Random Diffusion Coefficient Research Articles

Parabolic PDE-constrained optimal control under uncertainty with entropic risk measure using quasi-Monte Carlo integration

We study the application of a tailored quasi-Monte Carlo (QMC) method to a class of optimal control problems subject to parabolic partial differential equation (PDE) constraints under uncertainty: the state in our setting is the solution of a parabolic PDE with a random thermal diffusion coefficient, steered by a control function. To account for the presence of uncertainty in the optimal control problem, the objective function is composed with a risk measure. We focus on two risk measures, both involving high-dimensional integrals over the stochastic variables: the expected value and the (nonlinear) entropic risk measure. The high-dimensional integrals are computed numerically using specially designed QMC methods and, under moderate assumptions on the input random field, the error rate is shown to be essentially linear, independently of the stochastic dimension of the problem—and thereby superior to ordinary Monte Carlo methods. Numerical results demonstrate the effectiveness of our method.

Comparison of Statistical Approaches for Reconstructing Random Coefficients in the Problem of Stochastic Modeling of Air–Sea Heat Flux Increments

This paper compares two statistical methods for parameter reconstruction (random drift and diffusion coefficients of the Itô stochastic differential equation, SDE) in the problem of stochastic modeling of air–sea heat flux increment evolution. The first method relates to a nonparametric estimation of the transition probabilities (wherein consistency is proven). The second approach is a semiparametric reconstruction based on the approximation of the SDE solution (in terms of distributions) by finite normal mixtures using the maximum likelihood estimates of the unknown parameters. This approach does not require any additional assumptions for the coefficients, with the exception of those guaranteeing the existence of the solution to the SDE itself. It is demonstrated that the corresponding conditions hold for the analyzed data. The comparison is carried out on the simulated samples, modeling the case where the SDE random coefficients are represented in trigonometric form, which is related to common climatic models, as well as on the ERA5 reanalysis data of the sensible and latent heat fluxes in the North Atlantic for 1979–2022. It is shown that the results of these two methods are close to each other in a quantitative sense, but differ somewhat in temporal variability and spatial localization. The differences during the observed period are analyzed, and their geophysical interpretations are presented. The semiparametric approach seems promising for physics-informed machine learning models.

An efficient approximation to the stochastic Allen-Cahn equation with random diffusion coefficient field and multiplicative noise

An efficient approximation to the stochastic Allen-Cahn equation with random diffusion coefficient field and multiplicative noise

An ensemble scheme for the numerical solution of a random transient heat equation with uncertain inputs

An ensemble-based time stepping scheme is applied to solving a transient heat equation with random Robin boundary and diffusion coefficients. By introducing two ensemble means of Robin boundary and diffusion coefficients, we propose a new ensemble Monte Carlo (EMC) scheme for the transient heat equation. The EMC scheme solves a single linear system including several right-side vectors at each time step. Stability analysis and error estimates are derived. Two numerical examples verify the theoretical results and the validity of the EMC method.

Extended Milstein Approximation to the Stochastic Allen-Cahn Equation with Random Diffusion Coefficient Field and Multiplicative Noise

Extended Milstein Approximation to the Stochastic Allen-Cahn Equation with Random Diffusion Coefficient Field and Multiplicative Noise

An MLMCE-HDG method for the convection diffusion equation with random diffusivity

In this paper, a multilevel Monte Carlo ensemble hybridizable discontinuous Galerkin (MLMCE-HDG) method is proposed to solve the convection diffusion equation with random diffusion coefficients. This method combines the MLMC method in probability space, an ensemble approach in time space and the HDG method in physical space. The proposed method possesses a second order accuracy in time space and optimal L 2 convergence rate in physical space. The stability analysis, error estimates and computational cost are provided. Finally, to illustrate the theoretical analyses, several numerical experiments are performed.

An adaptive stochastic Galerkin method based on multilevel expansions of random fields: Convergence and optimality

The subject of this work is a new stochastic Galerkin method for second-order elliptic partial differential equations with random diffusion coefficients. It combines operator compression in the stochastic variables with tree-based spline wavelet approximation in the spatial variables. Relying on a multilevel expansion of the given random diffusion coefficient, the method is shown to achieve optimal computational complexity up to a logarithmic factor. In contrast to existing results, this holds in particular when the achievable convergence rate is limited by the regularity of the random field, rather than by the spatial approximation order. The convergence and complexity estimates are illustrated by numerical experiments.

An efficient numerical approach for stochastic evolution PDEs driven by random diffusion coefficients and multiplicative noise

<abstract><p>In this paper, we investigate the stochastic evolution equations (SEEs) driven by a bounded $ \log $-Whittle-Mat$ \acute{{\mathrm{e}}} $rn (W-M) random diffusion coefficient field and $ Q $-Wiener multiplicative force noise. First, the well-posedness of the underlying equations is established by proving the existence, uniqueness, and stability of the mild solution. A sampling approach called approximation circulant embedding with padding is proposed to sample the random coefficient field. Then a spatio-temporal discretization method based on semi-implicit Euler-Maruyama scheme and finite element method is constructed and analyzed. An estimate for the strong convergence rate is derived. Numerical experiments are finally reported to confirm the theoretical result.</p></abstract>

MDFEM: Multivariate decomposition finite element method for elliptic PDEs with uniform random diffusion coefficients using higher-order QMC and FEM

We introduce the multivariate decomposition finite element method (MDFEM) for solving elliptic PDEs with uniform random diffusion coefficients. We show that the MDFEM can be used to reduce the computational complexity of estimating the expected value of a linear functional of the solution of the PDE. The proposed algorithm combines the multivariate decomposition method, to compute infinite-dimensional integrals, with the finite element method, to solve different instances of the PDE. The strategy of the MDFEM is to decompose the infinite-dimensional problem into multiple finite-dimensional ones which lends itself to easier parallelization than to solve a single large dimensional problem. Our first result adjusts the analysis of the multivariate decomposition method to incorporate the \((\ln (n))^d\)-factor which typically appears in error bounds for d-dimensional n-point cubature formulae and we take care of the fact that n needs to come, e.g., in powers of 2 for higher order approximations. For the further analysis we specialize the cubature methods to be two types of quasi-Monte Carlo (QMC) rules, being digitally shifted polynomial lattice rules and interlaced polynomial lattice rules. The second and main contribution then presents a bound on the error of the MDFEM and shows higher-order convergence w.r.t. the total computational cost in case of the interlaced polynomial lattice rules in combination with a higher-order finite element method. We show that the cost to achieve an error \(\epsilon \) is of order \(\epsilon ^{-a_{\mathrm {MDFEM}}}\) with \(a_{\mathrm {MDFEM}} = 1/\lambda + d'/\tau \) if the QMC cubature errors can be bounded by \(n^{-\lambda }\) and the FE approximations converge like \(h^\tau \) with cost \(h^{d'}\), where \(\lambda = \tau (1-p^*) / (p^* (1+d'/\tau ))\) and \(p^*\) is a parameter representing the “sparsity” of the random field expansion. A comparison with a dimension truncation algorithm shows that the MDFEM will perform better than the truncation algorithm if \(p^*\) is sufficiently small, i.e., the representation of the random field is sufficiently sparse.

A Multiscale Method for Heterogeneous Bulk-Surface Coupling

In this paper, we construct and analyze a multiscale (finite element) method for parabolic problems with heterogeneous dynamic boundary conditions. As the origin, we consider a reformulation of the system in order to decouple the discretization of bulk and surface dynamics. This allows us to combine multiscale methods on the boundary with standard Lagrangian schemes in the interior. We prove convergence and quantify explicit rates for low-regularity solutions, which are independent of the oscillatory behavior of the heterogeneities. As a result, coarse discretization parameters, which do not resolve the fine scales, can be considered. The theoretical findings are justified by a number of numerical experiments including dynamic boundary conditions with random diffusion coefficients.

The Uniqueness of Stationary Solution for Nonlinear Random Reaction-diffusion Equation

We study a nonlinear random reaction-diffusion problem in abstract Banach spaces, driven by a real noise, with random diffusion coefficient and random initial condition. We consider a polynomial non linear term. The reaction-diffusion equation belongs to the class of parabolic stochastic partial differential equations. We assume that the initial condition is an element of Hilbert space. The real noise is a Wiener process. We construct a suitable stochastic basis and define the solution of reaction-diffusion problem in the weak sense. We define the stationary process in abstract Banach spaces in the strong sense of Doob-Rozanov. That is, the probability density function of the stochastic process is independent of time shift. We define the invariant measure for random reaction-diffusion equation in the sense of Arnold, DaPrato, and Zabczyk [1,2]. In other words, we define the invariant measure for random dynamical system, associated with random reaction-diffusion problem.
 Using the Variation Inequalities Theory, we prove the uniqueness of stationary solution for nonlinear random reaction-diffusion problem. The obtained theoretical results have several applications in Quantum Physics, Biology, Medicine, and Economic Sciences. Especially, we can study the existence of stationary solution for the stochastic models of tumor growth.

A mathematical model for Alzheimer’s disease: An approach via stochastic homogenization of the Smoluchowski equation

In this note, we apply the theory of stochastic homogenization to find the asymptotic behavior of the solution of a set of Smoluchowski's coagulation-diffusion equations with non-homogeneous Neumann boundary conditions. This system is meant to model the aggregation and diffusion of β-amyloid peptide (Aβ) in the cerebral tissue, a process associated with the development of Alzheimer's disease. In contrast to the approach used in our previous works, in the present paper we account for the non-periodicity of the cellular structure of the brain by assuming a stochastic model for the spatial distribution of neurons. Further, we consider non-periodic random diffusion coefficients for the amyloid aggregates and a random production of Aβ in the monomeric form at the level of neuronal membranes.

Test particle dynamics in low-frequency tokamak turbulence

We study the evolution of one million test particles in a turbulent plasma simulation, using the gyrokinetic code Trapped Element REduction in Semi-Lagrangian Approach (TERESA), as a method to get insights into the type of transport governing the plasma. TERESA (Trapped Element REduction in Semi-Lagrangian Approach) is a collisionless global 4D code which treats the trapped particles kinetically, while the passing particles are considered adiabatic. The Vlasov-Poisson system of equations is averaged over the cyclotron and the trapped particle's bounce motion, and thus, the model focuses on slow phenomena of the order of the toroidal precession motion of the banana orbits. We initialize the test particles, which are de facto “test banana-centers,” at a time of the simulation when the plasma is turbulent. We impose an initial temperature and density gradients, and only the Trapped Ion Mode (TIM) instability can develop in this system. We then calculate the Mean Squared Displacement of the test particles as a function of time in order to obtain a random walk diffusion coefficient. We observe that the radial diffusion of the test particles depends on their toroidal precession kinetic energy (E), in such a way that the transport of particles is dominated by a strong, relatively narrow peak at the resonant energies. A radial particle diffusion flux is then calculated and compared to the total radial particle flux accounting for all the transport processes such as diffusion and advection which is obtained directly from the TERESA code. We can thus compare the diffusive contribution to the particle flux against the nondiffusive contributions. The results show that the total flux is essentially diffusive which is consistent with our simulation setup aiming for “global turbulence.” Both fluxes present a peak around a resonance energy ER≈1.74Ti between the TIM and the particles. Both thermal and high-energy particles do not contribute significantly to radial transport.

Multilevel methods for uncertainty quantification of elliptic PDEs with random anisotropic diffusion

We consider elliptic diffusion problems with a random anisotropic diffusion coefficient, where, in a notable direction given by a random vector field, the diffusion strength differs from the diffusion strength perpendicular to this notable direction. The Karhunen–Loeve expansion then yields a parametrisation of the random vector field and, therefore, also of the solution of the elliptic diffusion problem. We show that, given regularity of the elliptic diffusion problem, the decay of the Karhunen–Loeve expansion entirely determines the regularity of the solution’s dependence on the random parameter, also when considering this higher spatial regularity. This result then implies that multilevel quadrature methods may be used to lessen the computation complexity when approximating quantities of interest, like the solution’s mean or its second moment, while still yielding the expected rates of convergence. Numerical examples in three spatial dimensions are provided to validate the presented theory.

Spatiotemporal pattern formation of a ratio-dependent invasion-diffusion predator-prey system with disease in the predator

This paper is purported to investigate a ratio-dependent invasion-diffusion predator-prey epidemic system in a bounded domain with no flux boundary condition. The local and global stabilities of positive equilibrium are investigated to this system without invasion-diffusion. The sufficient conditions to nonexistence and existence of non-constant positive solution are found for this invasion-diffusion epidemic system, which implies the existence of spatiotemporal pattern formation. The results show that in wide space for the predator and the prey to diffuse by the self pressure and with a little tendency of predator to catch prey, they can’t coexist and there isn’t endemic disease extensively, but they can coexist and endemic disease exists extensively when the coefficient of invasion-diffusion for predator is big enough with other random diffusion coefficients being fixed and satisfying curtain conditions.

Stability Properties of Systems of Linear Stochastic Differential Equations with Random Coefficients

This work is concerned with the stability properties of linear stochastic differential equations with random (drift and diffusion) coefficient matrices, and the stability of a corresponding random transition matrix (or exponential semigroup). We consider a class of random matrix drift coefficients that involves random perturbations of an exponentially stable flow of deterministic (time-varying) drift matrices. In contrast with more conventional studies, our analysis is not based on the existence of Lyapunov functions, and it does not rely on any ergodic properties. These approaches are often difficult to apply in practice when the drift/diffusion coefficients are random. We present rather weak and easily checked perturbation-type conditions for the asymptotic stability of time-varying and random linear stochastic differential equations. We provide new log-Lyapunov estimates and exponential contraction inequalities on any time horizon as soon as the fluctuation parameter is sufficiently small. These seem to be the first results of this type for this class of linear stochastic differential equations with random coefficient matrices.

Ensemble Time-Stepping Algorithm for the Convection-Diffusion Equation with Random Diffusivity

In this paper, we develop two ensemble time-stepping algorithms to solve the convection-diffusion equation with random diffusion coefficients, forcing terms and initial conditions based on the pseudo-spectral stochastic collocation method. The key step of the pseudo-spectral stochastic collocation method is to solve a number of deterministic problems derived from the original stochastic convection-diffusion equation. In general, a common way to solve the set of deterministic problems is by using the backward differentiation formula, which requires us to store the coefficient matrix and right-hand-side vector multiple times, and solve them one by one. However, the proposed algorithm only need to solve a single linear system with one shared coefficient matrix and multiple right-hand-side vectors, reducing both storage required and computational cost of the solution process. The stability and error analysis of the first- and second-order ensemble time-stepping algorithms are provided. Several numerical experiments are presented to confirm the theoretical analyses and verify the feasibility and effectiveness of the proposed method.

Haven Correlation Parameter for Diffusion of Fluorine in Superionic Conductors La1 – ySryF3 – y

The processes of electric charge transfer (conductivity) and mass transfer (diffusion) in La1 ‒ ySryF3 – y superionic conductors are determined by mobile fluorine ions. The fluorine random-diffusion coefficients Dσ have been calculated from experimental data on the ionic conductivity σdc for La1 ‒ ySryF3 – y single crystals at the SrF2 dopant contents of 1, 3, 5, 7.5, 10, and 15 mol %. A maximum in the region of 3–5 mol % SrF2 is observed in the dependence Dσ(y). The value of the Haven correlation parameter Hr = DNMR/Dσ (DNMR is the fluorine diffusion coefficient measured by NMR on 19F nuclei) is determined; it characterizes the ion transport mechanism in La1 – ySryF3 – y crystals. The Hr values are 0.75 ± 0.15, 0.45 ± 0.15, and 0.65 ± 0.15 at 400–800 K for 1, 3, and 15–16 mol % SrF2, respectively. Superionic conductor La0.97Sr0.03F2.97 with maximum σdc and Dσ values has a minimum Haven parameter. The obtained Hr values indicate that fluorine diffusion in superionic conductors La1 – ySryF3 – y goes not according to the vacancy mechanism involving single vacancies but by cooperative motion of F– conduction ions.

Representations of Gaussian Random Fields and Approximation of Elliptic PDEs with Lognormal Coefficients

Approximation of elliptic PDEs with random diffusion coefficients typically requires a representation of the diffusion field in terms of a sequence \(y=(y_j)_{j\ge 1}\) of scalar random variables. One may then apply high-dimensional approximation methods to the solution map \(y\mapsto u(y)\). Although Karhunen–Loeve representations are commonly used, it was recently shown, in the relevant case of lognormal diffusion fields, that multilevel-type expansions may yield better approximation rates. Motivated by these results, we construct wavelet-type representations of stationary Gaussian random fields defined on arbitrary bounded domains. The size and localization properties of these wavelets are studied, and used to obtain polynomial approximation results for the related elliptic PDE which outperform those achievable when using Karhunen–Loeve representations. Our construction is based on a periodic extension of the stationary random field, and the expansion on the domain is then obtained by simple restriction. This makes the approach easily applicable even when the computational domain of the PDE has a complicated geometry. In particular, we apply this construction to the class of Gaussian processes defined by the family of Matern covariances. The proposed periodic continuation technique has other relevant applications such as fast simulation of trajectories. It can be regarded as a continuous analog of circulant embedding techniques introduced for Toeplitz matrices. One of its specific features is that the rate of decay of the eigenvalues of the covariance operator of the periodized process provably matches that of the Fourier transform of the covariance function of the original process.

Uncertainty Quantification for PDEs with Anisotropic Random Diffusion

In this article, we consider elliptic diffusion problems with an anisotropic random diffusion coefficient. We model the notable direction in terms of a random vector field and derive regularity results for the solution's dependence on the random parameter. It turns out that the decay of the vector field's Karhunen--Loeve expansion entirely determines this regularity. The obtained results allow for sophisticated quadrature methods, such as the quasi--Monte Carlo method or the anisotropic sparse grid quadrature, in order to approximate quantities of interest, like the solution's mean or the variance. Numerical examples in three spatial dimensions are provided to supplement the presented theory.