Nonsmooth Data Research Articles

Convergence of an Exponential Runge–Kutta Method for Non-smooth Initial Data

The paper presents error bounds for the second order exponential Runge-Kutta method for parabolic abstract linear time-dependent differential equations incorporating non-smooth initial data. As an example for this particular type of problems, the paper presents a spatial discretization of a partial integro-differential equation arising in financial mathematics, where non-smooth initial conditions occur in option pricing models. For this example, numerical studies of the convergence rate are given

Numerical approximation of stochastic time-fractional diffusion

We develop and analyze a numerical method for stochastic time-fractional diffusion driven by additive fractionally integrated Gaussian noise. The model involves two nonlocal terms in time,i.e.,a Caputo fractional derivative of orderα ∈ (0,1), and fractionally integrated Gaussian noise (with a Riemann-Liouville fractional integral of orderγ ∈ [0,1] in the front). The numerical scheme approximates the model in space by the standard Galerkin method with continuous piecewise linear finite elements and in time by the classical Grünwald-Letnikov method (for both Caputo fractional derivative and Riemann-Liouville fractional integral), and the noise by theL2-projection. Sharp strong and weak convergence rates are established, using suitable nonsmooth data error estimates for the discrete solution operators for the deterministic inhomogeneous problem. One- and two-dimensional numerical results are presented to support the theoretical findings.

A Split-Step Predictor–Corrector Method for Space-Fractional Reaction–Diffusion Equations with Nonhomogeneous Boundary Conditions

A split-step second-order predictor–corrector method for space-fractional reaction–diffusion equations with nonhomogeneous boundary conditions is presented and analyzed for the stability and convergence. The matrix transfer technique is used for spatial discretization of the problem. The method is shown to be unconditionally stable and second-order convergent. Numerical experiments are performed to confirm the stability and second-order convergence of the method. The split-step predictor–corrector method is also compared with an IMEX predictor–corrector method which is found to incur oscillatory behavior for some time steps. Our method is seen to produce reliable and oscillation-free results for any time step when implemented on numerical examples with nonsmooth initial data. We also present a priori reliability constraint for the IMEX predictor–corrector method to avoid unwanted oscillations and show its validity numerically.

Fast algorithms for convolution quadrature of Riemann-Liouville fractional derivative

Recently, the numerical schemes of the Fokker-Planck equations describing anomalous diffusion with two internal states have been proposed in Nie et al., [23], which use convolution quadrature to approximate the Riemann-Liouville fractional derivative; and the schemes need huge storage and computational cost because of the non-locality of fractional derivative and the large scale of the system. This paper first provides fast algorithms for computing the Riemann-Liouville derivative based on convolution quadrature with the generating function given by the backward Euler and second-order backward difference methods; the algorithms don't require the assumption of the regularity of the solution in time, while the computation time and the total memory requirement are greatly reduced. Then we apply fast algorithms to solve the homogeneous fractional Fokker-Planck equations with two internal states for nonsmooth data and get the first- and second-order accuracy in time. Lastly, numerical examples are presented to verify the convergence and the effectiveness of fast algorithms.

Convergence Analysis of a Petrov–Galerkin Method for Fractional Wave Problems with Nonsmooth Data

This paper analyzes the convergence of a Petrov–Galerkin method for time fractional wave problems with nonsmooth data. Well-posedness and regularity of the weak solution to the time fractional wave problem are firstly established. Then an optimal convergence analysis with nonsmooth data is derived. Moreover, several numerical experiments are presented to validate the theoretical results.

Fourier spectral exponential time differencing methods for multi-dimensional space-fractional reaction–diffusion equations

In this paper, we propose two numerical methods for solving the space-fractional reaction–diffusion equations, which are based on a Fourier spectral approach in space and exponential time differencing schemes in time. The advantages of the approaches are that they attain spectral convergence, produce a full diagonal representation of the fractional operator, and the extension to multiple spatial dimensions is the same as the one-dimensional space. That can overcome the constraints associated with many of the numerical schemes for these equations such as the computational efficiency caused by the non-locality of the fractional operator, which results in full, dense matrices. Moreover, the proposed schemes are second-order convergent, unconditionally stable, and highly efficient due to the predictor–corrector feature when comparing them with the existing method. It is observed that the scheme based on using Padé(1,1) approximations to the matrix exponential function introduces oscillations with non-smooth initial data for some time steps due to high frequency components present in the solution which diminish as the fractional order decreases. However, the scheme based on Padé(0,2) approximations to the matrix exponential function is oscillation-free for any time step. Numerical experiments for well-known models from the literature are performed to show the reliability and effectiveness of the proposed methods.

Spectral methods in the presence of discontinuities

Spectral methods provide an elegant and efficient way of numerically solving differential equations of all kinds. For smooth problems, truncation error for spectral methods vanishes exponentially in the infinity norm and L2-norm. However, for non-smooth problems, convergence is significantly worse—the L2-norm of the error for a discontinuous problem will converge at a sub-linear rate and the infinity norm will not converge at all. We explore and improve upon a post-processing technique—optimally convergent mollifiers—to recover exponential convergence from a poorly-converging spectral reconstruction of non-smooth data. This is an important first step towards using these techniques for simulations of realistic systems.

An Exponentially Convergent Scheme in Time for Time Fractional Diffusion Equations with Non-smooth Initial Data

In this paper we aim at developing highly accurate and stable method in temporal direction for time-fractional diffusion equations with initial data $$v\in L^2(\varOmega )$$ . To this end we begin with a kind of (time-)fractional ODEs, and a hybrid multi-domain Petrov–Galerkin spectral method is proposed. To match the singularity at the beginning of time, we use geometrically graded mesh together with fractional power Jacobi functions as basis on the first interval. Jacobi polynomials are then chosen to approximate the solution in temporal direction on the intervals hereafter. The algorithm is motivated by the discovery that the solution in temporal direction possesses high regularity in proper weighted Sobolev space on a special mesh in piecewise sense. Combining standard finite element method, we extend it to solve time-fractional diffusion equations with initial data $$v\in L^2(\varOmega )$$ , in which the mesh in temporal direction is determined by spatial mesh size and finial time T. Numerical tests show that the scheme is stable and converges exponentially in time.

Composite Finite Element Approximation for Parabolic Problems in Nonconvex Polygonal Domains

Abstract The purpose of this paper is to generalize known a priori error estimates of the composite finite element (CFE) approximations of elliptic problems in nonconvex polygonal domains to the time dependent parabolic problems. This is a new class of finite elements which was introduced by [W. Hackbusch and S. A. Sauter, Composite finite elements for the approximation of PDEs on domains with complicated micro-structures, Numer. Math. 75 1997, 4, 447–472] and subsequently modified by [M. Rech, S. A. Sauter and A. Smolianski, Two-scale composite finite element method for Dirichlet problems on complicated domains, Numer. Math. 102 2006, 4, 681–708] for the approximations of stationery problems on complicated domains. The basic idea of the CFE procedure is to work with fewer degrees of freedom by allowing finite element mesh to resolve the domain boundaries and to preserve the asymptotic order convergence on coarse-scale mesh. We analyze both semidiscrete and fully discrete CFE methods for parabolic problems in two-dimensional nonconvex polygonal domains and derive error estimates of order 𝒪 ( H s Log ^ ( H h ) s 2 ) {\mathcal{O}(H^{s}\widehat{\mathrm{Log}}{}^{\frac{s}{2}}(\frac{H}{h}))} and 𝒪 ( H 2 ⁢ s Log ^ ( H h ) s ) {\mathcal{O}(H^{2s}\widehat{\mathrm{Log}}{}^{s}(\frac{H}{h}))} in the L ∞ ⁢ ( H 1 ) {L^{\infty}(H^{1})} -norm and L ∞ ⁢ ( L 2 ) {L^{\infty}(L^{2})} -norm, respectively. Moreover, for homogeneous equations, error estimates are derived for nonsmooth initial data. Numerical results are presented to support the theoretical rates of convergence.

Subdiffusion with a time-dependent coefficient: Analysis and numerical solution

In this work, a complete error analysis is presented for fully discrete solutions of the subdiffusion equation with a time-dependent diffusion coefficient, obtained by the Galerkin finite element method with conforming piecewise linear finite elements in space and backward Euler convolution quadrature in time. The regularity of the solutions of the subdiffusion model is proved for both nonsmooth initial data and incompatible source term. Optimal-order convergence of the numerical solutions is established using the proven solution regularity and a novel perturbation argument via freezing the diffusion coefficient at a fixed time. The analysis is supported by numerical experiments.

Analysis of a Time-Stepping Scheme for Time Fractional Diffusion Problems with Nonsmooth Data

This paper establishes the convergence of a time-steeping scheme for time fractional diffusion problems with nonsmooth data. We first analyze the regularity of the model problem with nonsmooth data, and then prove that the time-steeping scheme possesses optimal convergence rates in $ L^2(0,T;L^2(\Omega)) $-norm and $ L^2(0,T;H_0^1(\Omega)) $-norm with respect to the regularity of the solution. Finally, numerical results are provided to verify the theoretical results.

Điều kiện tối ưu tập chấp nhận được lồi xác định bởi vô hạn ràng buộc bất đẳng thức

The paper deals with the necessary and sufficient optimality conditions for the convex optimization problem with convex feasible set defined by infinite inequality constraints in the both cases, smooth and nonsmooth data. The results enhance some recent KKT type theorems by Lasserre for differentiable functions and by Dutta and Lalitha for Lipschitz functions.

On the role of compressibility in poroviscoelastic models.

In this article we conduct an analytical study of a poroviscoelastic mixture model stemming from the classical Biot's consolidation model for poroelastic media, comprising a fluid component and a solid component, coupled with a viscoelastic stress-strain relationship for the total stress tensor. The poroviscoelastic mixture is studied in the one-dimensional case, corresponding to the experimental conditions of confined compression. Upon assuming (i) negligible inertial effects in the balance of linear momentum for the mixture, (ii) a Kelvin-Voigt model for the effective stress tensor and (iii) a constant hydraulic permeability, we obtain an initial value/boundary value problem of pseudo-parabolic type for the spatial displacement of the solid component of the mixture. The dimensionless form of the differential equation is characterized by the presence of two positive parameters γ and η, representing the contributions of compressibility and structural viscoelasticity, respectively. Explicit solutions are obtained for different functional forms characterizing the boundary traction. The main result of our analysis is that the compressibility of the components of a poroviscoelastic mixture does not give rise to unbounded responses to non-smooth traction data. Interestingly, compressibility allows the system to store potential energy as its components are elastically compressed, thereby providing an additional mechanism that limits the maximum of the discharge velocity when the imposed boundary traction is irregular in time.

Parallel two-grid finite element method for the time-dependent natural convection problem with non-smooth initial data

In this paper, we consider the parallel two-grid finite element method for the transient natural convection problem with non-smooth initial data. Our numerical scheme involves solving a nonlinear natural convection problem on the coarse grid and solving a linear natural convection problem on the fine grid. The linear natural convection problem can be split into two subproblems which can be solved in parallel: a linearized Navier–Stokes problem and a linear parabolic problem. We firstly provide the stability and convergence of standard Galerkin finite element method with non-smooth initial data. Secondly, we develop optimal error estimates of two-grid finite element method for velocity and temperature in H1-norm and for pressure in L2-norm. In order to overcome the difficulty posed by the loss of regularity, some suitable weight functions are introduced in our stability and convergence analysis for the natural convection equations. Finally, some numerical results are presented to verify the established theoretical results.

Numerical methods for time-fractional evolution equations with nonsmooth data: A concise overview

Over the past few decades, there has been substantial interest in evolution equations that involve a fractional-order derivative of order α∈(0,1) in time, commonly known as subdiffusion, due to their many successful applications in engineering, physics, biology and finance. Thus, it is of paramount importance to develop and to analyze efficient and accurate numerical methods for reliably simulating such models, and the literature on the topic is vast and fast growing. The present paper gives a concise overview on numerical schemes for the subdiffusion model with nonsmooth problem data, which are important for the numerical analysis of many problems arising in optimal control, inverse problems and stochastic analysis. We focus on the following topics of the subdiffusion model: regularity theory, Galerkin finite element discretization in space, time-stepping schemes (including convolution quadrature and L1 type schemes), and space–time variational formulations, and compare the results with that for standard parabolic problems. Further, these aspects are showcased with illustrative numerical experiments and complemented with perspectives and pointers to relevant literature.

A numerical treatment for the system of singularly perturbed convection-reaction-diffusion two point boundary value problems with non-smooth data

Two point Singularly Perturbed Weakly Coupled System of Convection-Reaction-Diffusion Problems(WCSCRDPs), having small parameters (ε1, ε2) multiplying the highest derivative (diffusion) and the next highest derivative (convection) term with discontinuity over the source term is studied. Generally, for adequately small values of the perturbation parameters, the exact solution involves boundary and interior layers. Hence, the analysis of the problem splits into two cases, in accordance with the ratio of the convection and diffusion parameters. The continuous bounds of the solution and its derivatives are derived. A numerical method which is parameter uniform is constructed, on a suitably fitted piecewise uniform Shishkin mesh. The error estimation analysis is discussed and the numerical method applied here exhibits almost first order convergence. Numerical examples and the corresponding tables and figures demonstrate the efficiency of the numerical scheme.

Two-level value function approach to non-smooth optimistic and pessimistic bilevel programs

ABSTRACTThe authors' paper in Dempe et al. [Necessary optimality conditions in pessimistic bilevel programming. Optimization. 2014;63:505–533], was the first one to provide detailed optimality conditions for pessimistic bilevel optimization. The results there were based on the concept of the two-level optimal value function introduced and analysed in Dempe et al. [Sensitivity analysis for two-level value functions with applications to bilevel programming. SIAM J. Optim. 22 (2012), 1309–1343], for the case of optimistic bilevel programs. One of the basic assumptions in both of these papers is that the functions involved in the problems are at least continuously differentiable. Motivated by the fact that many real-world applications of optimization involve functions that are non-differentiable at some points of their domain, the main goal of the current paper is to extend the two-level value function approach by deriving new necessary optimality conditions for both optimistic and pessimistic versions in bilevel programming with non-smooth data.

Galerkin FEM for a time-fractional Oldroyd-B fluid problem

We consider the numerical approximation of a generalized fractional Oldroyd-B fluid problem involving two Riemann-Liouville fractional derivatives in time. We establish regularity results for the exact solution which play an important role in the error analysis. A semidiscrete scheme based on the piecewise linear Galerkin finite element method in space is analyzed, and optimal with respect to the data regularity error estimates are established. Further, two fully discrete schemes based on convolution quadrature in time generated by the backward Euler and the second-order backward difference methods are investigated and related error estimates for smooth and nonsmooth data are derived. Numerical experiments are performed with different values of the problem parameters to illustrate the efficiency of the method and confirm the theoretical results.

Second-order predictor-corrector schemes for nonlinear distributed-order space-fractional differential equations with non-smooth initial data

ABSTRACTWe propose second-order linearly implicit predictor-corrector schemes for diffusion and reaction-diffusion equations of distributed-order. For diffusion equations of distributed order, we propose an analytical solution based on the spectral representation of the fractional Laplacian. Numerically, we approximate the integral term of the equation by the midpoint quadrature rule to obtain a multi-term space-fractional differential equation. The matrix transfer technique is used for spatial discretization of the resulting differential equation and methods based on Padé approximations to the exponential function are used in time. In particular, we discuss the (0,2)- and (1,1)-Padé approximations to the exponential function. The method based on the (1,1)-Padé approximation to the exponential function are seen to produce oscillations for some time steps and we propose a constraint on the choice of the time step to avoid these unwanted oscillations. Stability and convergence of the schemes are discussed. Numerical experiments are performed to support our theoretical observations.

Sequence-based necessary second-order optimality conditions for semilinear elliptic optimal control problems with nonsmooth data

The main goal of this note is to formulate sequence-based necessary second-order optimality conditions for a semilinear elliptic optimal control problem, with a pointwise pure state constraint and a pointwise mixed state-control constraint, governed by nonsmooth functions. Using a modification of the Dubovitskii–Milyutin scheme, we develop necessary second-order conditions for optimality, which depend on given sequences, in terms of sequential tangent sets and sequential directional derivatives for not necessarily twice differentiable mappings.