Nonsmooth Data Research Articles

Two Time-Stepping Schemes for Sub-Diffusion Equations with Singular Source Terms

Singular source terms in sub-diffusion equations may lead to the unboundedness of solutions, which will bring a severe reduction of convergence order of existing time-stepping schemes. In this work, we propose two efficient time-stepping schemes for solving sub-diffusion equations with a class of source terms mildly singular in time. One discretization is based on the Gr{\"u}nwald-Letnikov and backward Euler methods. First-order error estimate with respect to time is rigorously established for singular source terms and nonsmooth initial data. The other scheme derived from the second-order backward differentiation formula (BDF) is proved to possess second-order accuracy in time. Further, piecewise linear finite element and lumped mass finite element discretizations in space are applied and analyzed rigorously. Numerical investigations confirm our theoretical results.

A new groundwater depth prediction model based on EMD-LSTM

Abstract Groundwater resources play an important role in life, but the lack of effective management of groundwater resources has led to a regional decline in groundwater levels. How to curb the over-exploitation of groundwater and realize the sustainable use of groundwater resources has become an important issue for ecological environmental protection. Therefore, accurate prediction of groundwater depth is an important foundation for the rational use of groundwater resources. Based on the significant advantages of empirical model decomposition (EMD) in dealing with non-smooth and nonlinear data and the long-term memory function of long and short-term memory (LSTM) network, a coupled EMD-LSTM-based groundwater prediction model is constructed and apply to groundwater depth prediction of three professional observation wells. The results show that the maximum relative error of the prediction results of the EMD-LSTM model for the three professional observation wells is 5.00% and the minimum is 0.07%. The prediction passing rate is 100% and the prediction accuracy of the coupled model for groundwater depth is higher than that of the single LSTM model and back-propagation (BP) model. In conclusion, the model has high prediction accuracy and provides an effective method for the prediction of groundwater depth. HIGHLIGHTS Empirical mode decomposition (EMD) is a relatively novel data preprocessing method that can effectively reduce the non-smoothness of time series. LSTM is considered to be superior in terms of learning rate and generalization ability. The EMD-LSTM coupled model has better nonlinear and complex process learning ability in groundwater forecasting.

A fast difference scheme on a graded mesh for time-fractional and space distributed-order diffusion equation with nonsmooth data

The time-fractional and space distributed-order diffusion equation with a singularity at the initial time is investigated in this paper. For the solution of this equation, we propose a new difference scheme on a graded mesh. The theoretical underpinning for the suggested procedure is presented, which includes the stability and convergence of the difference scheme. Furthermore, the difference strategy proves to be unconditionally stable in this research. Additionally, we demonstrate that the difference technique is convergent and that the temporal convergence rate is faster than when a uniform mesh uses. Finally, a test example is provided to ensure that the theoretical analysis is accurate.

Some approximation results for mild solutions of stochastic fractional order evolution equations driven by Gaussian noise

We investigate the quality of space approximation of a class of stochastic integral equations of convolution type with Gaussian noise. Such equations arise, for example, when considering mild solutions of stochastic fractional order partial differential equations but also when considering mild solutions of classical stochastic partial differential equations. The key requirement for the equations is a smoothing property of the deterministic evolution operator which is typical in parabolic type problems. We show that if one has access to nonsmooth data estimates for the deterministic error operator together with its derivative of a space discretization procedure, then one obtains error estimates in pathwise Hölder norms with rates that can be read off the deterministic error rates. We illustrate the main result by considering a class of stochastic fractional order partial differential equations and space approximations performed by spectral Galerkin methods and finite elements. We also improve an existing result on the stochastic heat equation.

Weak convergence of the L1 scheme for a stochastic subdiffusion problem driven by fractionally integrated additive noise

The weak convergence of a fully discrete scheme for approximating a stochastic subdiffusion problem driven by fractionally integrated additive noise is studied. The Caputo fractional derivative is approximated by the L1 scheme and the Riemann-Liouville fractional integral is approximated with the first order convolution quadrature formula. The noise is discretized by using the Euler method and the spatial derivative is approximated with the linear finite element method. Based on the nonsmooth data error estimates of the corresponding deterministic problem, the weak convergence orders of the fully discrete schemes for approximating the stochastic subdiffusion problem driven by fractionally integrated additive noise are proved by using the Kolmogorov equation approach. Numerical experiments are given to show that the numerical results are consistent with the theoretical results.

Error estimation of a discontinuous Galerkin method for time fractional subdiffusion problems with nonsmooth data

This paper is devoted to the numerical analysis of a piecewise constant discontinuous Galerkin method for time fractional subdiffusion problems. The regularity of weak solution is firstly established by using variational approach and Mittag-Leffler function. Then several optimal error estimates are derived with low regularity data. Finally, numerical experiments are conducted to verify the theoretical results.

Fully decoupled, linear and unconditional stability implicit/explicit scheme for the natural convection problem

The aim of this paper is to develop an unconditionally stable Euler implicit/explicit scheme for the natural convection equations based on an introduced exponential scalar auxiliary variable. Our numerical scheme involves treating the linear terms in the implicit scheme and nonlinear terms in the explicit scheme. Then, the considered problem is decoupled into a series of linearized subproblems, and these algebraic systems with constant coefficient matrixes can be solved efficiently. Compared with the existing theoretical findings of the implicit/explicit scheme for the incompressible flow [He YN. The Euler implicit/explicit scheme for the 2D time-dependent Navier-Stokes equations with smooth or nonsmooth initial data. Math Comput. 2008;77(264):2097–2124.; He YN, Huang PZ, Feng XL. H2-stability of first order fully discrete schemes for the time-dependent Navier-Stokes equations. J Sci Comput. 2015;62:230–264; He YN, Li J. A penalty finite element method based on the Euler implicit/explicit scheme for the time-dependent Navier-Stokes equations. J Comput Appl Math. 2010;235(3):708–725; Jin JJ, Zhang T, Li J. H2 stability of the first order Galerkin method for the Boussinesq equations with smooth and nonsmooth initial data. Comput Math Appl. 2018;75(1):248–288; Su J, He YN. The almost unconditional convergence of the Euler implicit/explicit scheme for the three dimensional nonstationary Navier-Stokes equations. Discrete Continuous Dyn Syst Ser B. 2017;22(9):3421–3438], our analysis removes the restriction on time step , and obtains the unconditional stability and convergence theoretical findings. Some numerical results are also provided to verify the established theoretical analysis and show the performances of the considered numerical scheme.

Numerical Valuation of European and American Options under Fractional Black-Scholes Model

In this paper, we investigate the numerical valuation of European and American options under the time fractional Black-Scholes model. We first apply a coordinate stretching transformation to the asset price so that the spatial region can focus on the vicinity of singularities, which are usually found in the payoff function. The radial basis function finite difference method is used for the spatial discretization, and the improved L1 method is used to deal with the reduced order of convergence for the nonsmooth initial data. We use the operator splitting method for solving the linear complementary problem of American options. The proposed scheme leads to a sparse linear system which is trivial to solve. Moreover, the stability of the proposed numerical scheme is analyzed using Fourier analysis. Numerical experiments demonstrate the accuracy and efficiency of the proposed method.

Asymptotic regularity for Lipschitzian nonlinear optimization problems with applications to complementarity constrained and bilevel programming

Asymptotic stationarity and regularity conditions turned out to be quite useful to study the qualitative properties of numerical solution methods for standard nonlinear and complementarity constrained programs. In this paper, we first extend these notions to nonlinear optimization problems with nonsmooth but Lipschitzian data functions in order to find reasonable notions of asymptotic stationarity and regularity in terms of Clarke's and Mordukhovich's subdifferential construction. Particularly, we compare the associated novel asymptotic constraint qualifications with already existing ones. The second part of the paper presents two applications of the obtained theory. On the one hand, we specify our findings for complementarity constrained optimization problems and recover recent results from the literature which demonstrates the power of the approach. Furthermore, we hint at potential extensions to or- and vanishing constrained optimization. On the other hand, we demonstrate the usefulness of asymptotic regularity in the context of bilevel optimization. More precisely, we justify a well-known stationarity system for affinely constrained bilevel optimization problems in a novel way. Afterwards, we suggest a solution algorithm for this class of bilevel optimization problems which combines a penalty method with ideas from DC-programming. After a brief convergence analysis, we present the results of some numerical experiments.

Finite Element Penalty Method for the Oldroyd Model of Order One with Non-smooth Initial Data

Abstract In this paper, a penalty formulation is proposed and analyzed in both continuous and finite element setups, for the two-dimensional Oldroyd model of order one, when the initial velocity is in 𝐇 0 1 {\mathbf{H}_{0}^{1}} . New regularity results which are valid uniformly in time as t → ∞ {t\to\infty} and in the penalty parameter ε as ε → 0 {\varepsilon\to 0} are derived for the solution of the penalized problem. Then, based on conforming finite elements to discretize the spatial variables and keeping temporal variable continuous, a semidiscrete problem is discussed and a uniform-in-time a priori bound of the discrete velocity in Dirichlet norm is derived with the help of a penalized discrete Stokes operator and a modified uniform Gronwall’s lemma. Further, optimal error estimates for the penalized velocity in 𝐋 2 {\mathbf{L}^{2}} as well in 𝐇 1 {\mathbf{H}^{1}} -norms and for the penalized pressure in L 2 {L^{2}} -norm have been established for the semidiscrete problem with non-smooth data. These error estimates hold uniformly in time under uniqueness assumption and also in the penalty parameter as it goes to zero. Our analysis relies on the suitable use of the inverse of the penalized Stokes operator, penalized Stokes–Volterra projection and judicious application of weighted time estimates with positivity property of the memory term. Finally, several numerical experiments are conducted on benchmark problems which confirm our theoretical findings.

The virtual element method for the time fractional convection diffusion reaction equation with non-smooth data

In this paper, we develop an efficient virtual element method to solve the two-dimension time fractional convection diffusion reaction equation involving the Caputo fractional derivative with non-smooth solutions in the time direction. The equation exhibits a weak singularity near the initial time. We first use the exponential transformation to eliminate the convection terms, and obtain the time fractional diffusion reaction equation. By utilizing properties of the energy projection operator and continuous Grönwall inequality, we derive estimates of the approximation error in the L2-norm and H1-norm under semi-discrete on polygonal meshes. Based on the L1 scheme on graded mesh in time, the fully-discrete scheme is proved to be unconditionally stable and the optimal convergence results are derived with regards to the L2-norm and H1-norm. Finally, some numerical experiments are implemented to verify the theoretical results.

Spatial-Temporal Convolutional Transformer Network for Multivariate Time Series Forecasting.

Multivariate time series forecasting has long been a research hotspot because of its wide range of application scenarios. However, the dynamics and multiple patterns of spatiotemporal dependencies make this problem challenging. Most existing methods suffer from two major shortcomings: (1) They ignore the local context semantics when modeling temporal dependencies. (2) They lack the ability to capture the spatial dependencies of multiple patterns. To tackle such issues, we propose a novel Transformer-based model for multivariate time series forecasting, called the spatial–temporal convolutional Transformer network (STCTN). STCTN mainly consists of two novel attention mechanisms to respectively model temporal and spatial dependencies. Local-range convolutional attention mechanism is proposed in STCTN to simultaneously focus on both global and local context temporal dependencies at the sequence level, which addresses the first shortcoming. Group-range convolutional attention mechanism is designed to model multiple spatial dependency patterns at graph level, as well as reduce the computation and memory complexity, which addresses the second shortcoming. Continuous positional encoding is proposed to link the historical observations and predicted future values in positional encoding, which also improves the forecasting performance. Extensive experiments on six real-world datasets show that the proposed STCTN outperforms the start-of-the-art methods and is more robust to nonsmooth time series data.

Evolution of interface singularities in shallow water equations with variable bottom topography

AbstractWave front propagation with nontrivial bottom topography is studied within the formalism of hyperbolic long wave models. Evolution of nonsmooth initial data is examined, and, in particular, the splitting of singular points and their short time behavior is described. In the opposite limit of longer times, the local analysis of wave fronts is used to estimate the gradient catastrophe formation and how this is influenced by the topography. The limiting cases when the free surface intersects the bottom boundary, belonging to the so‐called “physical” and “nonphysical” vacuum classes, are examined. Solutions expressed by power series in the spatial variable lead to a hierarchy of ordinary differential equations for the time‐dependent series coefficients, which are shown to reveal basic differences between the two vacuum cases: for nonphysical vacuums, the equations of the hierarchy are recursive and linear past the first two pairs, whereas for physical vacuums, the hierarchy is nonrecursive, fully coupled, and nonlinear. The former case may admit solutions that are free of singularities for nonzero time intervals, whereas the latter is shown to develop nonstandard velocity shocks instantaneously. Polynomial bottom topographies simplify the hierarchy, as they contribute only a finite number of inhomogeneous forcing terms to the equations in the recursion relations. However, we show that truncation to finite‐dimensional systems and polynomial solutions is in general only possible for the case of a quadratic bottom profile. In this case, the system's evolution can reduce to, and is completely described by, a low‐dimensional dynamical system for the time‐dependent coefficients. This system encapsulates all the nonlinear properties of the solution for general power series initial data, and, in particular, governs the loss of regularity in finite times at the dry point. For the special case of parabolic bottom topographies, an exact, self‐similar solution class is introduced and studied to illustrate via closed‐form expressions the general results.

Data assimilation in 2D viscous Burgers equation using a stabilized explicit finite difference scheme run backward in time

The 2D viscous Burgers equation is a system of two nonlinear equations in two unknowns, . This paper considers the data assimilation problem of finding initial values that can evolve into a close approximation to a desired target result , at some realistic T>0. Highly nonsmooth target data are considered, that may not correspond to actual solutions at time T. Such an ill-posed 2D viscous Burgers problem has not previously been studied. An effective approach is discussed and demonstrated based on recently developed stabilized explicit finite difference schemes that can be run backward in time. Successful data assimilation experiments are presented involving 8 bit, pixel grey-scale images, defined by nondifferentiable intensity data. An instructive example of failure is also included.

Study of BP neural network based on improved particle swarm algorithm

This paper uses first-order difference to transform non-smooth data into smooth time series data, determines the p and q parameters in the model by judging the trailing and truncated nature of ACF, PACF, and finally establishes the ARIMA model after ACI, BCI detection. According to the parameters of the neural network randomly selected similar to the initial spatial position of the particles in the particle swarm algorithm, the improved particle swarm algorithm is used instead of the gradient correction method to precisely adjust the parameters and establish the BP neural network, which improves the robustness and accuracy of the prediction model.

Backward Euler method for the equations of motion arising in Oldroyd model of order one with nonsmooth initial data

Abstract In this paper, a backward Euler method combined with finite element discretization in spatial direction is discussed for the equations of motion arising in the two-dimensional Oldroyd model of viscoelastic fluids of order one with the forcing term independent of time or in $L^{\infty }$ in time. It is shown that the estimates of the discrete solution in Dirichlet norm is bounded uniformly in time. Optimal a priori error estimate in $\textbf {L}^2$-norm is derived for the discrete problem with nonsmooth initial data. This estimate is shown to be uniform in time, under the assumption of uniqueness condition. Finally, we present some numerical results to validate our theoretical results.

Efficient Temporal Third/Fourth-Order Finite Element Method for a Time-Fractional Mobile/Immobile Transport Equation with Smooth and Nonsmooth Data.

In recent years, the numerical theory of fractional models has received more and more attention from researchers, due to the broad and important applications in materials and mechanics, anomalous diffusion processes and other physical phenomena. In this paper, we propose two efficient finite element schemes based on convolution quadrature for solving the time-fractional mobile/immobile transport equation with the smooth and nonsmooth data. In order to deal with the weak singularity of solution near , we choose suitable corrections for the derived schemes to restore the third/fourth-order accuracy in time. Error estimates of the two fully discrete schemes are presented with respect to data regularity. Numerical examples are given to illustrate the effectiveness of the schemes.

Spectrally accurate option pricing under the time fractional Black-Scholes model

We propose a Legendre–Laguerre spectral approximation to price the European and double barrier options in the time-fractional framework. By choosing an appropriate basis function, the spectral discretization is used for the approximation of the spatial derivatives of the time-fractional Black–Scholes equation. For the time discretization, we consider the popular \(L1\) finite difference approximation, which converges with order \(\mathcal{O}((\Delta \tau)^{2-\alpha})\) for functions which are twice continuously differentiable. However, when using the \(L1\) scheme for problems with nonsmooth initial data, only the first-order accuracy in time is achieved. This low-order accuracy is also observed when solving the time-fractional Black–Scholes European and barrier option pricing problems for which the payoffs are all nonsmooth. To increase the temporal convergence rate, we therefore consider a Richardson extrapolation method, which when combined with the spectral approximation in space, exhibits higher order convergence such that high accuracies over the whole discretization grid are obtained. Compared with the traditional finite difference scheme, numerical examples clearly indicate that the spectral approximation converges exponentially over a small number of grid points. Also, as demonstrated, such high accuracies can be achieved in much fewer time steps using the extrapolation approach. doi:10.1017/S1446181121000286

The uniform [formula omitted] long-time behavior of time discretization for time-fractional partial differential equations with nonsmooth data

The goal of this paper is to prove the theory that the L1 scheme for solving time fractional partial differential equations with nonsmooth data has the uniform l1 optimal order error estimate. For the L1 scheme combined with the first-order convolution quadrature scheme, by using Laplace transform rules we obtain the uniform l1 long time convergence of the L1 scheme for smooth and nonsmooth initial data with of Lubich with first-order accuracy in the homogeneous case. In earlier work, various authors studied the convergence properties of the L1 scheme for smooth and nonsmooth initial data in both the homogeneous and inhomogeneous cases. However, their convergence does not apply to the uniform l1 long time convergence behavior.

On nonlinear singular BVPs with nonsmooth data. Part 2: Convergence of collocation methods

We discuss numerical solution of boundary value problems for systems of nonlinear ordinary differential equations with a time singularity,x′(t)=M(t)tx(t)+f(t,x(t))t,t∈(0,1],b(x(0),x(1))=0, where M:[0,1]→Rn×n and f:[0,1]×Rn→Rn are continuous matrix-valued and vector-valued functions, respectively. Moreover, b:Rn×Rn→Rn is a continuous nonlinear mapping which is specified according to a spectrum of the matrix M(0) to guarantee the BVP to be well-posed. For the case where M(0) has eigenvalues with nonzero real parts, we prove new convergence results for the collocation method and analytical results about the necessary smoothness of the solution to the problem required in the numerical analysis.We illustrate the theory by means of numerical examples.