Isogeometric Analysis on Implicit Domains:Approximation,Stability and Error Estimates

The gap between finite element analysis and computer-aided design derives the development of isogeometric analysis(IGA),which uses the same representation in the geometry and the analysis.However,the parameterization in IGA is non-trivial.Weighted extended B-splines(WEB)method replaces grid generation and parameterization with weight function construction(R-function or distance function).By using implicit spline representation,isogeometric analysis on implicit domains(IGAID)adopts the merits of the“isoparametric”in IGA and“weight function generation”in WEB.But the theoretical properties have not been fully studied yet.In this paper,we study the theoretical aspects of IGAID using tensor-product B-splines.Both the approximation and stability properties of IGAID are considered.By setting appropriate constraints on the weight function,we can derive the optimal approximation order and stability.Numerical examples show the effectiveness of the approach and validate the theoretical results.

Improvement of photogrammetric joint roughness coefficient value by integrating automatic shooting parameter selection and composite error model

In order to improve the accuracy of the photogrammetric joint roughness coefficient(JRC)value,the present study proposed a novel method combining an autonomous shooting parameter selection algorithm with a composite error model.Firstly,according to the depth map-based photogrammetric theory,the estimation of JRC from a three-dimensional(3D)digital surface model of rock discontinuities was presented.Secondly,an automatic shooting parameter selection algorithm was novelly proposed to establish the 3D model dataset of rock discontinuities with varying shooting parameters and target sizes.Meanwhile,the photogrammetric tests were performed with custom-built equipment capable of adjusting baseline lengths,and a total of 36 sets of JRC data was gathered via a combination of laboratory and field tests.Then,by combining the theory of point cloud coordinate computation error with the equation of JRC calculation,a composite error model controlled by the shooting parameters was proposed.This newly proposed model was validated via the 3D model dataset,demonstrating the capability to correct initially obtained JRC values solely based on shooting parameters.Furthermore,the implementation of this correction can significantly reduce errors in JRC values obtained via photographic measurement.Subsequently,our proposed error model was integrated into the shooting parameter selection algorithm,thus improving the rationality and convenience of selecting suitable shooting parameter combinations when dealing with target rock masses with different sizes.Moreover,the optimal combination of three shooting parameters was offered.JRC values resulting from various combinations of shooting parameters were verified by comparing them with 3D laser scan data.Finally,the application scope and limitations of the newly proposed approach were further addressed.

Least Square Estimation for Multiple Functional Linear Model with Autoregressive Errors

As an extension of linear regression in functional data analysis,functional linear regression has been studied by many researchers and applied in various fields.However,in many cases,data is collected sequentially over time,for example the financial series,so it is necessary to consider the autocorrelated structure of errors in functional regression background.To this end,this paper considers a multiple functional linear model with autoregressive errors.Based on the functional principal component analysis,we apply the least square procedure to estimate the functional coeficients and autoregression coeficients.Under some regular conditions,we establish the asymptotic properties of the proposed estimators.A simulation study is conducted to investigate the finite sample performance of our estimators.A real example on China's weather data is applied to illustrate the validity of our model.

Risk factors for biometry prediction error by Barrett Universal II intraocular lens formula in Chinese patients

AIM:To investigate the influence of postoperative intraocular lens(IOL)positions on the accuracy of cataract surgery and examine the predictive factors of postoperative biometry prediction errors using the Barrett Universal II(BUII)IOL formula for calculation.METHODS:The prospective study included patients who had undergone cataract surgery performed by a single surgeon from June 2020 to April 2022.The collected data included the best-corrected visual acuity(BCVA),corneal curvature,preoperative and postoperative central anterior chamber depths(ACD),axial length(AXL),IOL power,and refractive error.BUII formula was used to calculate the IOL power.The mean absolute error(MAE)was calculated,and all the participants were divided into two groups accordingly.Independent t-tests were applied to compare the variables between groups.Logistic regression analysis was used to analyze the influence of age,AXL,corneal curvature,and preoperative and postoperative ACD on MAE.RESULTS:A total of 261 patients were enrolled.The 243(93.1%)and 18(6.9%)had postoperative MAE<1 and>1 D,respectively.The number of females was higher in patients with MAE>1 D(χ^(2)=3.833,P=0.039).The postoperative BCVA(logMAR)of patients with MAE>1 D was significantly worse(t=-2.448;P=0.025).After adjusting for gender in the logistic model,the risk of postoperative refractive errors was higher in patients with a shallow postoperative anterior chamber[odds ratio=0.346;95% confidence interval(CI):0.164,0.730,P=0.005].CONCLUSION:Risk factors for biometry prediction error after cataract surgery include the patient’s sex and postoperative ACD.Patients with a shallow postoperative anterior chamber are prone to have refractive errors.

Topology Data Analysis-Based Error Detection for Semantic Image Transmission with Incremental Knowledge-Based HARQ

Semantic communication(SemCom)aims to achieve high-fidelity information delivery under low communication consumption by only guaranteeing semantic accuracy.Nevertheless,semantic communication still suffers from unexpected channel volatility and thus developing a re-transmission mechanism(e.g.,hybrid automatic repeat request[HARQ])becomes indispensable.In that regard,instead of discarding previously transmitted information,the incremental knowledge-based HARQ(IK-HARQ)is deemed as a more effective mechanism that could sufficiently utilize the information semantics.However,considering the possible existence of semantic ambiguity in image transmission,a simple bit-level cyclic redundancy check(CRC)might compromise the performance of IK-HARQ.Therefore,there emerges a strong incentive to revolutionize the CRC mechanism,thus more effectively reaping the benefits of both SemCom and HARQ.In this paper,built on top of swin transformer-based joint source-channel coding(JSCC)and IK-HARQ,we propose a semantic image transmission framework SC-TDA-HARQ.In particular,different from the conventional CRC,we introduce a topological data analysis(TDA)-based error detection method,which capably digs out the inner topological and geometric information of images,to capture semantic information and determine the necessity for re-transmission.Extensive numerical results validate the effectiveness and efficiency of the proposed SC-TDA-HARQ framework,especially under the limited bandwidth condition,and manifest the superiority of TDA-based error detection method in image transmission.

The Exact Limits and Improved Decay Estimates for All Order Derivatives of the Global Weak Solutions of n-Dimensional Incompressible Navier-Stokes Equations

We couple together existing ideas,existing results,special structure and novel ideas to accomplish the exact limits and improved decay estimates with sharp rates for all order derivatives of the global weak solutions of the Cauchy problem for an n-dimensional incompressible Navier-Stokes equations.We also use the global smooth solution of the corresponding heat equation to approximate the global weak solutions of the incompressible Navier-Stokes equations.

Error estimates of H^1-Galerkin mixed finite element method for Schrdinger equation

An H^1-Galerkin mixed finite element method is discussed for a class of second order SchrSdinger equation. Optimal error estimates of semidiscrete schemes are derived for problems in one space dimension. At the same time, optimal error estimates are derived for fully discrete schemes. And it is showed that the H1-Galerkin mixed finite element approximations have the same rate of convergence as in the classical mixed finite element methods without requiring the LBB consistency condition.

A Priori and A Posteriori Error Estimates of Streamline Diffusion Finite Element Method for Optimal Control Problem Governed by Convection Dominated Diffusion Equation

In this paper,we investigate a streamline diffusion finite element approxi- mation scheme for the constrained optimal control problem governed by linear con- vection dominated diffusion equations.We prove the existence and uniqueness of the discretized scheme.Then a priori and a posteriori error estimates are derived for the state,the co-state and the control.Three numerical examples are presented to illustrate our theoretical results.

Error Estimates for Mixed Finite Element Methods for Sobolev Equation

The purpose of this paper is to investigate the convergence of the mixed finite element method for the initial-boundary value problem for the Sobolev equation Ut-div{aut + b1 u} = f based on the Raviart-Thomas space Vh × Wh H(div; × L2(). Optimal order estimates are obtained for the approximation of u, ut, the associated velocity p and divp respectively in L(0,T;L2()), L(0,T;L2()), L(0,T;L2()2), and L2(0, T; L2()). Quasi-optimal order estimates are obtained for the approximations of u, ut in L(0, T; L()) and p in L(0,T; L()2).

Adaptive Finite Element Method Based on Optimal Error Estimates for Linear Elliptic Problems on Nonconvex Polygonal Domains

The subject of this work is to propose adaptive finite element methods based on an optimal maximum norm error control estimate.Using estimators of the local regularity of the unknown exact solution derived from computed approximate solutions,the proposed procedures are analyzed in detail for a non-trivial class of corner problems and shown to be efficient in the sense that they generate the correct type of refinement and lead to the desired control under consideration.

OPTIMAL ERROR ESTIMATES OF A DECOUPLED SCHEME BASED ON TWO-GRID FINITE ELEMENT FOR MIXED NAVIER-STOKES/DARCY MODEL

Although the two-grid finite element decoupled scheme for mixed Navier-Stokes/ Darcy model in literatures has given the numerical results of optimal convergence order, the theoretical analysis only obtain the optimal error order for the porous media flow and a non-optimal error order for the fluid flow. In this article, we give a more rigorous of the error analysis for the fluid flow and obtain the optimal error estimates of the velocity and the pressure.

A POSTERIORI ERROR ESTIMATES OF FINITEELEMENT METHOD FOR PARABOLIC PROBLEMS

This paper deals with a-posteriori error estimates for piecewise linear finite element approximations of parabolic problems in two space dimensions. The analysis extends previous results for elliptic problems to the parabolic context.

Error Estimates of H^1-Galerkin Mixed Methods for the Viscoelasticity Wave Equation

H1-Galerkin mixed methods are proposed for viscoelasticity wave equation.Depending on the physical quantities of interest,two methods are discussed.The optimal error estimates and the proof of the existence and uniqueness of semidiscrete solutions are derived for problems in one space dimension.And the methods don＇t require the LBB condition.

The Error Estimates of Direct Discontinuous Galerkin Methods Based on Upwind-Baised Fluxes

In this paper, we study the error estimates for direct discontinuous Galerkin methods based on the upwind-biased fluxes. We use a newly global projection to obtain the optimal error estimates. The numerical experiments imply that L2 norms error estimates can reach to order k + 1 by using time discretization methods.

A PRIORI L_2 ERROR ESTIMATES FOR A NONLINEAR PARABOLIC SYSTEM BASED ON COMBINING THE METHOD OF CHARACTERISTICS WITH MIXED FINITE ELEMENT PROCEDURE

A nonlinear parabolic system is derived to describe incompressible nuclear waste-disposal contamination in porous media. A sequential implicit tirne-stepping is defined, in which the pressure and Darcy velocity of the mixture are approximated simultaneously by a mixed finite element method and the brine, radionuclid and heat are treated by a combination of a Galerkin finite element method and the method of characteristics. Optimal-order convergence in L2 is proved. Time-truncation errors of standard procedures are reduced by time stepping along the characteristics of the hyperbolic part of the brine, radionuclide and heal equalios, temporal and spatial error are lossened by direct compulation of the velocity in the mixed method, as opposed to differentiation of the pressure.

ADAPTIVE FINITE ELEMENT METHOD BASED ON OPTIMAL ERROR ESTIMATES FOR LINEAR ELLIPTIC PROBLEMS ON CONCAVE CORNER DOMAINS

An adaptive finite element procedure designed for specific computational goals is presented,using mesh refinement strategies based on optimal or nearly optimal a priori error estimates for the finite element method and using estimators of the local regularity of the unknown exact solution derived from computed approximate solutions.The proposed procedure is analyzed in detail for a non-trivial class of corner problems and shown to be efficient in the sense that the method can generate the correct type of refinements and lead to the desired control under consideration.

AN ASYMPTOTIC BEHAVIOR AND A POSTERIORI ERROR ESTIMATES FOR THE GENERALIZED SCHWARTZ METHOD OF ADVECTION-DIFFUSION EQUATION

In this paper, a posteriori error estimates for the generalized Schwartz method with Dirichlet boundary conditions on the interfaces for advection-diffusion equation with second order boundary value problems are proved by using the Euler time scheme combined with Galerkin spatial method. Furthermore, an asymptotic behavior in Sobolev norm is de- duced using Benssoussau-Lions＇ algorithm. Finally, the results of some numerical experiments are presented to support the theory.

Optimal error estimates for Fourier spectral approximation of the generalized KdV equation

A Fourier spectral method for the generalized Korteweg-de Vries equation with periodic boundary conditions is analyzed, and a corresponding optimal error estimate in L^2-norm is obtained. It improves the result presented by Maday and Quarteroni. A modified Fourier pseudospectral method is also presented, with the same convergence properties as the Fourier spectral method.

LOCAL ERROR ESTIMATES FOR METHODS OF CHARACTERISTICS INCORPORATING STREAMLINE DIFFUSION

Allen and Liu (1995) introduced a new method for a time-dependent convection dominated diffusion problem, which combines the modified method of characteristics and method of streamline diffusion. But they ignored the fact that the accuracy of time discretization decays at half an order when the characteristic line goes out of the domain. In present paper, the author shows that, as a remedy, a simple lumped scheme yields a full accuracy approximation. Forthermore, some local error bounds independent of the small viscosity axe derived for this scheme outside the boundary layers.

A COUPLING METHOD OF DIFFERENCE WITH HIGH ORDER ACCURACY AND BOUNDARY INTEGRAL EQUATION FOR EVOLUTIONARY EQUATION AND ITS ERROR ESTIMATES

In the present paper, a new numerical method for solving initial-boundary value problems of evolutionary equations is proposed and studied, combining difference method with high accuracy with boundary integral equation method. The numerical approximate schemes for both problems on a bounded or unbounded domain in R3 are proposed and their prior error estimates are obtained.