the residual r of order ${\displaystyle \alpha }$ higher than 2 ${\displaystyle (i.e.,\left\vert \mathbf {r} (t_{n},\mathbf {z} _{n};h)-{\widetilde {\mathbf {r} }}(t_{n},\mathbf {z} _{n};h)\right\vert \propto h^{\alpha }).}$  The HOLL discretization (4) converges with order � to the solution of nonlinear ODEs, but it match the solution of the linear ODEs.

HOLL discretizations can be derived in two ways: 1) by approximating the integral representation (2) of r; and 2) by using a numerical integrator for the di§erential representation of r deffined by

${\displaystyle {\frac {d\mathbf {r} \left(t\right)}{dt}}=\mathbf {q} (t_{n},\mathbf {z} _{n};t\mathbf {,\mathbf {r} } \left(t\right)\mathbf {),} \qquad with\qquad \mathbf {r} \left(t_{n}\right)=\mathbf {0,} \qquad \qquad \qquad (5)}$ 

for all ${\displaystyle t\in \lbrack t_{k},t_{k+1}]}$ , where

${\displaystyle \mathbf {q} (t_{n},\mathbf {z} _{n};s\mathbf {,\xi } )=\mathbf {f} (s,\mathbf {z} _{n}+\%\mathbf {\phi } \left(t_{n},\mathbf {z} _{n};s-t_{n}\right)+\mathbf {\xi } )-\%\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {z} _{n})\mathbf {\phi } \left(t_{n},\%\mathbf {z} _{n};s-t_{n}\right)-\mathbf {f} _{t}\left(t_{n},\mathbf {z} \%_{n}\right)(s-t_{n})-\mathbf {f} \left(t_{n},\mathbf {z} _{n}\right).}$ 

The resulting approximation is often called Locally Linearized discretization.

Known HOLL discretizations are the following.

Locally Linearized Runge Kutta discretization

${\displaystyle \mathbf {z} _{n+1}=\mathbf {z} _{n}+\mathbf {\phi } (t_{n},\mathbf {z} _{n};h_{n})+h_{n}\sum _{j=1}^{s}b_{j}\mathbf {k} _{j},\quad with\quad \mathbf {k} _{i}=\mathbf {q} (t_{n},\mathbf {z} _{n};{\text{ }}t_{n}+c_{i}h_{n}\mathbf {,} \mathbf {} h_{n}\sum _{j=1}^{i-1}a_{ij}\mathbf {k} _{j}),}$ 

which is obtained by solving (5) via a s-stage RK scheme with coefficients ${\displaystyle \mathbf {c} =\left[c_{i}\right],\mathbf {A} =\left[a_{ij}\right]\quad and\quad \mathbf {b} =\left[b_{j}\right]}$ 

Local Linear Taylor discretization

${\textstyle \mathbf {z} _{n+1}=\mathbf {z} _{n}+\mathbf {\phi } (t_{n},\mathbf {z} \%_{n};h_{n})+\int _{0}^{h_{n}}e^{\left(h_{n}-s\right)\mathbf {f} _{\mathbf {x} \%}\left(t_{n},\mathbf {z} _{n}\right)}\sum _{j=2}^{p}{\frac {\mathbf {c} _{n,j}}{j!\%}}s^{j}ds,{\text{ with }}\mathbf {c} _{n,j}=\left({\frac {d^{j+1}\mathbf {x} \left(t\right)}{dt^{j+1}}}-\mathbf {f} _{\mathbf {x} }\left(t_{n},\mathbf {z} \%_{n}\right){\frac {d^{j}\mathbf {x} \left(t\right)}{dt^{j}}}\right)\mid _{t=\%\mathbf {z} _{n}},}$ 

which results from the approximation of ${\displaystyle \mathbf {g} _{n}}$ in (2) by its order-p truncated Taylor expansion.

Exponential Rosembrock discretization (poner link) is obtained by approximating the integral (2) by aquadrature rule.

Linealized Exponential Adams discretization

${\textstyle \mathbf {z} _{n+1}=\mathbf {z} _{n}+\mathbf {\phi } (t_{n},\mathbf {z} _{n};h_{n})+h_{n}\sum _{j=1}^{p}\sum _{l=1}^{j}{\frac {\gamma _{j+1}}{l}}\nabla ^{l}\mathbf {g} _{n}(t_{n},\mathbf {z} _{n}),\quad with\quad \gamma _{j+1}=(-1)^{j+1}\int \limits _{0}^{1}e^{(1-\theta )h_{n}\mathbf {f} _{\mathbf {x} }\left(t_{n},\mathbf {z} _{n}\right)}\theta \left({\begin{array}{c}-\theta \\j\%\end{array}}\%\right)d\theta ,}$ 

which results from the interpolation of ${\displaystyle \mathbf {g} _{n}}$ in (2) by a Hermite polynomial of degree p, where ${\displaystyle \nabla ^{l}\mathbf {g} \%_{n}(t_{m},\mathbf {z} _{m})}$  denotes the l-th backward di§erence of ${\displaystyle \mathbf {g} _{n}(t_{m},\mathbf {z} _{m})}$ .

Local Linearization schemes

All numerical implementation ${\displaystyle \mathbf {y} _{n}}$  of the LL (or of a HOLL) discretization ${\displaystyle \mathbf {z} _{n}}$  involves approximations ${\displaystyle {\widetilde {\phi }}_{j}}$  to

integrals ${\displaystyle \phi _{j}}$  of the form

${\displaystyle \phi _{j}(\mathbf {A} ,h)=\int \limits _{0}^{h}e^{(h-s)\mathbf {A} }s^{j-1}ds,\qquad j=1,2...,}$ 

where A is an d ${\displaystyle \times }$  d matrix. Every numerical implementation ${\displaystyle \mathbf {y} _{n}}$  of a Local Linear discretization ${\displaystyle \mathbf {z} _{n}}$  of any order is generically called Local Linearization scheme.

Computing integrals involving matrix exponential

Among a number of algorithms to compute the integrals ${\displaystyle \phi _{j}}$ , those based on rational Padé and Krylov subspaces

approximations for exponential matrix are preferred. For this, a central role is playing by the expression

${\displaystyle \sum \nolimits _{i=1}^{l}\phi _{i}(\mathbf {A} ,h)\mathbf {a} _{i}=\mathbf {L} e^{h\mathbf {H} }\mathbf {r,} }$ 

where ${\displaystyle \mathbf {a} _{i}}$  are d-dimensional vectors,

${\displaystyle \mathbf {H} ={\begin{bmatrix}\mathbf {A} &\mathbf {v} _{l}&\mathbf {v} _{l-1}&\cdots &\mathbf {v} _{1}\\\mathbf {0} &\mathbf {0} &1&\cdots &0\\\mathbf {0} &\mathbf {0} &0&\ddots &0\\\vdots &\vdots &\vdots &\ddots &1\\\mathbf {0} &\mathbf {0} &0&\cdots &0\%\end{bmatrix}}\in \mathbb {R} ^{(d+lr)\times (d+lr)},}$ 

${\displaystyle \mathbf {L} =[\mathbf {I} \mathbf {0} _{d\times l)}],\mathbf {r} =[\mathbf {0} _{1\times (d+l-1)}\quad 1]^{\intercal }\quad ,and\quad \mathbf {v} _{i}=\mathbf {a} _{i}(i-1)!}$ 

If ${\displaystyle \mathbf {P} _{p,q}(2^{-k}\mathbf {H} h)}$  denotes the (p; q)-Padé approximation of ${\displaystyle e^{2^{-k}\mathbf {H} h}}$  and k is the smallest integer number such that ${\displaystyle \left\Vert 2^{-k}\mathbf {H} h\right\Vert \leq {\frac {1}{2}},}$