Deterministic Partial Differential Equations Research Articles

Geometric shape of invariant manifolds for a class of stochastic partial differential equations

Invariant manifolds play an important role in the study of the qualitative dynamical behaviors for nonlinear stochastic partial differential equations. However, the geometric shape of these manifolds is largely unclear. The purpose of the present paper is to try to describe the geometric shape of invariant manifolds for a class of stochastic partial differential equations with multiplicative white noises. The local geometric shape of invariant manifolds is approximated, which holds with significant likelihood. Furthermore, the result is compared with that for the corresponding deterministic partial differential equations.

Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs

Partial differential equations (PDEs) with random input data, such as random loadings and coefficients, are reformulated as parametric, deterministic PDEs on parameter spaces of high, possibly infinite dimension. Tensorized operator equations for spatial and temporalk-point correlation functions of their random solutions are derived. Parametric, deterministic PDEs for the laws of the random solutions are derived. Representations of the random solutions' laws on infinite-dimensional parameter spaces in terms of ‘generalized polynomial chaos’ (GPC) series are established. Recent results on the regularity of solutions of these parametric PDEs are presented. Convergence rates of bestN-term approximations, for adaptive stochastic Galerkin and collocation discretizations of the parametric, deterministic PDEs, are established. Sparse tensor products of hierarchical (multi-level) discretizations in physical space (and time), and GPC expansions in parameter space, are shown to converge at rates which are independent of the dimension of the parameter space. A convergence analysis of multi-level Monte Carlo (MLMC) discretizations of PDEs with random coefficients is presented. Sufficient conditions on the random inputs for superiority of sparse tensor discretizations over MLMC discretizations are established for linear elliptic, parabolic and hyperbolic PDEs with random coefficients.

ASSESSMENT OF COLLOCATION AND GALERKIN APPROACHES TO LINEAR DIFFUSION EQUATIONS WITH RANDOM DATA

We compare the performance of two methods, the stochastic Galerkin method and the stochastic collocation method, for solving partial differential equations (PDEs) with random data. The stochastic Galerkin method requires the solution of a single linear system that is several orders larger than linear systems associated with deterministic PDEs. The stochastic collocation method requires many solves of deterministic PDEs, which allows the use of existing software. However, the total number of degrees of freedom in the stochastic collocation method can be considerably larger than the number of degrees of freedom in the stochastic Galerkin system. We implement both methods using the Trilinos software package and we assess their cost and performance. The implementations in Trilinos are known to be efficient, which allows for a realistic assessment of the computational complexity of the methods. We also develop a cost model for both methods which allows us to examine asymptotic behavior.

Observability estimate for stochastic Schrödinger equations

In this Note, we present an observability estimate for stochastic Schrödinger equations with nonsmooth lower order terms. The desired inequality is derived by a global Carleman estimate which is based on a fundamental weighted identity for stochastic Schrödinger-like operator. As an interesting byproduct, starting from this identity, one can deduce all the known controllability/observability results for several stochastic and deterministic partial differential equations that are derived before via Carleman estimate in the literature.

Stochastic projection schemes for deterministic linear elliptic partial differential equations on random domains

AbstractWe present stochastic projection schemes for approximating the solution of a class of deterministic linear elliptic partial differential equations defined on random domains. The key idea is to carry out spatial discretization using a combination of finite element methods and stochastic mesh representations. We prove a result to establish the conditions that the input uncertainty model must satisfy to ensure the validity of the stochastic mesh representation and hence the well posedness of the problem. Finite element spatial discretization of the governing equations using a stochastic mesh representation results in a linear random algebraic system of equations in a polynomial chaos basis whose coefficients of expansion can be non‐intrusively computed either at the element or the global level. The resulting randomly parametrized algebraic equations are solved using stochastic projection schemes to approximate the response statistics. The proposed approach is demonstrated for modeling diffusion in a square domain with a rough wall and heat transfer analysis of a three‐dimensional gas turbine blade model with uncertainty in the cooling core geometry. The numerical results are compared against Monte–Carlo simulations, and it is shown that the proposed approach provides high‐quality approximations for the first two statistical moments at modest computational effort. Copyright © 2010 John Wiley & Sons, Ltd.

A deterministic partial differential equation model for dose calculation in electron radiotherapy

High-energy ionizing radiation is a prominent modality for the treatment of many cancers. The approaches to electron dose calculation can be categorized into semi-empirical models (e.g. Fermi–Eyges, convolution-superposition) and probabilistic methods (e.g. Monte Carlo). A third approach to dose calculation has only recently attracted attention in the medical physics community. This approach is based on the deterministic kinetic equations of radiative transfer. We derive a macroscopic partial differential equation model for electron transport in tissue. This model involves an angular closure in the phase space. It is exact for the free streaming and the isotropic regime. We solve it numerically by a newly developed HLLC scheme based on Berthon et al (2007 J. Sci. Comput. 31 347–89) that exactly preserves the key properties of the analytical solution on the discrete level. We discuss several test cases taken from the medical physics literature. A test case with an academic Henyey–Greenstein scattering kernel is considered. We compare our model to a benchmark discrete ordinate solution. A simplified model of electron interactions with tissue is employed to compute the dose of an electron beam in a water phantom, and a case of irradiation of the vertebral column. Here our model is compared to the PENELOPE Monte Carlo code. In the academic example, the fluences computed with the new model and a benchmark result differ by less than 1%. The depths at half maximum differ by less than 0.6%. In the two comparisons with Monte Carlo, our model gives qualitatively reasonable dose distributions. Due to the crude interaction model, these so far do not have the accuracy needed in clinical practice. However, the new model has a computational cost that is less than one-tenth of the cost of a Monte Carlo simulation. In addition, simulations can be set up in a similar way as a Monte Carlo simulation. If more detailed effects such as coupled electron–photon transport, bremsstrahlung, Compton scattering and the production of δ electrons are added to our model, the computation time will only slightly increase. Its margin of error, on the other hand, will decrease and should be within a few per cent of the actual dose. Therefore, the new model has the potential to become useful for dose calculations in clinical practice.

A Lax equivalence theorem for stochastic differential equations

In this paper, a stochastic mean square version of Lax’s equivalence theorem for Hilbert space valued stochastic differential equations with additive and multiplicative noise is proved. Definitions for consistency, stability, and convergence in mean square of an approximation of a stochastic differential equation are given and it is shown that these notions imply similar results as those known for approximations of deterministic partial differential equations. Examples show that the assumptions made are met by standard approximations.

A Hybrid Particle-Continuum Method for Hydrodynamics of Complex Fluids

A previously developed hybrid particle-continuum method [J. B. Bell, A. Garcia, and S. A. Williams, Multiscale Model. Simul., 6 (2008), pp. 1256–1280] is generalized to dense fluids and two- and three-dimensional flows. The scheme couples an explicit fluctuating compressible Navier–Stokes solver with the isotropic direct simulation Monte Carlo (DSMC) particle method [A. Donev, A. L. Garcia, and B. J. Alder, J. Stat. Mech. Theory Exp., 2009 (2009), article P11008]. To achieve bidirectional dynamic coupling between the particle (microscale) and continuum (macroscale) regions, the continuum solver provides state-based boundary conditions to the particle subdomain, while the particle solver provides flux-based boundary conditions for the continuum subdomain. This type of coupling ensures both state and flux continuity across the particle-continuum interface analogous to coupling approaches for deterministic parabolic partial differential equations; here, when fluctuations are included, a small ($<1\%$) mismat...

Iterative Solvers for the Stochastic Finite Element Method

This paper presents an overview and comparison of iterative solvers for linear stochastic partial differential equations (PDEs). A stochastic Galerkin finite element discretization is applied to transform the PDE into a coupled set of deterministic PDEs. Specialized solvers are required to solve the very high-dimensional systems that result after a finite element discretization of the resulting set. This paper discusses one-level iterative methods, based on matrix splitting techniques; multigrid methods, which apply a coarsening in the spatial dimension; and multilevel methods, which make use of the hierarchical structure of the stochastic discretization. Also Krylov solvers with suitable preconditioning are addressed. A local Fourier analysis provides quantitative convergence properties. The efficiency and robustness of the methods are illustrated on two nontrivial numerical problems. The multigrid solver with block smoother yields the most robust convergence properties, though a cheaper point smoother performs as well in most cases. Multilevel methods based on coarsening the stochastic dimension perform in general poorly due to a large computational cost per iteration. Moderate size problems can be solved very quickly by a Krylov method with a mean-based preconditioner. For larger spatial and stochastic discretizations, however, this approach suffers from its nonoptimal convergence properties.

A finite element method for elliptic problems with stochastic input data

We compute the expectation and the two-point correlation of the solution to elliptic boundary value problems with stochastic loadings. In case of elliptic problems on stochastic domains or with stochastic coefficients analogous expressions hold to leading order in the size of the stochastic perturbation. The solution's two-point correlation satisfies a deterministic tensor product partial differential equation on the twofold product domain. For its numerical solution we apply a sparse tensor product approximation by multilevel frames. This way standard finite element techniques can be used. Numerical examples illustrate feasibility and scope of the method.

Symmetry in Nonlinear Hydrologic Dynamics under Uncertainty: Ensemble Modeling of 2D Boussinesq Equation for Unsteady Flow in Heterogeneous Aquifers

In this paper, the probabilistic symmetry analysis approach proposed in the previous paper by Cayar and Kavvas in 2009 is adopted and applied to the analysis of nonlinear two-dimensional (2D) groundwater flow subject to random hydraulic conductivity field. The resulting model has the form of a two-dimensional Fokker-Planck equation (FPE). Following the methodology reported by Cayar and Kavvas in 2009, the 2D Boussinessq equation for unconfined groundwater flow in a heterogeneous aquifer is transformed to a system of ordinary differential equations (ODEs) through Lie group symmetry analysis in order to eliminate the spatial derivatives occurring in the governing equation. Next, these derived stochastic ODEs are ensemble averaged with second-order cumulant expansion and converted into a linear, deterministic partial differential equation. This new equation is called the FPE, and describes the evolution of the probability density function (PDF) of the hydraulic head, thereby, describing the ensemble behavior...

Discrete stochastic evolution rules and continuum evolution equations

The continuum evolution equations are derived from updating rules for three classes of stochastic models. The first class corresponds to models whose stochastic continuum equations are of the Langevin type obtained after carrying out a “local average” known as coarse-graining. The second class consists of a hierarchy of continuum equations for the correlations of the dynamical variables obtained after making an average over realizations. This average generates a hierarchy of deterministic partial differential equations except when the dynamical variables do not depend on the values of the neighboring dynamical variables, in which case a hierarchy of ordinary differential equations is obtained. The third class of evolution equations for the correlations of the dynamical variable constitutes another hierarchy after calculating an average over both realizations and all the sites of the lattice. This double average generates a hierarchy of deterministic ordinary differential equations. The second and third classes of equations are truncated using a mean field ( m , n ) -closure approximation in order to obtain a finite set of equations. Illustrative examples of every class are given.

A stochastic approach to nonlinear unconfined flow subject to multiple random fields

In this study, the KLME approach, a moment-equation approach based on the Karhunen–Loeve decomposition developed by Zhang and Lu (Comput Phys 194(2):773–794, 2004), is applied to unconfined flow with multiple random inputs. The log-transformed hydraulic conductivity F, the recharge R, the Dirichlet boundary condition H, and the Neumann boundary condition Q are assumed to be Gaussian random fields with known means and covariance functions. The F, R, H and Q are first decomposed into finite series in terms of Gaussian standard random variables by the Karhunen–Loeve expansion. The hydraulic head h is then represented by a perturbation expansion, and each term in the perturbation expansion is written as the products of unknown coefficients and Gaussian standard random variables obtained from the Karhunen–Loeve expansions. A series of deterministic partial differential equations are derived from the stochastic partial differential equations. The resulting equations for uncorrelated and perfectly correlated cases are developed. The equations can be solved sequentially from low to high order by the finite element method. We examine the accuracy of the KLME approach for the groundwater flow subject to uncorrelated or perfectly correlated random inputs and study the capability of the KLME method for predicting the head variance in the presence of various spatially variable parameters. It is shown that the proposed numerical model gives accurate results at a much smaller computational cost than the Monte Carlo simulation.

A quasilinear stochastic partial differential equation driven by fractional white noise

The objective of the paper is to give the representation of a solution of a quasilinear stochastic partial differential equation driven by scalar fractional Brownian motion B H ( t ), H ∈ (½, 1), in the white noise framework for fractional Brownian motion. The solution is represented as a Wick product between a fractional Wick exponential and the solution of a path wise deterministic parabolic partial differential equation. Thereby a fractional theory of fractional translation operators is developed and used in the spirit of Benth and Gjessing [F. E. Benth and H. Gjessing. A nonlinear parabolic equation with noise. Potential Analysis 12 (2000), 385–401] who used it in the pure Brownian motion case.

Fractal behavior in space and time in a simplified model of fluvial landform evolution

Two general approaches have been applied to understanding the fractal structure of fluvial topography: (1) deterministic, process-based models, and (2) stochastic partial differential equations (PDE). Deterministic models reproduce the fractal behavior of fluvial topography but have two limitations: they often underestimate the amount of lateral valley and ridge migration that occurs in nature, and the complexity has made it difficult to identify the precise origin of fractal behavior in fluvial landscapes. The simplicity of stochastic PDE models has made them useful for investigating fractal behavior, but they incorrectly suggest that fractal behavior is only possible with stochastic forcing. In this paper I investigate whether simplified, deterministic PDE models of landform evolution also exhibit fractal behavior and other features of complexity (i.e. deterministic chaos). These models are based on the KPZ equation, well known in the physics literature. This equation combines diffusion (i.e. hillslope processes) and nonlinear advection (i.e. bedrock or alluvial channel incision). Two models are considered: (1) a deterministic model with uniform erodibility and random initial topography, and (2) a deterministic model with random erodibility and uniform initial topography. Results illustrate that both of these deterministic models exhibit fractal behavior and deterministic chaos. In this context, chaotic behavior means that valley and ridge migration and nonlinear amplification of small perturbations in these models prevent an ideal steady state landscape from ever developing in the large-system limit. These results suggest that fractal structure and deterministic chaos are intrinsic features of the evolution of fluvial landforms, and that these features result from an inverse cascade of energy from small to large wavelengths in drainage basins. This inverse cascade differs from the direct cascade of three-dimensional turbulence in which energy flows from large to small wavelengths.

On the Cauchy-Dirichlet Problem in the Half Space for Parabolic SPDEs in Weighted Hoelder Spaces

We study the Cauchy-Dirichlet problem for a second order linear parabolic stochastic differential equation (SPDE) with constant coefficients in a half-space. Considering its solution as a function with values in a probability space and using the methods of deterministic partial differential equations, we establish the existence and uniqueness of a strong solution in Holder classes with weights.

Reactive boundary conditions for stochastic simulations of reaction–diffusion processes

Many cellular and subcellular biological processes can be described in terms of diffusing and chemically reacting species (e.g. enzymes). Such reaction–diffusion processes can be mathematically modelled using either deterministic partial-differential equations or stochastic simulation algorithms. The latter provide a more detailed and precise picture, and several stochastic simulation algorithms have been proposed in recent years. Such models typically give the same description of the reaction–diffusion processes far from the boundary of the simulated domain, but the behaviour close to a reactive boundary (e.g. a membrane with receptors) is unfortunately model-dependent. In this paper, we study four different approaches to stochastic modelling of reaction–diffusion problems and show the correct choice of the boundary condition for each model. The reactive boundary is treated as partially reflective, which means that some molecules hitting the boundary are adsorbed (e.g. bound to the receptor) and some molecules are reflected. The probability that the molecule is adsorbed rather than reflected depends on the reactivity of the boundary (e.g. on the rate constant of the adsorbing chemical reaction and on the number of available receptors), and on the stochastic model used. This dependence is derived for each model.

On Cauchy—Dirichlet Problem for Parabolic Quasilinear SPDEs

We study the Cauchy–Dirichlet problem for a second-order quasilinear parabolic stochastic differential equation (SPDE) in a domain with a zero order noise term driven by a cylindrical Brownian motion. Considering its solution as a function with values in a probability space and using the methods of deterministic partial differential equations, we establish the existence and uniqueness of a strong solution in Holder classes with weights.

A Hybrid Deterministic−Stochastic Algorithm for Modeling Cell Signaling Dynamics in Spatially Inhomogeneous Environments and under the Influence of External Fields

Cell signaling dynamics mediate myriad processes in biology. It has become increasingly clear that inter- and intracellular signaling reactions often occur in a spatially inhomogeneous environment and that it is important to account for stochastic fluctuations of certain species involved in signaling reactions. The importance of these effects enhances the difficulty of gleaning mechanistic information from observations of a few experimental reporters and highlights the significance of synergistic experimental and computational studies. When both stochastic fluctuations and spatial inhomogeneity must be included in a model simultaneously, however, the resulting computational demands quickly become overwhelming. In many situations the failure of standard coarse-graining methods (i.e., ignoring spatial variation or stochastic fluctuations) when applied to all components of a complex system does not exclude the possibility of successfully applying such coarse-graining to some components of the system. Following this approach alleviates computational cost but requires "hybrid" algorithms where some variables are treated at a coarse-grained level while others are not. We present an efficient algorithm for simulation of stochastic, spatially inhomogeneous reaction-diffusion kinetics coupled to coarse-grained fields described by (stochastic or deterministic) partial differential equations (PDEs). The PDEs could represent mean-field descriptions of reactive species present in large copy numbers or evolution of hydrodynamic variables that influence signaling (e.g., membrane shape or cytoskeletal motion). We discuss the approximations made to derive our algorithm and test its efficacy by applying it to problems that include many features typical of realistic cell signaling processes.

Spatial modeling of dimerization reaction dynamics in the plasma membrane: Monte Carlo vs. continuum differential equations

Bimolecular reactions in the plasma membrane, such as receptor dimerization, are a key signaling step for many signaling systems. For receptors to dimerize, they must first diffuse until a collision happens, upon which a dimerization reaction may occur. Therefore, study of the dynamics of cell signaling on the membrane may require the use of a spatial modeling framework. Despite the availability of spatial simulation methods, e.g., stochastic spatial Monte Carlo (MC) simulation and partial differential equation (PDE) based approaches, many biological models invoke well-mixed assumptions without completely evaluating the importance of spatial organization. Whether one is to utilize a spatial or non-spatial simulation framework is therefore an important decision. In order to evaluate the importance of spatial effects a priori, i.e., without performing simulations, we have assessed the applicability of a dimensionless number, known as second Damköhler number ( Da), defined here as the ratio of time scales of collision and reaction, for 2-dimensional bimolecular reactions. Our study shows that dimerization reactions in the plasma membrane with Da ∼> 0.1 (tested in the receptor density range of 10 2–10 5/μm 2) require spatial modeling. We also evaluated the effective reaction rate constants of MC and simple deterministic PDEs. Our simulations show that the effective reaction rate constant decreases with time due to time dependent changes in the spatial distribution of receptors. As a result, the effective reaction rate constant of simple PDEs can differ from that of MC by up to two orders of magnitude. Furthermore, we show that the fluctuations in the number of copies of signaling proteins (noise) may also depend on the diffusion properties of the system. Finally, we used the spatial MC model to explore the effect of plasma membrane heterogeneities, such as receptor localization and reduced diffusivity, on the dimerization rate. Interestingly, our simulations show that localization of epidermal growth factor receptor (EGFR) can cause the diffusion limited dimerization rate to be up to two orders of magnitude higher at higher average receptor densities reported for cancer cells, as compared to a normal cell.