Approximation Of Differential Equations Research Articles

Pathwise convergence under Knightian uncertainty

In this note, we will obtain the first quasi-surely convergence rate of approximation of stochastic differential equations driven by G-Brownian motion. We carry it out by making a relation between Lp and quasi-surely convergences by considering this point that in general Lp convergence does not imply quasi-surely one and neither does quasi-surely convergence. This result will show that the rate of quasi-surely convergence can not exceed that of p-th mean.

CutFEM based on extended finite element spaces

We develop a general framework for construction and analysis of discrete extension operators with application to unfitted finite element approximation of partial differential equations. In unfitted methods so called cut elements intersected by the boundary occur and these elements must in general by stabilized in some way. Discrete extension operators provides such a stabilization by modification of the finite element space close to the boundary. More, precisely the finite element space is extended from the stable interior elements over the boundary in a stable way which also guarantees optimal approximation properties. Our framework is applicable to all standard nodal based finite elements of various order and regularity. We develop an abstract theory for elliptic problems and associated parabolic time dependent partial differential equations and derive a priori error estimates. We finally apply this to some examples of partial differential equations of different order including the interface problems, the biharmonic operator and the sixth order triharmonic operator.

Numerical approximation of nonlinear SPDE’s

The numerical analysis of stochastic parabolic partial differential equations of the form du+A(u)dt=fdt+gdW,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} du + A(u)\\, dt = f \\,dt + g \\, dW, \\end{aligned}$$\\end{document}is surveyed, where A is a nonlinear partial operator and W a Brownian motion. This manuscript unifies much of the theory developed over the last decade into a cohesive framework which integrates techniques for the approximation of deterministic partial differential equations with methods for the approximation of stochastic ordinary differential equations. The manuscript is intended to be accessible to audiences versed in either of these disciplines, and examples are presented to illustrate the applicability of the theory.

Numerical computation of probabilities for nonlinear SDEs in high dimension using Kolmogorov equation

Stochastic Differential Equations (SDEs) in high dimension, having the structure of finite dimensional approximation of Stochastic Partial Differential Equations (SPDEs), are considered. The aim is to numerically compute the expected values and probabilities associated to their solutions, by solving the corresponding Kolmogorov equations, with a partial use of Monte Carlo strategy - precisely, using Monte Carlo only for the linear part of the SDE. The basic idea was presented in [16], but here we strongly improve the numerical results by means of a shift of the auxiliary Gaussian process. For relatively simple nonlinearities, we have good results in dimension of the order of 100.

On the extreme eigenvalues and asymptotic conditioning of a class of Toeplitz matrix-sequences arising from fractional problems

The analysis of the spectral features of a Toeplitz matrix-sequence , generated by the function , real-valued almost everywhere (a.e.), has been provided in great detail in the last century, as well as the study of the conditioning, when f is nonnegative a.e. Here we consider a novel type of problem arising in the numerical approximation of distributed-order fractional differential equations (FDEs), where the matrices under consideration take the form belong to the interval with independent of n, , , and for every . For nonnegative functions or sequences, the notation means that there exist positive constants c, d, independent of the variable x in the definition domain such that for any x. Since the resulting generating function depends on n, the standard theory cannot be applied and the analysis has to be performed using new ideas. Few selected numerical experiments are presented, also in connection with matrices that come from distributed-order FDE problems, and the adherence with the theoretical analysis is discussed, together with open questions and future investigations.

Fractional Romanovski–Jacobi tau method for time-fractional partial differential equations with nonsmooth solutions

In this paper, we introduce a new finite class of non-polynomial basis functions for the spectral tau approximation of time-fractional partial differential equations on a semi-infinite interval. Different from many other spectral tau approaches, the singular basis functions are based on a finite class of Romanovski–Jacobi polynomials through a fractional coordinate transform. As such, the singularity of the new basis can be tailored to suit that of the singular solutions to a class of time-fractional partial differential equations, leading to spectrally accurate approximations. We introduce the fractional Romanovski-Jacobi-Gauss-type quadrature formulae along with basic approximation results for some weighted projection operators defined in suitable weighted Sobolev spaces. We discuss the relationship between such kind of finite orthogonal functions and other classes of infinite orthogonal functions such as fractional Laguerre functions and fractional modified generalized Laguerre functions. Moreover, we derive a new formula expressing explicitly the fractional integral of the fractional Romanovski-Jacobi functions in terms of the same functions themselves. This family of orthogonal systems offers great flexibility to match a wide range of time-fractional differential equations with Robin boundary conditions and with nonsmooth solutions.

Delay-dependent elliptic reconstruction and optimal 𝐿^{∞}(𝐿²) a posteriori error estimates for fully discrete delay parabolic problems

We derive optimal order a posteriori error estimates for fully discrete approximations of linear parabolic delay differential equations (PDDEs), in the L ∞ ( L 2 ) L^\infty (L^2) -norm. For the discretization in time we use Backward Euler and Crank-Nicolson methods, while for the space discretization we use standard conforming finite element methods. A novel space-time reconstruction operator is introduced, which is a generalization of the elliptic reconstruction operator, and we call it as delay-dependent elliptic reconstruction operator. The related a posteriori error estimates for the delay-dependent elliptic reconstruction play key roles in deriving optimal order a posteriori error estimates in the L ∞ ( L 2 ) L^\infty (L^2) -norm. Numerical experiments verify and complement our theoretical results.

Rate of convergence for particle approximation of PDEs in Wasserstein space

AbstractWe prove a rate of convergence for the N-particle approximation of a second-order partial differential equation in the space of probability measures, such as the master equation or Bellman equation of the mean-field control problem under common noise. The rate is of order $1/N$ for the pathwise error on the solution v and of order $1/\sqrt{N}$ for the $L^2$ -error on its L-derivative $\partial_\mu v$ . The proof relies on backward stochastic differential equation techniques.

Moment estimation for parameters in high-order uncertain differential equations

As a type of differential equations with high-order derivatives of uncertain processes, high-order uncertain differential equations are widely applied to modelling dynamic systems in uncertain environment, which usually involve unknown parameters to be estimated. Since observations are always discrete in practice, based on these discrete observations of solution processes, we propose moment estimations for unknown parameters by Euler method approximation of high-order uncertain differential equations. Matching sample moments with corresponding population moments, a system of equations whose solution is the moment estimation of the set of unknown parameters is derived. Finally, some examples illustrate our method in detail.

Overcoming the curse of dimensionality in the numerical approximation of backward stochastic differential equations

Abstract Backward stochastic differential equations (BSDEs) belong nowadays to the most frequently studied equations in stochastic analysis and computational stochastics. BSDEs in applications are often nonlinear and high-dimensional. In nearly all cases such nonlinear high-dimensional BSDEs cannot be solved explicitly and it has been and still is a very active topic of research to design and analyze numerical approximation methods to approximatively solve nonlinear high-dimensional BSDEs. Although there are a large number of research articles in the scientific literature which analyze numerical approximation methods for nonlinear BSDEs, until today there has been no numerical approximation method in the scientific literature which has been proven to overcome the curse of dimensionality in the numerical approximation of nonlinear BSDEs in the sense that the number of computational operations of the numerical approximation method to approximatively compute one sample path of the BSDE solution grows at most polynomially in both the reciprocal 1 / ε $ 1 / \varepsilon $ of the prescribed approximation accuracy ε ∈ ( 0 , ∞ ) $ \varepsilon \in(0, \infty) $ and the dimension d ∈ N = { 1 , 2 , 3 , … } $ d\in {\mathbb{N}}=\{1,2,3,\ldots\} $ of the BSDE. It is the key contribution of this article to overcome this obstacle by introducing a new Monte Carlo-type numerical approximation method for high-dimensional BSDEs and by proving that this Monte Carlo-type numerical approximation method does indeed overcome the curse of dimensionality in the approximative computation of solution paths of BSDEs.

Numerical approximation of partial differential equations by a variable projection method with artificial neural networks

We present a method for solving linear and nonlinear partial differential equations (PDE) based on the variable projection framework and artificial neural networks. For linear PDEs, enforcing the boundary/initial value problem on the collocation points gives rise to a separable nonlinear least squares problem about the network coefficients. We reformulate this problem by the variable projection approach to eliminate the linear output-layer coefficients, leading to a reduced problem about the hidden-layer coefficients only. The reduced problem is solved first by the nonlinear least squares method to determine the hidden-layer coefficients, and then the output-layer coefficients are computed by the linear least squares method. For nonlinear PDEs, enforcing the boundary/initial value problem on the collocation points gives rise to a nonlinear least squares problem that is not separable, which precludes the variable projection strategy for such problems. To enable the variable projection approach for nonlinear PDEs, we first linearize the problem with a Newton iteration, using a particular linearization formulated in terms of the updated approximation field. The linearized system is solved by the variable projection framework together with artificial neural networks. Upon convergence of the Newton iteration, the neural-network coefficients provide the representation of the solution field to the original nonlinear problem. We present ample numerical examples to demonstrate the performance of the method developed herein. For smooth field solutions, the errors of the current method decrease exponentially as the number of collocation points or the number of output-layer coefficients increases. We compare extensively the current method with the extreme learning machine (ELM) method from a previous work. Under identical conditions and configurations, the current method exhibits an accuracy significantly superior to the ELM method.

BDDC Preconditioners for Divergence Free Virtual Element Discretizations of the Stokes Equations

The virtual element method (VEM) is a new family of numerical methods for the approximation of partial differential equations, where the geometry of the polytopal mesh elements can be very general. The aim of this article is to extend the balancing domain decomposition by constraints preconditioner to the solution of the saddle-point linear system arising from a VEM discretization of the two-dimensional Stokes equations. Under suitable hypotesis on the choice of the primal unknowns, the preconditioned linear system results symmetric and positive definite, thus the preconditioned conjugate gradient method can be used for its solution. We provide a theoretical convergence analysis estimating the condition number of the preconditioned linear system. Several numerical experiments validate the theoretical estimates, showing the scalability and quasi-optimality of the method proposed. Moreover, the solver exhibits a robust behavior with respect to the shape of the polygonal mesh elements. We also show that a faster convergence could be achieved with an easy to implement coarse space, slightly larger than the minimal one covered by the theory.

APPROXIMATE SOLUTIONS OF WEAKLY NONLINEAR DIFFERENTIAL EQUATIONS

In this work, we study the most useful approximation methods for proving solutions to analytic approximation for solving a weak second-order nonlinear differential equation in a power series with small parameters. We prove the second-order periodic approximation solution and also the best third-order approximation of the weak nonlinear differential equation.

An analytical approach for Shehu transform on fractional coupled 1D, 2D and 3D Burgers’ equations

Abstract Obtaining the numerical approximation of fractional partial differential equations (PDEs) is a cumbersome task. Therefore, more researchers regarding approximated-analytical solutions of such complex-natured fractional PDEs (FPDEs) are required. In this article, analytical-approximated solutions of the fractional-order coupled Burgers’ equation are provided in one-, two-, and three-dimensions. The proposed technique is named as Iterative Shehu Transform Method (ISTM). The simplicity and accurateness of the method are affirmed through five examples. Graphical representation and tabular discussion are provided to compare the exact and approximated results. The robustness of the proposed regime is also validated by error analysis. In the present work, approximated and exact solutions are compared to verify the validity of the proposed scheme. Error analysis is also provided through which the efficiency of the proposed scheme can be assured. Obtained errors are lesser than the compared results.

A Model for Cell Proliferation in a Developing Organism

In mathematical biology, there is a great deal of interest in producing continuum models by scaling discrete agent-based models governed by local stochastic rules. We discuss a particular example of this approach: a model for the proliferation of neural crest cells that can help us understand the development of Hirschprung’s disease, a potentially-fatal condition in which the enteric nervous system of a new-born child does not extend all the way through the intestine and colon. Our starting point is a discrete-state, continuous-time Markov chain model proposed by Hywood et al. (2013a) for the location of the neural crest cells that make up the enteric nervous system. Hywood et al. (2013a) scaled their model to derive an approximate second order partial differential equation describing how the limiting expected number of neural crest cells evolve in space and time. In contrast, we exploit the relationship between the above-mentioned Markov chain model and the well-known Yule-Furry process to derive the exact form of the scaled version of the process. Furthermore, we provide expressions for other features of the domain agent occupancy process, such as the variance of the marginal occupancy at a particular site, the distribution of the number of agents that are yet to reach a given site and a stochastic description of the process itself.

Genetic algorithm optimization to model business investment in fashion design

ABSTRACT In the field of marketing, business and finance, the rise and fall and the associated bifurcations can be better interpreted with the aid of differential equations and the dynamical systems. The differential equations, when Incorporated with the delayed dynamics can interpret the financial delays in more realistic manner. The parametric approximation of such differential equations can be achieved with the aid of robust optimization tool, that is termed as the genetic algorithm. In this manuscript, the concept of delay is invoked with the aid of specific factors linked with the raise and fall of investment frequency in the field of fashion industry and the corresponding stability is modeled with the aid of detailed dynamical analysis. The stability and instability criteria of delay differential equations, governing the dynamics of the investment trends is reported in this manuscript. The bifurcation and their relative criteria of occurrence is discussed. The Hopf bifurcation is important since with the aid of the bifurcation analysis, one can obtain a region of instability, and can thus invest in a safer manner by avoiding such circumstances.

Stationary Wong–Zakai Approximation of Fractional Brownian Motion and Stochastic Differential Equations with Noise Perturbations

In this article, we introduce a Wong–Zakai type stationary approximation to the fractional Brownian motions and provide a sharp rate of convergence in Lp(Ω). Our stationary approximation is suitable for all values of H∈(0,1). As an application, we consider stochastic differential equations driven by a fractional Brownian motion with H>1/2. We provide sharp rate of convergence in a certain fractional-type Sobolev space of the approximation, which in turn provides rate of convergence for the solution of the approximated equation. This generalises some existing results in the literature concerning approximation of the noise and the convergence of corresponding solutions.

Multilevel Picard approximations of high-dimensional semilinear partial differential equations with locally monotone coefficient functions

The full history recursive multilevel Picard approximation method for semilinear parabolic partial differential equations (PDEs) is the only method which provably overcomes the curse of dimensionality for general time horizons if the coefficient functions and the nonlinearity are globally Lipschitz continuous and the nonlinearity is gradient-independent. In this article we extend this result to locally monotone coefficient functions. Our results cover a range of semilinear PDEs with polynomial coefficient functions.

The Cădariu–Radu method for existence, uniqueness and Gauss Hypergeometric stability of a class of Ξ-Hilfer fractional differential equations

Abstract In this paper, we apply the Cădariu–Radu method derived from the Diaz–Margolis theorem to investigate existence, uniqueness approximation of Ξ-Hilfer fractional differential equations, and Hypergeometric stability for both finite and infinite domains. An example is given to illustrate the main result for a fractional system.

A Decision-Making Machine Learning Approach in Hermite Spectral Approximations of Partial Differential Equations

The accuracy and effectiveness of Hermite spectral methods for the numerical discretization of partial differential equations on unbounded domains are strongly affected by the amplitude of the Gaussian weight function employed to describe the approximation space. This is particularly true if the problem is under-resolved, i.e., there are no enough degrees of freedom. The issue becomes even more crucial when the equation under study is time-dependent, forcing in this way the choice of Hermite functions where the corresponding weight depends on time. In order to adapt dynamically the approximation space, it is here proposed an automatic decision-making process that relies on machine learning techniques, such as deep neural networks and support vector machines. The algorithm is numerically tested with success on a simple 1D problem, but the main goal is its exportability in the context of more serious applications. As a matter of fact we also show at the end an application in the framework of plasma physics.