Local linearization method

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Local Linearization Method

1 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.

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

3 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.

4 LL methods for ODEs

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

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

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 ${\displaystyle t_{n}<t_{n+1}\$and\$h_{n}=t_{n+1}-t_{n}\leq h}$ . 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

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

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 ${\displaystyle \mathbf {g} _{n}(s,\mathbf {u} )=\mathbf {f} (s,\mathbf {u} )-\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {z} _{n})\mathbf {u} -\mathbf {f} _{t}\left(t_{n},\mathbf {z} _{n}\right)(s-t_{n})-\mathbf {f} \left(t_{n},\mathbf {z} _{n}\right)+\mathbf {f} _{\mathbf {x} }(t_{n},\mathbf {z} _{n})\mathbf {z} _{n}}$