Splitting-up Method Research Articles

An application of the splitting-up method for the computation of a neural network representation for the solution for the filtering equations

The filtering equations govern the evolution of the conditional distribution of a signal process given partial, and possibly noisy, observations arriving sequentially in time. Their numerical approximation plays a central role in many real-life applications, including numerical weather prediction [Llopis et al. (SIAM J Sci Comput 40(3):A1544–A1565, 2018), Galanis et al. (Geophysicae 24(10): 2451–2460, 2006)], finance [Brigo and Hanzon (Insurance Math Econom 22(1):53–64, 1998), Date and Ponomareva (IMA J Manag Math 22(3): 195–211, 2011), Crisan and Rozovskii (The Oxford handbook of nonlinear filtering, 2011)] and engineering [Myötyri et al. (Reliability Eng Syst Saf 91(2):200–208, 2005)]. One of the classical approaches to approximate the solution of the filtering equations is to use a PDE inspired method, called the splitting-up method, initiated by Gyongy, Krylov, LeGland, among other contributors, see e.g., Gyöngy and Krylov (Stochastic inequalities and applications, Progr. Probab. 56:301–321, 2003), Le Gland(Stochastic partial differential equations and their applications (Charlotte, NC, 1991), Lect. Notes Control Inf. Sci. 176:177–187, 1992). This method, and other PDE based approaches, have particular applicability for solving low-dimensional problems. In this work we combine this method with a neural network representation inspired by [Han et al. (Proc Natl acad Sci 115(34):8505–8510, 2018)]. The new methodology is used to produce an approximation of the unnormalised conditional distribution of the signal process. We further develop a recursive normalisation procedure to recover the normalised conditional distribution of the signal process. The new scheme can be iterated over multiple time steps whilst keeping its asymptotic unbiasedness property intact. We test the neural network approximations with numerical approximation results for the Kalman and Benes filter.

Solution to a stochastic 3D nonlocal Cahn–Hilliard–Navier–Stokes model with shear dependent viscosity via a splitting-up method

Solution to a stochastic 3D nonlocal Cahn–Hilliard–Navier–Stokes model with shear dependent viscosity via a splitting-up method

Approximation of a stochastic two-phase flow model by a splitting-up method

<p style='text-indent:20px;'>In this paper, we consider a stochastic Allen-Cahn Navier-Stokes system in a bounded domain of <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^d, $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M2">\begin{document}$ d = 2,3 $\end{document}</tex-math></inline-formula>. The system models the evolution of an incompressible isothermal mixture of binary fluids under the influence of stochastic external forces. We prove the existence of a global weak martingale solution. The proof is based on splitting-up method as well as some compactness method.

Approximation of backward stochastic partial differential equations by a splitting-up method

This paper develops a splitting-up method to solution of backward stochastic parabolic partial differential equations. Both splitting equations are simpler for the numerical computations. Two different approximation schemes are given, and their convergence results are proved. The first-order convergence rate is proved in the linear case.

Existence of a solution to the stochastic nonlocal Cahn–Hilliard Navier–Stokes model via a splitting-up method

We consider a stochastic diffuse interface model which describes the motion of an incompressible isothermal mixture of two immiscible fluids under the influence of stochastic external forces in a bounded domain of , d = 2, 3. The model consists of the stochastic Navier–Stokes equations coupled with a nonlocal Cahn–Hilliard equation. We prove the existence of a global weak martingale solution via a numerical scheme based on splitting-up method.

Splitting-up scheme for the stochastic Cahn–Hilliard Navier–Stokes model

In this paper, we consider a stochastic Cahn–Hilliard Navier–Stokes system in a bounded domain of [Formula: see text] [Formula: see text]. The system models the evolution of an incompressible isothermal mixture of binary fluids under the influence of stochastic external forces. We prove the existence of a global weak martingale solution. The proof is based on the splitting-up method as well as some compactness method.

Nonlinear edge-preserving diffusion with adaptive source for document images binarization

This paper proposes a nonlinear edge-preserving diffusion equation with an adaptive source term for binarization of degraded document images. The role of nonlinear diffusion term is to smooth images with preservation of text edges and corners, while the source term is responsible for the desired binarization. Unlike other binarization techniques (such as clustering-based and threshold-based), the idea behind the proposed method is that a sequence of gradually binarized images is obtained by solving the evolution equation starting with the image to be binarized, and tends to the slightly smoothed version of the desired binary image at infinity. A semi-implicit parallel splitting-up method is developed for solving the proposed model effectively. The proposed model with algorithm is tested on the DIBCO (Document Image Binarization Competitions) series datasets. The results show that it has generally the best performance, compared to four PDE (partial differential equation)-based binarization models, and six recent and benchmark binarization algorithms (non-PDE based).

Numerical approximation of the inviscid 3D primitive equations in a limited domain

A new set of nonlocal boundary conditions is proposed for the higher modes of the 3D inviscid primitive equations. Numerical schemes using the splitting-up method are proposed for these modes. Numerical simulations of the full nonlinear primitive equations are performed on a nested set of domains, and the results are discussed.

On the splitting-up method for rough (partial) differential equations

This article introduces the splitting method to systems driven by rough paths. The focus is on (nonlinear) partial differential equations with rough noise but we also cover rough differential equations. Applications to stochastic partial differential equations arising in control theory and nonlinear filtering are given.

Convergence and optimization of the parallel method of simultaneous directions for the solution of elliptic problems

For the solution of elliptic problems, fractional step methods and in particular alternating directions (ADI) methods are iterative methods where fractional steps are sequential. Therefore, they only accept parallelization at low level. In [T. Lu, P. Neittaanmäki, X.C. Tai, A parallel splitting-up method for partial differential equations and its applications to Navier–Stokes equations, RAIRO Modél. Math. Anal. Numér. 26 (6) (1992) 673–708], Lu et al. proposed a method where the fractional steps can be performed in parallel. We can thus speak of parallel fractional step (PFS) methods and, in particular, simultaneous directions (SDI) methods. In this paper, we perform a detailed analysis of the convergence and optimization of PFS and SDI methods, complementing what was done in [T. Lu, P. Neittaanmäki, X.C. Tai, A parallel splitting-up method for partial differential equations and its applications to Navier–Stokes equations, RAIRO Modél. Math. Anal. Numér. 26 (6) (1992) 673–708]. We describe the behavior of the method and we specify the good choice of the parameters. We also study the efficiency of the parallelization. Some 2D, 3D and high-dimensional tests confirm our results.

Filtering and identification of Heston's stochastic volatility model and its market risk

We study the filtering problem for the stochastic volatility model of Heston by using the nonlinear estimation theory. To solve the estimation problem for the stochastic volatility process, we use the random time change method. The derived basic equation for the filtering is the so-called Zakai equation and its numerically realized algorithm is proposed with the aid of the splitting-up method. Regarding the European call option problem, the identification of the market price of the volatility risk is also studied. Some numerical simulation studies are demonstrated to show the advantage of the proposed method.

Wavelet-Taylor Galerkin Method for the Burgers Equation

In this paper, we propose a wavelet-Taylor Galerkin method for the numerical solution of the Burgers equation. In deriving the computational scheme, Taylor-generalized Euler time discretization is performed prior to wavelet-based Galerkin spatial approximation. The linear system of equations obtained in the process are solved by approximate-factorization-based simple explicit schemes, and the resulting solution is compared with that from regular methods. To deal with transient advection-diffusion situations that evolve toward a convective steady state, a splitting-up strategy is known to be very effective. So the Burgers equation is also solved by a splitting-up method using a wavelet-Taylor Galerkin approach. Here, the advection and diffusion terms in the Burgers equation are separated, and the solution is computed in two phases by appropriate wavelet-Taylor Galerkin schemes. Asymptotic stability of all the proposed schemes is verified, and the L∞ errors relative to the analytical solution together with the numerical solution are reported.

Existence result for a doubly degenerate quasilinear stochastic parabolic equation

Using the splitting-up method, we establish a new existence result for an initial boundary value problem for the doubly degenerate stochastic quasilinear parabolic equation \begin{equation*} d \bigl( \left|y\right|^{\alpha-2}y\bigr) -\left[ \sum_{i=1}^{n} \frac{\partial}{\partial x_{i}} \left( \left| \frac{\partial y}{\partial x}\right|^{p-2} \frac{\partial y}{\partial x_{i}}\right) - g(t,y) \right] dt = \sum_{l=0}^{d}h_{l} (t,y) dW_{t}^{l}, \end{equation*} where $W_{t}^{l}$ are one-dimensional Wiener process defined on a complete probability space, $p$, $\alpha$ and the functions $g$ and $h_{l}$ satisfy appropriate restrictions.

An Accelerated Splitting-up Method for Parabolic Equations

We approximate the solution u of the Cauchy problem \begin{gather*} \frac{\partial}{\partial t} u(t,x)=Lu(t,x)+f(t,x), \quad (t,x)\in(0,T]\times\bR^d,\\ u(0,x)=u_0(x),\quad x\in\bR^d, \end{gather*} by splitting the equation into the system $$ \frac{\partial}{\partial t} v_r(t,x)=L_rv_r(t,x)+f_r(t,x), \qquad r=1,2,\ldots,d_1, $$ where $L,L_r$ are second order differential operators; f, $f_r$ are functions of $t,x$ such that $L=\sum_r L_r$, $f=\sum_r f_r$. Under natural conditions on solvability in the Sobolev spaces $W^m_p$, we show that for any $k>1$ one can approximate the solution u with an error of order $\delta^k$, by an appropriate combination of the solutions $v_r$ along a sequence of time discretization, where $\delta$ is proportional to the step size of the grid. This result is obtained by using the time change introduced in [I. Gyöngy and N. Krylov, Ann. Probab., 31 (2003), pp. 564-591], together with Richardson's method and a power series expansion of the error of splitting-up approximations in terms of $\delta$.

Filtering of Stochastic Volatility Model

We study the filtering problem for the stochastic volatility model of Heston by using the nonlinear estimation theory. To solve the estimation problem for the stochastic volatility process, we use the random time change method. The derived basic equation for the filtering is the so-called Zakai equation and its numerically realized algorithm is proposed with the aid of the splitting-up method. Some numerical simulation studies are demonstrated to show the advantage of the proposed method.

On the splitting-up method and stochastic partial differential equations

We consider two stochastic partial differential equations du_{\varepsilon}(t)= (L_ru_{\varepsilon}(t)+f_{r}(t)) \,dV_{\varepsilon t}^r+(M_{k}u_{\varepsilon}(t)+g_k(t))\, \circ dY_t^k, \qquad\hspace*{-5pt} \varepsilon=0,1, driven by the same multidimensional martingale Y=(Yk) and by different increasing processes Vr0, Vr1, r=1,2,…,d1, where Lr and Mk are second-and first-order partial differential operators and ∘ stands for the Stratonovich differential. We estimate the moments of the supremum in t of the Sobolev norms of u1(t)−u0(t) in terms of the supremum of the differences\break |Vr0t−Vr1t|. Hence, we obtain moment estimates for the error of a multistage splitting-up method for stochastic PDEs, in particular, for the equation of the unnormalized conditional density in nonlinear filtering.

Splitting-up method and adjoint equation method in the ocean dynamics problem

Splitting-up method and adjoint equation method in the ocean dynamics problem

Remarks on Nonlinear Stochastic Partial Differential Equations: An Application of the Splitting-Up Method

The objective of this article is to apply the splitting-up method to the existence theorem for nonlinear stochastic partial differential equations. With use of this method an approximating sequence is constructed. By the compactness argument a convergent subsequence can be extracted, and this fact provides the solution.

Stochastic Navier-Stokes Equations

The purpose of this article is to survey some results related to the theory of stochastic Navier-Stokes equations (SNSE). The interest of SNSE arises from modelling turbulence. We begin to show how SNSE can be introduced intuitively from the random motion of particles. We then review briefly the deterministic theory and present the main core of existence theory for NSE. We also discuss uniqueness issues. We end up by showing how the splitting-up method provides a useful constructive approach to existence, and by presenting some extensions, like weakening assumptions or considering the special case of small initial data.

A numerical technique for geophysical data assimilation problems using Pontryagin’s principle and splitting-up method

Article A numerical technique for geophysical data assimilation problems using Pontryagin’s principle and splitting-up method was published on January 1, 1993 in the journal Russian Journal of Numerical Analysis and Mathematical Modelling (volume 8, issue 4).