Unconditional Error Research Articles

Unconditional error analysis of the linearized transformed [formula omitted] virtual element method for nonlinear coupled time-fractional Schrödinger equations

This paper constructs a linearized transformed L1 virtual element method for the generalized nonlinear coupled time-fractional Schrödinger equations. The solutions to such problems typically exhibit singular behavior at the beginning. To avoid this pitfall, we introduce an identical s-fractional differential system derived from a smoothing transformation of variables t=s1/α, 0<α<1. By utilizing the discrete complementary convolution kernels, we prove the boundedness and error estimates of the solution of time-discrete system. Moreover, the unconditionally optimal error bounds of the proposed fully discrete scheme are derived in L2-norm without restriction on the grid ratio. Finally, numerical tests on a set of polygonal meshes are presented to verify the theoretical results.

A new linearized second-order energy-stable finite element scheme for the nonlinear Benjamin-Bona-Mahony-Burgers equation

In this paper, a novel linearized second-order energy-stable fully discrete scheme for the nonlinear Benjamin-Bona-Mahony-Burgers (BBMB) equation is introduced, along with a superconvergence analysis of the conforming finite element method (FEM). Through skillful decomposition of the nonlinear term u∇⋅u, a new linearized second-order fully discrete scheme is developed. This scheme requires fewer iterations and exhibits higher computational efficiency compared to the nonlinear approach. Furthermore, it maintains energy stability, enabling direct numerical solution boundedness in the H1-norm, which represents an improvement over the L∞-norm boundedness reported in previous literature. Secondly, relying on this boundedness and the high-precision results of the bilinear element, we derive the unconditional error estimates for superclose and global superconvergence without any restrictions on the time step size Δt and spatial mesh size h. Finally, numerical examples are presented to validate the theoretical analysis.

Unconditional Error Analysis of VEMs for a Generalized Nonlinear Schrödinger Equation

Unconditional Error Analysis of VEMs for a Generalized Nonlinear Schrödinger Equation

Error estimates for the finite element method of the Navier-Stokes-Poisson-Nernst-Planck equations

In this paper, we consider the linearized backward Euler finite element scheme for the Navier-Stokes-Poisson-Nernst-Planck (NSPNP) equations. We aim to derive the unconditional error estimates of all variables in l∞(L2) and l∞(H1) norms. Compared with some previous results, we remove the mesh restriction. In addition, we derive two important physical properties for the proposed numerical scheme. Finally, three numerical examples are implemented to verify the theoretical analysis.

Unconditional error estimates of linearized BDF2-Galerkin FEMs for nonlinear coupled Schrödinger-Helmholtz equations

Unconditional error estimates of linearized BDF2-Galerkin FEMs for nonlinear coupled Schrödinger-Helmholtz equations

Cluster Index Modulation for mmWave Communication Systems

In this study, a novel cluster index modulation (CIM) scheme, which is based on indexing the available clusters in the environment, is proposed for future mmWave communication systems. Exploiting the fact that the available clusters in the system are well separated in terms of their angular distribution, we selected the best path for each of them and then performed IM in an algorithmic manner to convey information bits. It is shown that by means of large antenna arrays and analog RF beamforming with the indexed clusters, the destructive effect of inter-beam/cluster interference can be remarkably mitigated. Also, we designed a hybrid beamforming architecture at the transmitter to further reduce the effect of residual inter-beam/cluster interference, where the analog RF beamformer is followed by a digital baseband precoder using the zero-forcing technique. Computer simulations reveal that the proposed scheme can provide better error performance than traditional mmWave communication, and the proposed hybrid architecture outperforms beam index modulation (BIM) for a point-to-point scenario. Semi-analytical derivations and closed-form unconditional pairwise error probability (UPEP) expressions are derived for both analog and hybrid architectures, which confirm the validity and superiority of our proposed scheme.

Unconditionally optimal H1-norm error estimates of a fast and linearized Galerkin method for nonlinear subdiffusion equations

In this paper, the unconditionally optimal H1-norm error estimate for a two-step fast L1 scheme is considered for a nonlinear subdiffusion equation, which involves the considerations of both the initial singularity of the solution and the computational complexity to evaluate the Caputo derivative. For the fully discrete scheme, we use the fast L1 scheme developed in [28] to speed up the evaluation of the Caputo derivative on nonuniform time step, employ the Galerkin finite element (FE) method to discretize the spatial direction, and take the Newton linearized method to approximate the nonlinear term. While the theory of the L2-norm error estimate is well studied, there are few works on the unconditional convergence of H1-norm error estimate of the linearized Galerkin FE scheme for nonlinear subdiffusion equations. The main reason is that it requires to find a suitable test function in the FE space. To this end, we here respectively take the time-discrete operator and Laplace operator as the suitable test function in the FE space, and combine an improved discrete fractional Grönwall inequality to achieve the unconditional error estimate in H1-norm. Numerical results are provided to demonstrate the theoretical results.

Unconditionally optimal error estimates of two linearized Galerkin FEMs for the two-dimensional nonlinear fractional Rayleigh–Stokes problem

In this paper, two linearized Galerkin finite element methods, which are based on the L1 approximation and the WSGD operator, respectively, are proposed to solve the nonlinear fractional Rayleigh-Stokes problem. In order to obtain the unconditionally optimal error estimate, we firstly introduce a time-discrete elliptic equation, and derive the unconditional error estimate between the exact solution and the solution of the time-discrete system in H2-norm. Secondly, we obtain the boundedness of the fully discrete finite element solution in L∞-norm through the more detailed study of the error equation. Then, the optimal L2-norm error estimate is derived for the fully discrete system without any restriction conditions on the time step size. Finally, some numerical experiments are presented to confirm the theoretical results, showing that the two linearized schemes given in this paper are efficient and reliable.

Implicit MAC scheme for compressible Navier–Stokes equations: low Mach asymptotic error estimates

AbstractWe investigate the error between any discrete solution of the implicit marker-and-cell (MAC) numerical scheme for compressible Navier–Stokes equations in the low Mach number regime and an exact strong solution of the incompressible Navier–Stokes equations. The main tool is the relative energy method suggested on the continuous level in Feireisl et al. (2012, Relative entropies, suitable weak solutions, and weak–strong uniqueness for the compressible Navier–Stokes system. J. Math. Fluid Mech., 14, 717–730). Our approach highlights the fact that numerical and mathematical analyses are not two separate fields of mathematics. The result is achieved essentially by exploiting in detail the synergy of analytical and numerical methods. We get an unconditional error estimate in terms of explicitly determined positive powers of the space–time discretization parameters and Mach number in the case of well-prepared initial data and in terms of the boundedness of the error if the initial data are ill prepared. The multiplicative constant in the error estimate depends on a suitable norm of the strong solution but it is independent of the numerical solution itself (and of course, on the discretization parameters and the Mach number). This is the first proof that the MAC scheme is unconditionally and uniformly asymptotically stable in the low Mach number regime.

A consistent projection finite element method for the incompressible MHD equations

ABSTRACT In this paper, we introduce a consistent projection finite element method for the impressible MHD (MagnetoHydroDynamics) equations. It's a fully discrete projection method, which associates the virtues of both the gauge and Uzawa methods with a variational framework. We also testify the unconditional stability and error estimates of the velocity, pressure and magnetic field by the variational method using some assumptions. Finally, we present some numerical experiments to support the validity of the consistent projection finite element method.

Speed and Conditional Noise as Information for Illiquidity and Fixed Income Arbitrage

Previous studies suggest that trading conditions in the secondary market for nominal U.S. Treasury (UST) coupon securities embeds critical information about global financial market liquidity and the limits to arbitrage. We propose three new general measures based on deviations of observed nominal UST yields from their fitted values derived from standard estimates of the term structure. These metrics extend beyond unconditional fitting errors to include conditional mispricings net explicit estimates of premiums for liquidity and collateral value at the security-level. In addition, we consider not only the size of unconditional and conditional errors but also the speed at which arbitragers eliminate implied prospective gains. Based on daily CRSP data on individual UST securities from June 1961 through December 2013, broad inferences from these alternative measures contradict conclusions in previous studies that relate the size of UST fitting errors to overall global market liquidity. We also analyze market-neutral trading strategies that produce absolute returns that on average often exceed those on the S&P 500, with a small fraction of the variance, no covariance with stock returns, and very positive as opposed to negative skew.

A high-order split-step finite difference method for the system of the space fractional CNLS

In this paper, the schemes based on the high-order quasi-compact split-step finite difference methods are derived for the one- and two-dimensional coupled fractional Schrodinger equations. In order to improve the computing efficiency, we adopt the split-step method for handling the nonlinearity. By using a high-order quasi-compact scheme in space, the numerical method improves the accuracy effectively. We prove the conservation laws, prior boundedness and unconditional error estimates of the quasi-compact finite difference scheme for the linear problem. Moreover, for the nonlinear problem, we show that the quasi-compact split-step finite difference method can also keep the conservation law in the mass sense. For solving the multi-dimensional problem, we combine the quasi-compact split-step method with the alternating direction implicit technique. At last, numerical examples are performed to illustrate our theoretical results and show the efficiency of the proposed schemes.

Misreporting of Government Transfers: How Important Are Survey Design and Geography?

Recent studies linking household surveys to administrative records reveal high rates of misreporting of program receipt. We use the FoodAPS survey to examine whether the findings of these studies of general household surveys using one or two states generalize to a survey with a narrow focus and across many states. First, we study how reporting errors differ from other surveys. We find a lower rate of false negatives (failures to report true receipt) in FoodAPS, likely partly due to the shorter recall period of FoodAPS. Misreporting varies with household characteristics and between interviewers. Second, we examine geographic heterogeneity in survey error to assess whether we can extrapolate from linked data from a few states. We find systematic differences between states in unconditional error rates but no evidence of substantial differences conditional on common covariates. Thus, extrapolating error rates across states may yield more accurate receipt estimates than uncorrected survey estimates.

Misreporting of Government Transfers: How Important are Survey Design and Geography?

Recent studies linking household surveys to administrative records reveal high rates of misreporting of program receipt. We use the FoodAPS survey to examine whether the findings of these studies of general household surveys using one or two states generalize to a survey with a narrow focus and across many states. First, we study how reporting errors differ from other surveys. We find a lower rate of false negatives (failures to report true receipt) in FoodAPS, likely partly due to the shorter recall period of FoodAPS. Misreporting varies with household characteristics and between interviewers. Second, we examine geographic heterogeneity in survey error to assess whether we can extrapolate from linked data from a few states. We find systematic differences between states in unconditional error rates but no evidence of substantial differences conditional on common covariates. Thus, extrapolating error rates across states may yield more accurate receipt estimates than uncorrected survey estimates.

Remarks on energy methods for structure-preserving finite difference schemes – Small data global existence and unconditional error estimate

In the previous article (Yoshikawa, 2017), the author proposes the energy method for structure-preserving finite difference schemes, which enable us to show global existence and uniqueness of solution for the schemes and error estimates. In this article, we give two extended remarks of the methods. One is related to the small data global existence results for schemes of which energy is not necessarily bounded from below. The other is an unconditional error estimate which holds globally in time and without smallness condition for split sizes. These results can be shown due to the structure-preserving property.

Modulation Set Optimization for the Improved Complex Quadrature SM

At each channel use, the complex quadrature spatial modulation (CQSM) transmits two signal symbols drawn from two disjoint modulation sets. The indices of the antennas from which symbols are transmitted also carry information. In the improved CQSM (ICQSM), an additional antenna is used to transmit the second signal symbol only when the indices of the antennas to be used for transmission are equal. Conventionally, the second modulation set is a rotated version of the first, where the rotation angle is optimized such that the average unconditional error probability (AUP) is reduced. In this paper, we propose a low‐complexity method to design the PSK modulation sets based on reducing the AUP. After introducing min‐BER and max‐dmin, two exhaustive search methods, we analytically show that the AUP depends on Euclidean distance between transmitted vectors, which in turn depends on the power of signal symbols, the Euclidean distance between the symbols of each modulation set, and the Euclidean distance between the symbols of the two sets. The optimal rotation angle is analytically derived for any modulation order and the radii of the modulation sets are optimized such that AUP is reduced for a wide range of system configurations. The simulation results show that more than 3 dB of power gain is achieved in the case of 16PSK, where higher gains are achieved for higher modulation orders. These gains are achieved at no computational cost because the optimization does not depend on the channel realization.

Galerkin finite element method for the nonlinear fractional Ginzburg–Landau equation

In this paper, we are concerned with the numerical solution of the nonlinear fractional Ginzburg–Landau equation. Galerkin finite element method is used for the spatial discretization, and an implicit midpoint difference method is employed for the temporal discretization. The boundedness, existence and uniqueness of the numerical solution, and the unconditional error estimates in the L2-norm are investigated in details. To numerically solve the nonlinear system, linearized iterative algorithms are also considered. Finally, some numerical examples are presented to illustrate the effectiveness of the algorithm.

Unconditional error analysis of Galerkin FEMs for nonlinear fractional Schrödinger equation

In this paper, we propose a conservative linearized Crank–Nicolson Galerkin FEMs for the nonlinear fractional Schrödinger equation. We construct5 a time-discrete system, to which, the mass conservation, semi-discrete error estimates and the suitable regularity of the numerical solution are obtained. With the spatial direction discreted by FEMs, the fully discrete conservative linearized finite element scheme is presented. Moreover, by a new error splitting technique, an unconditional -norm error estimates are derived by the boundedness of the fully-discrete numerical solution in -norm. Finally, some numerical examples are given to confirm the theoretical results.

The Term Structure of Interest Rates in an Estimated New Keynesian Policy Model

We jointly estimate a New Keynesian Policy Model with a Gaussian affine no-arbitrage specification of the term structure of interest rates, and assess how important inflation, output and monetary policy shocks are as sources of fluctuations in interest rates and the term premium. To mitigate computational difficulties, we work with observable pricing factors only and utilize the computationally convenient normalization of Joslin et al. (2013b). This allows us to estimate the model without needing to restrict the parameters driving the market prices of risk. Using data for the U.S. from 1962:Q1 to 2014:Q2, we find that inflation and the output gap account for around 80% of the unconditional forecast error variance of bond yields at the short and medium end of the term structure, while monetary policy shocks account for around 20%. Our impulse response function analysis suggests that bond yields respond to macroeconomic shocks only gradually, peaking after about 4 quarters. This is due to sizable monetary policy inertia estimates in our model. At the peak of the response, inflation shocks increase bond yields by more than one-to-one, while output shocks do so by less than one-to-one, which is consistent with a Taylor type monetary policy rule. Our estimated time-varying term premium is strongly counter-cyclical. Moreover, we show that it can capture salient features of the term structure that constitute a puzzle in the expectations hypothesis, that is, LPY(i) and LPY(ii).

Error estimates for a numerical approximation to the compressible barotropic Navier–Stokes equations

We present here a general method based on the investigation of the relative energy of the system, that provides an unconditional error estimate for the approximate solution of the barotropic Navier Stokes equations obtained by time and space discretization. We use this methodology to derive an error estimate for a specific DG/finite element scheme for which the convergence was proved in [26].