Relevant bibliographies by topics / Numerical scheme for SDEs

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  2. Dissertations / Theses
  3. Books
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Academic literature on the topic 'Numerical scheme for SDEs'

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Published: 9 September 2022

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Journal articles on the topic "Numerical scheme for SDEs"

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C. De Vecchi, Francesco, Andrea Romano, and Stefania Ugolini. "A symmetry-adapted numerical scheme for SDEs." Journal of Geometric Mechanics 11, no. 3 (2019): 325–59. http://dx.doi.org/10.3934/jgm.2019018.

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Yamada, Toshihiro. "High order weak approximation for irregular functionals of time-inhomogeneous SDEs." Monte Carlo Methods and Applications 27, no. 2 (February 20, 2021): 117–36. http://dx.doi.org/10.1515/mcma-2021-2085.

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Abstract This paper shows a general weak approximation method for time-inhomogeneous stochastic differential equations (SDEs) using Malliavin weights. A unified approach is introduced to construct a higher order discretization scheme for expectations of non-smooth functionals of solutions of time-inhomogeneous SDEs. Numerical experiments show the validity of the method.
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Ewald, Brian. "Weak Versions of Stochastic Adams-Bashforth and Semi-implicit Leapfrog Schemes for SDEs." Computational Methods in Applied Mathematics 12, no. 1 (2012): 23–31. http://dx.doi.org/10.2478/cmam-2012-0002.

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AbstractWe consider the weak analogues of certain strong stochastic numerical schemes, namely an Adams-Bashforth scheme and a semi-implicit leapfrog scheme. We show that the weak version of the Adams-Bashforth scheme converges weakly with order 2, and the weak version of the semi-implicit leapfrog scheme converges weakly with order 1. We also note that the weak schemes are computationally simpler and easier to implement than the corresponding strong schemes, resulting in savings in both programming and computational effort.
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Li, Xingjie Helen, Fei Lu, and Felix X. F. Ye. "ISALT: Inference-based schemes adaptive to large time-stepping for locally Lipschitz ergodic systems." Discrete & Continuous Dynamical Systems - S 15, no. 4 (2022): 747. http://dx.doi.org/10.3934/dcdss.2021103.

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<p style='text-indent:20px;'>Efficient simulation of SDEs is essential in many applications, particularly for ergodic systems that demand efficient simulation of both short-time dynamics and large-time statistics. However, locally Lipschitz SDEs often require special treatments such as implicit schemes with small time-steps to accurately simulate the ergodic measures. We introduce a framework to construct inference-based schemes adaptive to large time-steps (ISALT) from data, achieving a reduction in time by several orders of magnitudes. The key is the statistical learning of an approximation to the infinite-dimensional discrete-time flow map. We explore the use of numerical schemes (such as the Euler-Maruyama, the hybrid RK4, and an implicit scheme) to derive informed basis functions, leading to a parameter inference problem. We introduce a scalable algorithm to estimate the parameters by least squares, and we prove the convergence of the estimators as data size increases.</p><p style='text-indent:20px;'>We test the ISALT on three non-globally Lipschitz SDEs: the 1D double-well potential, a 2D multiscale gradient system, and the 3D stochastic Lorenz equation with a degenerate noise. Numerical results show that ISALT can tolerate time-step magnitudes larger than plain numerical schemes. It reaches optimal accuracy in reproducing the invariant measure when the time-step is medium-large.</p>
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Armstrong, J., and D. Brigo. "Intrinsic stochastic differential equations as jets." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 474, no. 2210 (February 2018): 20170559. http://dx.doi.org/10.1098/rspa.2017.0559.

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We explain how Itô stochastic differential equations (SDEs) on manifolds may be defined using 2-jets of smooth functions. We show how this relationship can be interpreted in terms of a convergent numerical scheme. We also show how jets can be used to derive graphical representations of Itô SDEs, and we show how jets can be used to derive the differential operators associated with SDEs in a coordinate-free manner. We relate jets to vector flows, giving a geometric interpretation of the Itô–Stratonovich transformation. We show how percentiles can be used to give an alternative coordinate-free interpretation of the coefficients of one-dimensional SDEs. We relate this to the jet approach. This allows us to interpret the coefficients of SDEs in terms of ‘fan diagrams’. In particular, the median of an SDE solution is associated with the drift of the SDE in Stratonovich form for small times.
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Mao, Xuerong, Aubrey Truman, and Chenggui Yuan. "Euler-Maruyama approximations in mean-reverting stochastic volatility model under regime-switching." Journal of Applied Mathematics and Stochastic Analysis 2006 (July 13, 2006): 1–20. http://dx.doi.org/10.1155/jamsa/2006/80967.

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Stochastic differential equations (SDEs) under regime-switching have recently been developed to model various financial quantities. In general, SDEs under regime-switching have no explicit solutions, so numerical methods for approximations have become one of the powerful techniques in the valuation of financial quantities. In this paper, we will concentrate on the Euler-Maruyama (EM) scheme for the typical hybrid mean-reverting θ-process. To overcome the mathematical difficulties arising from the regime-switching as well as the non-Lipschitz coefficients, several new techniques have been developed in this paper which should prove to be very useful in the numerical analysis of stochastic systems.
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Zhang, Wei. "Ergodic SDEs on submanifolds and related numerical sampling schemes." ESAIM: Mathematical Modelling and Numerical Analysis 54, no. 2 (February 12, 2020): 391–430. http://dx.doi.org/10.1051/m2an/2019071.

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In many applications, it is often necessary to sample the mean value of certain quantity with respect to a probability measure μ on the level set of a smooth function ξ : ℝd → ℝk, 1 ≤ k < d. A specially interesting case is the so-called conditional probability measure, which is useful in the study of free energy calculation and model reduction of diffusion processes. By Birkhoff’s ergodic theorem, one approach to estimate the mean value is to compute the time average along an infinitely long trajectory of an ergodic diffusion process on the level set whose invariant measure is μ. Motivated by the previous work of Ciccotti et al. (Commun. Pur. Appl. Math. 61 (2008) 371–408), as well as the work of Leliévre et al. (Math. Comput. 81 (2012) 2071–2125), in this paper we construct a family of ergodic diffusion processes on the level set of ξ whose invariant measures coincide with the given one. For the conditional measure, we propose a consistent numerical scheme which samples the conditional measure asymptotically. The numerical scheme doesn’t require computing the second derivatives of ξ and the error estimates of its long time sampling efficiency are obtained.
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Buckwar, Evelyn, Massimiliano Tamborrino, and Irene Tubikanec. "Spectral density-based and measure-preserving ABC for partially observed diffusion processes. An illustration on Hamiltonian SDEs." Statistics and Computing 30, no. 3 (November 5, 2019): 627–48. http://dx.doi.org/10.1007/s11222-019-09909-6.

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Abstract Approximate Bayesian computation (ABC) has become one of the major tools of likelihood-free statistical inference in complex mathematical models. Simultaneously, stochastic differential equations (SDEs) have developed to an established tool for modelling time-dependent, real-world phenomena with underlying random effects. When applying ABC to stochastic models, two major difficulties arise: First, the derivation of effective summary statistics and proper distances is particularly challenging, since simulations from the stochastic process under the same parameter configuration result in different trajectories. Second, exact simulation schemes to generate trajectories from the stochastic model are rarely available, requiring the derivation of suitable numerical methods for the synthetic data generation. To obtain summaries that are less sensitive to the intrinsic stochasticity of the model, we propose to build up the statistical method (e.g. the choice of the summary statistics) on the underlying structural properties of the model. Here, we focus on the existence of an invariant measure and we map the data to their estimated invariant density and invariant spectral density. Then, to ensure that these model properties are kept in the synthetic data generation, we adopt measure-preserving numerical splitting schemes. The derived property-based and measure-preserving ABC method is illustrated on the broad class of partially observed Hamiltonian type SDEs, both with simulated data and with real electroencephalography data. The derived summaries are particularly robust to the model simulation, and this fact, combined with the proposed reliable numerical scheme, yields accurate ABC inference. In contrast, the inference returned using standard numerical methods (Euler–Maruyama discretisation) fails. The proposed ingredients can be incorporated into any type of ABC algorithm and directly applied to all SDEs that are characterised by an invariant distribution and for which a measure-preserving numerical method can be derived.
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BRUTI-LIBERATI, NICOLA, and ECKHARD PLATEN. "STRONG PREDICTOR–CORRECTOR EULER METHODS FOR STOCHASTIC DIFFERENTIAL EQUATIONS." Stochastics and Dynamics 08, no. 03 (September 2008): 561–81. http://dx.doi.org/10.1142/s0219493708002457.

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This paper introduces a new class of numerical schemes for the pathwise approximation of solutions of stochastic differential equations (SDEs). The proposed family of strong predictor–corrector Euler methods are designed to handle scenario simulation of solutions of SDEs. It has the potential to overcome some of the numerical instabilities that are often experienced when using the explicit Euler method. This is of importance, for instance, in finance where martingale dynamics arise for solutions of SDEs with multiplicative diffusion coefficients. Numerical experiments demonstrate the improved asymptotic stability properties of the proposed symmetric predictor–corrector Euler methods.
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Kloeden, P. E., and S. Shott. "Linear-implicit strong schemes for Itô-Galkerin approximations of stochastic PDEs." Journal of Applied Mathematics and Stochastic Analysis 14, no. 1 (January 1, 2001): 47–53. http://dx.doi.org/10.1155/s1048953301000053.

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Linear-implicit versions of strong Taylor numerical schemes for finite dimensional Itô stochastic differential equations (SDEs) are shown to have the same order as the original scheme. The combined truncation and global discretization error of an γ strong linear-implicit Taylor scheme with time-step Δ applied to the N dimensional Itô-Galerkin SDE for a class of parabolic stochastic partial differential equation (SPDE) with a strongly monotone linear operator with eigenvalues λ1≤λ2≤… in its drift term is then estimated by K(λN+1−½+Δγ) where the constant K depends on the initial value, bounds on the other coefficients in the SPDE and the length of the time interval under consideration.
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Dissertations / Theses on the topic "Numerical scheme for SDEs"

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Kumar, Chaman. "Explicit numerical schemes of SDEs driven by Lévy noise with super-linear coeffcients and their application to delay equations." Thesis, University of Edinburgh, 2015. http://hdl.handle.net/1842/15946.

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We investigate an explicit tamed Euler scheme of stochastic differential equation with random coefficients driven by Lévy noise, which has super-linear drift coefficient. The strong convergence property of the tamed Euler scheme is proved when drift coefficient satisfies one-sided local Lipschitz condition whereas diffusion and jump coefficients satisfy local Lipschitz conditions. A rate of convergence for the tamed Euler scheme is recovered when local Lipschitz conditions are replaced by global Lipschitz conditions and drift satisfies polynomial Lipschitz condition. These findings are consistent with those of the classical Euler scheme. New methodologies are developed to overcome challenges arising due to the jumps and the randomness of the coefficients. Moreover, as an application of these findings, a tamed Euler scheme is proposed for the stochastic delay differential equation driven by Lévy noise with drift coefficient that grows super-linearly in both delay and non-delay variables. The strong convergence property of the tamed Euler scheme for such SDDE driven by Lévy noise is studied and rate of convergence is shown to be consistent with that of the classical Euler scheme. Finally, an explicit tamed Milstein scheme with rate of convergence arbitrarily close to one is developed to approximate the stochastic differential equation driven by Lévy noise (without random coefficients) that has super-linearly growing drift coefficient.
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Alnafisah, Yousef Ali. "First-order numerical schemes for stochastic differential equations using coupling." Thesis, University of Edinburgh, 2016. http://hdl.handle.net/1842/20420.

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We study a new method for the strong approximate solution of stochastic differential equations using coupling and we prove order one error bounds for the new scheme in Lp space assuming the invertibility of the diffusion matrix. We introduce and implement two couplings called the exact and approximate coupling for this scheme obtaining good agreement with the theoretical bound. Also we describe a method for non-invertibility case (Combined method) and we investigate its convergence order which will give O(h3/4 √log(h)j) under some conditions. Moreover we compare the computational results for the combined method with its theoretical error bound and we have obtained a good agreement between them. In the last part of this thesis we work out the performance of the multilevel Monte Carlo method using the new scheme with the exact coupling and we compare the results with the trivial coupling for the same scheme.
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Handari, Bevina D. "Numerical methods for SDEs and their dynamics /." [St. Lucia, Qld.], 2002. http://www.library.uq.edu.au/pdfserve.php?image=thesisabs/absthe17145.pdf.

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Tzitzili, Efthalia. "Numerical approximation of Stratonovich SDEs and SPDEs." Thesis, Heriot-Watt University, 2015. http://hdl.handle.net/10399/2883.

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We consider the numerical approximation of stochastic differential and partial differential equations S(P)DEs, by means of time-differencing schemes which are based on exponential integrator techniques. We focus on the study of two numerical schemes, both appropriate for the simulation of Stratonovich- interpreted S(P)DEs. The first, is a basic strong order 1=2 scheme, called Stratonovich Exponential Integrators (SEI). Motivated by SEI and aiming at benefiting both from the higher order of the standard Milstein scheme and the efficiency of the exponential schemes when dealing with stiff problems, we develop a new Milstein type scheme called Milstein Stratonovich Exponential Integrators (MSEI). We prove strong convergence of the SEI scheme for high-dimensional semilinear Stratonovich SDEs with multiplicative noise and we use SEI as well as the MSEI scheme to approximate solutions of the stochastic Landau-Lifschitz- Gilbert (LLG) equation in three dimensions. We examine the L2(Ω ) approximation error of the SEI and MSEI schemes numerically and we prove analytically that MSEI achieves a higher order of convergence than SEI. We generalise SEI so that it is suited not only for Stratonovich SDEs, but also for It^o and for SDEs interpreted by the 'in-between' calculi. Moreover, we provide a general expression for the predictor contained in SEI and we study the theoretical convergence for the generalised version of the scheme. We show that the order of the scheme used in order to obtain the predictor as well as the stochastic integral interpretation do not affect the overall order of the scheme. We extend the convergence results for SEI to a space-time context by considering a second order semilinear Stratonovich SPDE with multiplicative noise. We discretise in space with the nite element method and we use SEI for discretising in time. We consider the case where we have trace class noise and we examine analytically the strong order of convergence for SEI. We implement SEI as a time discretisation scheme and present the results when simulating SPDEs with stochastic travelling wave solutions. Then, we use an alternative method, called 'freezing' method, for approximating wave solutions and estimating the speed of the waves for the stochastic Nagumo and FitzHugh-Nagumo models. The wave position and hence the speed is found by minimising the L2 distance between a reference function and the travelling wave. While the results obtained from the two different approaches agree, we observe that the behaviour of the wave solution is captured in a smaller computational domain, when we use the freezing method, making it more efficient for long time simulations.
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Campana, Lorenzo. "Modélisation stochastique de particules non sphériques en turbulence." Thesis, Université Côte d'Azur, 2022. http://www.theses.fr/2022COAZ4019.

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Le mouvement de petites particules non-sphériques en suspension dans un écoulement turbulent a lieu dans une grande variété d’applications naturelles et industrielles. Par exemple, ces phénomènes impactent la dynamique des aérosols dans l’atmosphère et dans les voies respiratoires, le mouvement des globules rouges dans le sang, la dynamique du plancton dans l’océan, la glace dans les nuages ou bien la combustion. Les particules anisotropes réagissent aux écoulements turbulents de manière complexe. Leur dynamique dépend ainsi d’un large éventail de para- mètres (forme, inertie, cisaillement du fluide). Les particules sans inertie, dont la taille est inférieure à la longueur de Kolmogorov, suivent le mouvement du fluide avec une orientation généralement gouvernée par le gradient local de vitesse turbulente. Cette thèse est axée sur la dynamique de tels objets en turbulence en ayant recours à des méthodes Lagrangienes stochastiques. Le développement d’un modèle qui peut être utilisé comme outil prédictif dans le cadre de la dynamique de fluides numérique (CFD) au niveau industriel est d’un grand intérêt pour les applications concrètes en ingénierie. Par ailleurs, pour progresser dans le domaine de la médecine, de l’environnement et des procédés industriels, il est nécessaire que ces modèles atteignent un compromis acceptable entre simplicité et précision. La formulation d’un modèle stochastique pour l’orientation de telles particules est tout d’abord présentée dans le cadre d’un écoulement turbulent bidimensionnel avec un cisaillement homogène. Des simulations numériques directes (DNS) sont produites pour guider et évaluer la proposition de modèle. Les questions abordés dans ce travail portent sur la représentation de formes analytiques du modèle, sur les effets des anisotropies inclues dans le modèle, et sur l’extension de la notion de dynamique rotationnelle dans le cadre de cette approche stochastique. Les résultats obtenus avec le modèle, comparés avec la DNS, produisent une réponse qualitative acceptable, même si ce modèle diffusif n’est pas conçu pour reproduire les caractéristiques non-gaussiennes des expériences numériques (DNS). L’extension au cas tridimensionnel du modèle d’orientation pose le problème de son implé- mentation numérique efficace. Dans ce travail, un schéma numérique capable de simuler la dynamique d’orientation de telles particules, à un coût de calcul raisonnable, est introduit. La convergence de ce schéma est également analysée. Pour ce faire, un schéma fondé sur la décomposition de la dynamique a été développé pour résoudre les équations différen- tielles stochastiques (EDS) de rotation de ces particules. Cette décomposition permet de surmonter les problèmes d’instabilité typiques de la méthode Euler–Maruyama; on a ainsi obtenu une convergence en norme L2 d’ordre 1/2 et une convergence faible d’ordre 1, comme classiquement attendu. Enfin, le schéma numérique a été implémenté dans un code CFD industriel (Code_Saturne). Ce modèle a ensuite été utilisé pour étudier l’orientation et la rotation de particules anisotropes sans inertie dans le cas d’un écoulement turbulent inhomogène, à savoir un écoulement de canal plan turbulent. Cette application dans un cas pratique a permis de mettre en evidence deux difficultés liées au modèle : d’abord, l’implémentation numérique dans un code industriel, ensuite la capacité du modèle à reproduire les expériences numériques obtenues par DNS. Ainsi, le modèle stochastique Lagrangien pour l’orientation de sphéroïdes implémenté dans Code_Saturne permet de reproduire, avec certaines limites, les statistiques d’orientation et de rotation de sphéroïdes mesurées dans la DNS
The motion of small non- spherical particles suspended in a turbulent flow is relevant for a large variety of natural and industrial applications such as aerosol dynamics in respiration, red blood cells motion, plankton dynamics, ice in clouds, combustion, to name a few. Anisotropic particles react on turbulent flows in complex ways, which depend on a wide range of parameters (shape, inertia, fluid shear). Inertia-free particles, with size smaller than the Kolmogorov length, follow the fluid motion with an orientation generally defined by the local turbulent velocity gradient. Therefore, this thesis is focused on the dynamics of these objects in turbulence exploiting stochastic Lagrangian methods. The development of a model that can be used as predictive tool in industrial computational fluid dynamics (CFD) is highly valuable for practical applications in engineering. Models that reach an acceptable compromise between simplicity and accuracy are needed for progressing in the field of medical, environmental and industrial processes. The formulation of a stochastic orientation model is studied in two-dimensional turbulent flow with homogeneous shear, where results are compared with direct numerical simulations (DNS). Finding analytical results, scrutinising the effect of the anisotropies when they are included in the model, and extending the notion of rotational dynamics in the stochastic framework, are subjects addressed in our work. Analytical results give a reasonable qualitative response, even if the diffusion model is not designed to reproduce the non-Gaussian features of the DNS experiments. The extension to the three-dimensional case showed that the implementation of efficient numerical schemes in 3D models is far from straightforward. The introduction of a numerical scheme with the capability to preserve the dynamics at reasonable computational costs has been devised and the convergence analysed. A scheme of splitting decomposition of the stochastic differential equations (SDE) has been developed to overcome the typical instability problems of the Euler–Maruyama method, obtaining a mean-square convergence of order 1/2 and a weakly convergence of order 1, as expected. Finally, model and numerical scheme have been implemented in an industrial CFD code (Code_Saturne) and used to study the orientational and rotational behaviour of anisotropic inertia-free particles in an applicative prototype of inhomogeneous turbulence, i.e. a turbulent channel flow. This real application has faced two issues of the modelling: the numerical implementation in an industrial code, and whether and to which extent the model is able to reproduce the DNS experiments. The stochastic Lagrangian model for the orientation in the CFD code reproduces with some limits the orientation and rotation statistics of the DNS. The results of this study allows to predict the orientation and rotation of aspherical particles, giving new insight into the prediction of large scale motions both, in two-dimensional space, of interest for geophysical flows, and in three-dimensional industrial applications
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Herdiana, Ratna. "Numerical methods for SDEs - with variable stepsize implementation /." [St. Lucia, Qld.], 2003. http://www.library.uq.edu.au/pdfserve.php?image=thesisabs/absthe17638.pdf.

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Yannios, Nicholas, and mikewood@deakin edu au. "Computational aspects of the numerical solution of SDEs." Deakin University. School of Computing and Mathematics, 2001. http://tux.lib.deakin.edu.au./adt-VDU/public/adt-VDU20060817.123449.

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In the last 30 to 40 years, many researchers have combined to build the knowledge base of theory and solution techniques that can be applied to the case of differential equations which include the effects of noise. This class of ``noisy'' differential equations is now known as stochastic differential equations (SDEs). Markov diffusion processes are included within the field of SDEs through the drift and diffusion components of the Itô form of an SDE. When these drift and diffusion components are moderately smooth functions, then the processes' transition probability densities satisfy the Fokker-Planck-Kolmogorov (FPK) equation -- an ordinary partial differential equation (PDE). Thus there is a mathematical inter-relationship that allows solutions of SDEs to be determined from the solution of a noise free differential equation which has been extensively studied since the 1920s. The main numerical solution technique employed to solve the FPK equation is the classical Finite Element Method (FEM). The FEM is of particular importance to engineers when used to solve FPK systems that describe noisy oscillators. The FEM is a powerful tool but is limited in that it is cumbersome when applied to multidimensional systems and can lead to large and complex matrix systems with their inherent solution and storage problems. I show in this thesis that the stochastic Taylor series (TS) based time discretisation approach to the solution of SDEs is an efficient and accurate technique that provides transition and steady state solutions to the associated FPK equation. The TS approach to the solution of SDEs has certain advantages over the classical techniques. These advantages include their ability to effectively tackle stiff systems, their simplicity of derivation and their ease of implementation and re-use. Unlike the FEM approach, which is difficult to apply in even only two dimensions, the simplicity of the TS approach is independant of the dimension of the system under investigation. Their main disadvantage, that of requiring a large number of simulations and the associated CPU requirements, is countered by their underlying structure which makes them perfectly suited for use on the now prevalent parallel or distributed processing systems. In summary, l will compare the TS solution of SDEs to the solution of the associated FPK equations using the classical FEM technique. One, two and three dimensional FPK systems that describe noisy oscillators have been chosen for the analysis. As higher dimensional FPK systems are rarely mentioned in the literature, the TS approach will be extended to essentially infinite dimensional systems through the solution of stochastic PDEs. In making these comparisons, the advantages of modern computing tools such as computer algebra systems and simulation software, when used as an adjunct to the solution of SDEs or their associated FPK equations, are demonstrated.
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Adamu, Iyabo Ann. "Numerical approximation of SDEs and stochastic Swift-Hohenberg equation." Thesis, Heriot-Watt University, 2011. http://hdl.handle.net/10399/2460.

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We consider the numerical approximation of stochastic differential equations interpreted both in the It^o and Stratonovich sense and develop three stochastic time-integration techniques based on the deterministic exponential time differencing schemes. Two of the numerical schemes are suited for the simulations of It^o stochastic ordinary differential equations (SODEs) and they are referred to as the stochastic exponential time differencing schemes, SETD0 and SETD1. The third numerical scheme is a new numerical method we propose for the simulations of Stratonovich SODEs. We call this scheme, the Exponential Stratonovich Integrator (ESI). We investigate numerically the convergence of these three numerical methods, in addition to three standard approximation schemes and also compare the accuracy and efficiency of these schemes. The effect of small noise is also studied. We study the theoretical convergence of the stochastic exponential time differencing scheme (SETD0) for parabolic stochastic partial differential equations (SPDEs) with infinite-dimensional additive noise and one-dimensional multiplicative noise. We obtain a strong error temporal estimate of O(¢tµ + ²¢tµ + ²2¢t1=2) for SPDEs forced with a one-dimensional multiplicative noise and also obtain a strong error temporal estimate of O(¢tµ + ²2¢t) for SPDEs forced with an infinite-dimensional additive noise. We examine convergence for second-order and fourth-order SPDEs. We consider the effects of spatially correlated and uncorrelated noise on bifurcations for SPDEs. In particular, we study a fourth-order SPDE, the Swift-Hohenberg equation, and allow the control parameter to fluctuate. Numerical simulations show a shift in the pinning region with multiplicative noise.
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Gil, Gibin. "Hybrid Numerical Integration Scheme for Highly Oscillatory Dynamical Systems." Diss., The University of Arizona, 2013. http://hdl.handle.net/10150/306771.

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Computational efficiency of solving the dynamics of highly oscillatory systems is an important issue due to the requirement of small step size of explicit numerical integration algorithms. A system is considered to be highly oscillatory if it contains a fast solution that varies regularly about a slow solution. As for multibody systems, stiff force elements and contacts between bodies can make a system highly oscillatory. Standard explicit numerical integration methods should take a very small step size to satisfy the absolute stability condition for all eigenvalues of the system and the computational cost is dictated by the fast solution. In this research, a new hybrid integration scheme is proposed, in which the local linearization method is combined with a conventional integration method such as the fourth-order Runge-Kutta. In this approach, the system is partitioned into fast and slow subsystems. Then, the two subsystems are transformed into a reduced and a boundary-layer system using the singular perturbation theory. The reduced system is solved by the fourth-order Runge-Kutta method while the boundary-layer system is solved by the local linearization method. This new hybrid scheme can handle the coupling between the fast and the slow subsystems efficiently. Unlike other multi-rate or multi-method schemes, extrapolation or interpolation process is not required to deal with the coupling between subsystems. Most of the coupling effect can be accounted for by the reduced (or quasi-steady-state) system while the minor transient effect is taken into consideration by averaging. In this research, the absolute stability region for this hybrid scheme is derived and it is shown that the absolute stability region is almost independent of the fast variables. Thus, the selection of the step size is not dictated by the fast solution when a highly oscillatory system is solved, in turn, the computational efficiency can be improved. The advantage of the proposed hybrid scheme is validated through several dynamic simulations of a vehicle system including a flexible tire model. The results reveal that the hybrid scheme can reduce the computation time of the vehicle dynamic simulation significantly while attaining comparable accuracy.
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Alhojilan, Yazid Yousef M. "Higher-order numerical scheme for solving stochastic differential equations." Thesis, University of Edinburgh, 2016. http://hdl.handle.net/1842/15973.

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We present a new pathwise approximation method for stochastic differential equations driven by Brownian motion which does not require simulation of the stochastic integrals. The method is developed to give Wasserstein bounds O(h3/2) and O(h2) which are better than the Euler and Milstein strong error rates O(√h) and O(h) respectively, where h is the step-size. It assumes nondegeneracy of the diffusion matrix. We have used the Taylor expansion but generate an approximation to the expansion as a whole rather than generating individual terms. We replace the iterated stochastic integrals in the method by random variables with the same moments conditional on the linear term. We use a version of perturbation method and a technique from optimal transport theory to find a coupling which gives a good approximation in Lp sense. This new method is a Runge-Kutta method or so-called derivative-free method. We have implemented this new method in MATLAB. The performance of the method has been studied for degenerate matrices. We have given the details of proof for order h3/2 and the outline of the proof for order h2.
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Books on the topic "Numerical scheme for SDEs"

1

Kloeden, Peter E. Numerical solution of SDE through computer experiments. 2nd ed. Berlin: Springer, 1997.

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Eckhard, Platen, and Schurz Henri, eds. Numerical solution of SDE through computer experiments. Berlin: Springer-Verlag, 1994.

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Nicolaides, Roy A. Analysis and convergence of the MAC scheme. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, Institute for Computer Applications in Science and Engineering, 1991.

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Choo, Yung K. Generation of a composite grid for turbine flows and consideration of a numerical scheme. [Washington, D.C.]: National Aeronautics and Space Administration, 1987.

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Scott, James R. A new flux-conserving numerical scheme for the steady, incompressible Navier-Stokes equations. [Washington, DC: National Aeronautics and Space Administration, 1994.

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Feng, Wang. A conservative Eulerian numerical scheme for elasto-plasticity and application to plate impact problems. Stony Brook, N. Y: State University of New York at Stony Brook, Dept. of Applied Mathematics and Statistics, 1992.

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Yeffet, Amir. A non-dissipative staggered fourth-order accurate explicit finite difference scheme for the time-domain Maxwell's equations. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1999.

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Yeffet, Amir. A non-dissipative staggered fourth-order accurate explicit finite difference scheme for the time-domain Maxwell's equations. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1999.

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Yeffet, Amir. A non-dissipative staggered fourth-order accurate explicit finite difference scheme for the time-domain Maxwell's equations. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1999.

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Yeffet, Amir. A non-dissipative staggered fourth-order accurate explicit finite difference scheme for the time-domain Maxwell's equations. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1999.

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Book chapters on the topic "Numerical scheme for SDEs"

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Zhang, Zhongqiang, and George Em Karniadakis. "Balanced numerical schemes for SDEs with non-Lipschitz coefficients." In Numerical Methods for Stochastic Partial Differential Equations with White Noise, 135–60. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-57511-7_5.

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Zhang, Zhongqiang, and George Em Karniadakis. "Numerical schemes for SDEs with time delay using the Wong-Zakai approximation." In Numerical Methods for Stochastic Partial Differential Equations with White Noise, 103–33. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-57511-7_4.

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Cyganowski, Sasha, Peter Kloeden, and Jerzy Ombach. "Numerical Methods for SDEs." In Universitext, 277–302. Berlin, Heidelberg: Springer Berlin Heidelberg, 2002. http://dx.doi.org/10.1007/978-3-642-56144-3_10.

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Akdim, Khadija. "Reflected Backward SDEs in a Convex Polyhedron." In Applied and Numerical Harmonic Analysis, 21–31. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-35202-8_2.

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Tanguy, Jean-Michel. "Numerical-Scheme Study." In Numerical Methods, 235–65. Hoboken, NJ, USA: John Wiley & Sons, Inc., 2013. http://dx.doi.org/10.1002/9781118557877.ch10.

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Platen, Eckhard, and Nicola Bruti-Liberati. "Monte Carlo Simulation of SDEs." In Numerical Solution of Stochastic Differential Equations with Jumps in Finance, 477–505. Berlin, Heidelberg: Springer Berlin Heidelberg, 2010. http://dx.doi.org/10.1007/978-3-642-13694-8_11.

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Milstein, Grigori N., and Michael V. Tretyakov. "Numerical methods for SDEs with small noise." In Scientific Computation, 171–210. Berlin, Heidelberg: Springer Berlin Heidelberg, 2004. http://dx.doi.org/10.1007/978-3-662-10063-9_3.

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Milstein, Grigori N., and Michael V. Tretyakov. "Numerical Methods for SDEs with Small Noise." In Scientific Computation, 271–312. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-82040-4_4.

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Platen, Eckhard, and Nicola Bruti-Liberati. "Exact Simulation of Solutions of SDEs." In Numerical Solution of Stochastic Differential Equations with Jumps in Finance, 61–137. Berlin, Heidelberg: Springer Berlin Heidelberg, 2010. http://dx.doi.org/10.1007/978-3-642-13694-8_2.

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Chassagneux, Jean-François, Hinesh Chotai, and Mirabelle Muûls. "Numerical Approximation of FBSDEs." In A Forward-Backward SDEs Approach to Pricing in Carbon Markets, 59–74. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-63115-8_4.

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Conference papers on the topic "Numerical scheme for SDEs"

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Ryashko, Lev. "Approximation of stochastic attractors for nonlinear SDEs via confidence domains." In 11TH INTERNATIONAL CONFERENCE OF NUMERICAL ANALYSIS AND APPLIED MATHEMATICS 2013: ICNAAM 2013. AIP, 2013. http://dx.doi.org/10.1063/1.4825935.

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Chen, Lin. "Analysis of Stability for the Semi Implicit Scheme for SDEs with Polynomial Growth Condition." In 2018 3rd International Conference on Information Systems Engineering (ICISE). IEEE, 2018. http://dx.doi.org/10.1109/icise.2018.00024.

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M., Grigoriu. "Solution Stability and Phase Transition for Two SDEs by a Fixed Time Step Integration Scheme." In 6th International Conference on Computational Stochastic Mechanics. Singapore: Research Publishing Services, 2011. http://dx.doi.org/10.3850/978-981-08-7619-7_p031.

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Ladonkina, Marina, Olga Nekliudova, and Vladimir Tishkin. "Combined scheme based on Rusanov scheme and discontinuous Galerkin method." In INTERNATIONAL CONFERENCE OF NUMERICAL ANALYSIS AND APPLIED MATHEMATICS ICNAAM 2019. AIP Publishing, 2020. http://dx.doi.org/10.1063/5.0031578.

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Gil, Gibin, Ricardo G. Sanfelice, and Parviz E. Nikravesh. "Numerical Integration Scheme Using Singular Perturbation Method." In ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2013. http://dx.doi.org/10.1115/detc2013-13330.

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Some multi degree-of-freedom dynamical systems exhibit a response that contain fast and slow variables. An example of such systems is a multibody system with rigid and deformable bodies. Standard numerical integration of the resultant equations of motion must adjust the time step according to the frequency of the fastest variable. As a result, the computation time is sacrificed. The singular perturbation method is an analysis technique to deal with the interaction of slow and fast variables. In this study, a numerical integration scheme using the singular perturbation method is discussed, its absolute stability condition is derived, and its order of accuracy is investigated.
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Li Hui, Yuan Dongsheng, and Xu Lu. "Based on numerical simulation support scheme selection." In 2011 International Conference on Computer Science and Service System (CSSS). IEEE, 2011. http://dx.doi.org/10.1109/csss.2011.5974967.

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Righi, Marcello. "A Numerical Scheme for Hypersonic Turbulent Flow." In 45th AIAA Fluid Dynamics Conference. Reston, Virginia: American Institute of Aeronautics and Astronautics, 2015. http://dx.doi.org/10.2514/6.2015-3341.

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Srivastava, Shubham, Shivani Dixit, and M. Shukla. "Analysis of numerical interleaver for IDMA scheme." In 2017 7th International Conference on Communication Systems and Network Technologies (CSNT). IEEE, 2017. http://dx.doi.org/10.1109/csnt.2017.8418502.

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Boules, Adel N. "A Non-Adaptive Scheme for Numerical Integration." In 2018 International Conference on Computational Science and Computational Intelligence (CSCI). IEEE, 2018. http://dx.doi.org/10.1109/csci46756.2018.00044.

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Kulesza, Zbigniew, and Jerzy T. Sawicki. "Controlled Deflection Approach for Rotor Crack Detection." In ASME Turbo Expo 2012: Turbine Technical Conference and Exposition. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/gt2012-68960.

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A transverse shaft crack is a serious malfunction that can occur due to cyclic loading, creep, stress corrosion, and other mechanisms to which rotating machines are subjected. Though studied for many years, the problems of early crack detection and warning are still in the limelight of many researchers. This is due to the fact that the crack has subtle influence on the dynamic response of the machine and still there are no widely accepted, reliable methods of its early detection. This paper presents a new approach to these problems. The method utilizes the coupling mechanism between the bending and torsional vibrations of the cracked, non-rotating shaft. By applying an external lateral force of constant amplitude, a small shaft deflection is induced. Simultaneously, a harmonic torque is applied to the shaft inducing its torsional vibrations. By changing the angular position of the lateral force application, the position of the deflection also changes opening or closing of the crack. This changes the way the bending and torsional vibrations are being coupled. By studying the coupled lateral vibration response for each angular position of the lateral force one can assess the possible presence of the crack. The approach is demonstrated with a numerical model of a rotor. The model is based on the rigid finite element method (RFE), which has previously been successfully applied for the dynamic analysis of many complicated, mechanical structures. The RFE method is extended and adopted for the modeling of the cracked shafts. An original concept of crack modeling utilizing the RFE method is presented. The crack is modeled as a set of spring-damping elements (SDEs) of variable stiffness connecting two sections of the shaft. By calculating the axial deformations of the SDEs, the opening/closing mechanism of the crack is introduced. The results of numerical analysis demonstrate the potential of the suggested approach for effective shaft crack detection.
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Reports on the topic "Numerical scheme for SDEs"

1

Knio, Omar M., Habib N. Najm, and Phillip H. Paul. A numerical scheme for modelling reacting flow with detailed chemistry and transport. Office of Scientific and Technical Information (OSTI), September 2003. http://dx.doi.org/10.2172/918335.

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LeGrand, Sandra, Christopher Polashenski, Theodore Letcher, Glenn Creighton, Steven Peckham, and Jeffrey Cetola. The AFWA dust emission scheme for the GOCART aerosol model in WRF-Chem v3.8.1. Engineer Research and Development Center (U.S.), August 2021. http://dx.doi.org/10.21079/11681/41560.

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Airborne particles of mineral dust play a key role in Earth’s climate system and affect human activities around the globe. The numerical weather modeling community has undertaken considerable efforts to accurately forecast these dust emissions. Here, for the first time in the literature, we thoroughly describe and document the Air Force Weather Agency (AFWA) dust emission scheme for the Georgia Institute of Technology–Goddard Global Ozone Chemistry Aerosol Radiation and Transport (GOCART) aerosol model within the Weather Research and Forecasting model with chemistry (WRF-Chem) and compare it to the other dust emission schemes available in WRF-Chem. The AFWA dust emission scheme addresses some shortcomings experienced by the earlier GOCART-WRF scheme. Improved model physics are designed to better handle emission of fine dust particles by representing saltation bombardment. WRF-Chem model performance with the AFWA scheme is evaluated against observations of dust emission in southwest Asia and compared to emissions predicted by the other schemes built into the WRF-Chem GOCART model. Results highlight the relative strengths of the available schemes, indicate the reasons for disagreement, and demonstrate the need for improved soil source data.
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Russo, David, and William A. Jury. Characterization of Preferential Flow in Spatially Variable Unsaturated Field Soils. United States Department of Agriculture, October 2001. http://dx.doi.org/10.32747/2001.7580681.bard.

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Preferential flow appears to be the rule rather than the exception in field soils and should be considered in the quantitative description of solute transport in the unsaturated zone of heterogeneous formations on the field scale. This study focused on both experimental monitoring and computer simulations to identify important features of preferential flow in the natural environment. The specific objectives of this research were: (1) To conduct dye tracing and multiple tracer experiments on undisturbed field plots to reveal information about the flow velocity, spatial prevalence, and time evolution of a preferential flow event; (2) To conduct numerical experiments to determine (i) whether preferential flow observations are consistent with the Richards flow equation; and (ii) whether volume averaging over a domain experiencing preferential flow is possible; (3) To develop a stochastic or a transfer function model that incorporates preferential flow. Regarding our field work, we succeeded to develop a new method for detecting flow patterns faithfully representing the movement of water flow paths in structured and non-structured soils. The method which is based on application of ammonium carbonate was tested in a laboratory study. Its use to detect preferential flow was also illustrated in a field experiment. It was shown that ammonium carbonate is a more conservative tracer of the water front than the popular Brilliant Blue. In our detailed field experiments we also succeeded to document the occurrence of preferential flow during soil water redistribution following the cessation of precipitation in several structureless field soils. Symptoms of the unstable flow observed included vertical fingers 20 - 60 cm wide, isolated patches, and highly concentrated areas of the tracers in the transmission zone. Soil moisture and tracer measurements revealed that the redistribution flow became fingered following a reversal of matric potential gradient within the wetted area. Regarding our simulation work, we succeeded to develop, implement and test a finite- difference, numerical scheme for solving the equations governing flow and transport in three-dimensional, heterogeneous, bimodal, flow domains with highly contrasting soil materials. Results of our simulations demonstrated that under steady-state flow conditions, the embedded clay lenses (with very low conductivity) in bimodal formations may induce preferential flow, and, consequently, may enhance considerably both the solute spreading and the skewing of the solute breakthrough curves. On the other hand, under transient flow conditions associated with substantial redistribution periods with diminishing water saturation, the effect of the embedded clay lenses on the flow and the transport might diminish substantially. Regarding our stochastic modeling effort, we succeeded to develop a theoretical framework for flow and transport in bimodal, heterogeneous, unsaturated formations, based on a stochastic continuum presentation of the flow and a general Lagrangian description of the transport. Results of our analysis show that, generally, a bimodal distribution of the formation properties, characterized by a relatively complex spatial correlation structure, contributes to the variability in water velocity and, consequently, may considerably enhance solute spreading. This applies especially in formations in which: (i) the correlation length scales and the variances of the soil properties associated with the embedded soil are much larger than those of the background soil; (ii) the contrast between mean properties of the two subdomains is large; (iii) mean water saturation is relatively small; and (iv) the volume fraction of the flow domain occupied by the embedded soil is relatively large.
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