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== Weak LL methods for SDEs ==
Consider the ''d''-dimensional stochastic differential equation
<math>d\mathbf{x}(t)=\mathbf{f}(t,\mathbf{x}(t))dt+\sum\limits_{i=1}^{m}\mathbf{g}
_{i}(t)d\mathbf{w}^{i}(t),\qquad t\in \left[ t_{0},T\right] , \qquad \qquad (19)</math>
with initial condition <math>\mathbf{x}(t_{0})=\mathbf{x}_{0}</math>, where the drift coefficient <math>\mathbf{f}</math> and the diffusion coefficient <math>\mathbf{g}_{i}</math> are differentiable functions, and <math>\mathbf{w=(\mathbf{w}}^{1},\ldots ,\mathbf{w}^{m}\mathbf{)}</math> is an ''m''-dimensional standard Wiener process.
=== Local Linear discretization ===
For a time discretization <math>\left( t\right) _{h}</math>, the order-<math>\mathbb{\beta }</math> <math>(=1,2)</math> ''Weak Local Linear discretization'' of the solution of the SDE '''(19)''' is defined by the recursive relation
<math>\mathbf{z}_{n+1}=\mathbf{z}_{n}+\mathbf{\phi }_{\mathbb{\beta }}(t_{n},
\mathbf{z}_{n};h_{n})+\mathbf{\eta }(t_{n},\mathbf{z}_{n};h_{n}),\quad with \quad \mathbf{z}_{0}=\mathbf{x}_{0}, </math>
where
<math>\mathbf{\phi }_{\mathbb{\beta }}(t_{n},\mathbf{z}_{n};\delta
)=\int_{0}^{\delta }e^{\mathbf{f}_{\mathbf{x}}(t_{n},\mathbf{z}_{n})(\delta
-u)}(\mathbf{f(}t_{n},\mathbf{z}_{n})+\mathbf{b}^{\mathbb{\beta }}(t_{n},
\mathbf{z}_{n})u)du </math>
with
<math>\mathbf{b}^{\mathbb{\beta }}(t_{n},\mathbf{z}_{n})=
\begin{cases}{ll}
\mathbf{f}_{t}(t_{n},\mathbf{z}_{n}) & \text{for }\mathbb{\beta }=1 \\
\mathbf{f}_{t}(t_{n},\mathbf{z}_{n})\quad +\frac{1}{2}\sum
\limits_{j=1}^{m}\left( \mathbf{I}_{d\times d}\otimes \mathbf{g}
_{j}^{\intercal }\left( t_{n}\right) \right) \mathbf{f}_{\mathbf{xx}}(t_{n},
\mathbf{z}_{n})\mathbf{g}_{j}\left( t_{n}\right) & \text{for }\mathbb{\beta }
=2,
\end{cases}
.
</math>
and <math>\mathbf{\eta }(t_{n},\mathbf{z}_{n};\delta )</math> is a zero mean stochastic process with variance matrix
<math>\mathbf{\Sigma }(t_{n},\mathbf{z}_{n};\delta )=\int\limits_{0}^{\delta }e^{
\mathbf{f}_{\mathbf{x}}(t_{n},\mathbf{z}_{n})(\delta -s)}\mathbf{G}(t_{n}+s)
\mathbf{G}^{\intercal }(t_{n}+s)e^{\mathbf{f}_{\mathbf{x}}^{\intercal
}(t_{n},\mathbf{z}_{n})(\delta -s)}ds. </math>
Here, <math>\mathbf{f}_{\mathbf{x}}</math>, <math>\mathbf{f}_{t}</math> denote the partial derivatives of <math>\mathbf{f}</math> with respect to the variables <math>\mathbf{x}</math> and '''t''', respectively, <math>\mathbf{f}_{\mathbf{xx}}</math> the Hessian matrix of <math>\mathbf{f}</math> with respect to <math>\mathbf{x}</math>, and <math>\mathbf{G}(t)=[\mathbf{g}_{1}(t),..., \mathbf{g}_{m}(t)]</math>. The weak Local Linear discretization <math>\mathbf{z}_{n+1}</math> [[Convergence of random variables|converges]] with order <math>\mathbb{\beta }</math> (=1,2) to the solution of '''(19)'''.
=== Local Linearization schemes ===
Depending on the way of computing <math>\mathbf{\phi }_{\mathbb{\beta }}</math> and <math>\mathbf{\Sigma }</math> different numerical schemes could be obtained. Every numerical implementation <math>\mathbf{y}_{n}</math> of a Weak Local Linear discretization <math>\mathbf{z}_{n}</math> is generically called ''Weak Local Linearization scheme''.
==== Order 1 WLL scheme ====
<math>\mathbf{y}_{n+1}=\mathbf{y}_{n}+\mathbf{B}_{14}+(\mathbf{B}_{12}\mathbf{B}
_{11}^{\intercal })^{1/2}\mathbf{\xi }_{n} </math>
where, for SDEs with autonomous diffusion coefficients, <math>\mathbf{B}_{11}</math>, <math>\mathbf{B}_{12}</math> and <math>\mathbf{B}_{14}</math> are the submatrices defined by the [[Block matrix|partitioned matrix]] <math>\mathbf{B}=\mathbf{P}_{p,q}(2^{-k_{n}}\mathcal{M}_{n}h_{n}))^{2^{k_{n}}}</math>, with
<math>\mathcal{M}_{n}=\left[
\begin{array}{cccc}
\mathbf{f}_{\mathbf{x}}(t_{n},\mathbf{y}_{n}) & \mathbf{GG}^{\intercal } &
\mathbf{f}_{t}(t_{n},\mathbf{y}_{n}) & \mathbf{f}(t_{n},\mathbf{y}_{n}) \\
\mathbf{0} & -\mathbf{f}_{\mathbf{x}}^{\intercal }(t_{n},\mathbf{y}_{n}) &
\mathbf{0} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & 0 & 1 \\
\mathbf{0} & \mathbf{0} & 0 & 0%
\end{array}%
\right] \in \mathbb{R}^{(2d+2)\times (2d+2)}, </math>
and <math>\{\mathbf{\xi }_{n}\}</math> is a sequence of ''d''-dimensional independent [[Bernoulli distribution|two-points distributed random vectors]] satisfying
<math>P(\xi _{n}^{k}=\pm 1)=\frac{1}{2}. </math>
==== Order 2 WLL scheme ====
<math>\mathbf{y}_{n+1}=\mathbf{y}_{n}+\mathbf{B}_{16}+(\mathbf{B}_{14}\mathbf{B}
_{11}^{\intercal })^{1/2}\mathbf{\xi }_{n} </math>
where <math>\mathbf{B}_{11}</math>, <math>\mathbf{B}_{14}</math> and <math>\mathbf{B}_{16}</math> are the submatrices defined by the partitioned matrix <math>\mathbf{B}=\mathbf{P} _{p,q}(2^{-k_{n}}\mathcal{M}_{n}h_{n}))^{2^{k_{n}}}</math> with
<math>\mathcal{M}_{n}=\left[
\begin{array}{cccccc}
\mathbf{J} & \mathbf{H}_{2} & \mathbf{H}_{1} & \mathbf{H}_{0} & \mathbf{a}
_{2} & \mathbf{a}_{1} \\
\mathbf{0} & -\mathbf{J}^{\intercal } & \mathbf{I} & \mathbf{0} & \mathbf{0}
& \mathbf{0} \\
\mathbf{0} & \mathbf{0} & -\mathbf{J}^{\intercal } & \mathbf{I} & \mathbf{0}
& \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{0} & -\mathbf{J}^{\intercal } & \mathbf{0}
& \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & 0 & 1 \\
\mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & 0 & 0
\end{array}
\right] \in \mathbb{R}^{(4d+2)\times (4d+2)}, </math>
<math>\mathbf{J}=\mathbf{f}_{\mathbf{x}}(t_{n},\mathbf{y}_{n})\qquad
\mathbf{a}_{1}=\mathbf{f}(t_{n},\mathbf{y}_{n})\qquad \mathbf{a}
_{2}=\mathbf{f}_{t}(t_{n},\mathbf{y}_{n})+\frac{1}{2}\sum\limits_{i=1}^{m}(
\mathbf{I}\otimes (\mathbf{g}^{i}(t_{n}))^{\intercal })\mathbf{f}_{\mathbf{xx
}}(t_{n},\mathbf{y}_{n})\mathbf{g}^{i}(t_{n}) </math>
and
<math>\mathbf{H}_{0}=\mathbf{G}(t_{n})\mathbf{G}^{\intercal }(t_{n})\qquad
\mathbf{H}_{1}=\mathbf{G}(t_{n})\frac{d\mathbf{G}^{\intercal }(t_{n})}{dt}
+\frac{d\mathbf{G}(t_{n})}{dt}\mathbf{G}^{\intercal }(t_{n})\qquad
\mathbf{H}_{2}=\frac{d\mathbf{G}(t_{n})}{dt}\frac{d\mathbf{G}^{\intercal
}(t_{n})}{dt}\text{.} </math>
==== Stability and dynamics ====
By construction the weak LL and HOLL discretizations inherit the stability and [[Random dynamical system|dynamics]] of the linear ODEs, but it is not the case of the weak LL schemes in general. WLL schemes, with <math>p\leq q\leq p+2,</math> preserve the [[Moment (mathematics)|first two moments]] of the linear SDEs, and inherits the mean-square stability or instability that such solution may have. This includes, for instance, the equations of coupled harmonic oscillators driven by random force, and large systems of stiff linear SDEs that result from the method of lines for linear stochastic partial differential equations. Moreover these WLL schemes preserve the [[ergodicity]] of the linear equations, and are geometrically ergodic for some classes of nonlinear SDEs. For nonlinear SDEs with small noise (e.g., ('''19''') with <math>\mathbf{g}_{i}(t)\approx 0</math>), the solutions of these WLL schemes are basically the nonrandom paths of the LL scheme ('''6''') for ODEs plus a small disturbance related to the small noise. In this situation, the dynamical properties of that deterministic scheme, such as the linearization preserving and the preservation of the exact solution dynamics around hyperbolic equilibrium points and periodic orbits, become relevant for the mean of the WLL scheme.
== Historical notes ==
Below is a time line of the main developments of the LL method.
- D.A. Pope (1963) introduces the LL discretization for ODEs and the LL scheme based on Taylor expantion
http://dx.doi.org/10.1145/366707.367592
- T. Ozaki (1985) introduces the LL method for the integration and estimation of SDEs. The term\ "Local Linearization" (LL) is used for first time.
https://doi.org/10.1016/S0169-7161(85)05004-0
- R. Biscay et al. (1996) reformulate the strong LL method for SDEs.
http://dx.doi.org/10.1007/BF00052324
- I. Shoji \& T. Ozaki (1997) reformulate the weak LL method for SDEs
http://dx.doi.org/10.1111/1467-9892.00064
- M. Hochbruck et al. (1998) introduce the LL scheme for ODEs based on Krylov subspace approximation
http://dx.doi.org/10.1137/S1064827595295337
- J.C. Jimenez (2002) introduces the LL scheme for ODEs and SDEs based on rational Padé approximation
http://dx.doi.org/10.1016/S0893-9659(02)00041-1
- F.M. Carbonell et al. (2005) introduce the LL method for RDEs
http://dx.doi.org/10.1007/s10543-005-2645-9
- J.C. Jimenez et al. (2006) introduce the LL method for DDEs
http://dx.doi.org/10.1137/040607356
- de la Cruz et al. (2006, 2007) introduce the first two classes of HOLL integrators for ODEs: LLT and LLRK
http://dx.doi.org/10.1007/11758501\_22, http://dx.doi.org/10.1016/j.amc.2006.06.096
- de la Cruz et al. (2010) introduce the strong HOLL method for SDEs
http://dx.doi.org/10.1007/s10543-010-0272-6
== References ==
de la Cruz H., Biscay R.J., Carbonell F., Ozaki T., Jimenez J.C., A higher order Local Linearization method for solving ordinary differential equations, Appl. Math. Comput., 185 (2007) 197-212.
http://dx.doi.org/10.1016/j.amc.2006.06.096
de la Cruz H., Biscay R.J., Jimenez J.C. and Carbonell F. Local Linearization - Runge Kutta Methods: a class of A-stable explicit integrators for dynamical systems, Math. Comput. Modelling, 57 (2013) 720-740.
http://dx.doi.org/10.1016/j.mcm.2012.08.011
de la Cruz H., Biscay R.J., Jimenez J.C., Carbonell F., Ozaki T., High Order Local Linearization methods: an approach for constructing A-stable high order explicit schemes for stochastic differential equations with additive noise, BIT Numer. Math., 50 (2010) 509--539.
http://dx.doi.org/10.1007/s10543-010-0272-6
de la Cruz H., Jimenez J.C., Zubelli J.P., Locally Linearized methods for the simulation of stochastic oscillators driven by random forces, BIT Numer. Math., 57 (2017) 123-151.
http://dx.doi.org/10.1007/s10543-016-0620-2
M. Hochbruck, A. Ostermann, Exponential multistep methods of Adams-type, BIT Numer. Math. 51 (2011) 889--908.
http://dx.doi.org/10.1007/s10543-011-0332-6
Jimenez J.C., Local Linearization methods for the numerical integration of ordinary differential equations: An overview. ICTP Technical Report 035, 2009.
http://publications.ictp.it
Jimenez J.C., Biscay R., Mora C.M., Rodriguez L.M., Dynamic properties of the Local Linearization method for initial-value problems, Appl. Math. Comput., 126 (2002) 63-80.
http://dx.doi.org/10.1016/S0096-3003(00)00100-4
Jimenez J.C., Carbonell F., Rate of convergence of local linearization schemes for random differential equations, BIT Numer. Math., 49 (2009) 357--373.
http://dx.doi.org/10.1007/s10543-009-0225-0
Jimenez J.C., Carbonell F., Convergence rate of weak Local Linearization schemes for stochastic differential equations with additive noise, J. Comput. Appl. Math., 279 (2015) 106-122.
http://dx.doi.org/10.1016/j.cam.2014.10.021
Jimenez J.C., de la Cruz H., Convergence rate of strong Local Linearization schemes for stochastic differential equations with additive noise, BIT Numer. Math., 52 (2012) 357-382.
http://dx.doi.org/10.1007/s10543-011-0360-2
Jimenez J.C., Pedroso L., Carbonell F., Hernadez V., Local linearization method for numerical integration of delay differential equations, SIAM J. Numer. Analysis, 44 (2006) 2584-2609.
http://dx.doi.org/10.1137/040607356
Jimenez J.C., Sotolongo A. and Sanchez-Bornot J.M., Locally Linearized Runge Kutta method of Dormand and Prince, Appl. Math. Comput., 247 (2014) 589-606.
http://dx.doi.org/10.1016/j.amc.2014.09.001
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