Global Error Analysis Research Articles

Global error analysis of discontinuous Galerkin methods for systems of boundary value problems

Global error analysis of discontinuous Galerkin methods for systems of boundary value problems

Design and evaluation of an intelligent reduction robot system for the minimally invasive reduction in pelvic fractures

IntroductionPelvic fracture is a severe high-energy injury with the highest disability and mortality of all fractures. Traditional open surgery is associated with extensive soft tissue damages and many complications. Minimally invasive surgery potentially mitigates the risks of open surgical procedures and is becoming a new standard for pelvic fracture treatment. The accurate reduction has been recognized as the cornerstone of minimally invasive surgery for pelvic fracture. At present, the closed reduction in pelvic fractures is limited by the current sub-optimal 2D intra-operative imaging (fluoroscopy) and by the high forces of soft tissue involved in the fragment manipulation, which might result in fracture malreduction. To overcome these shortcomings and facilitate pelvic fracture reduction, we developed an intelligent robot-assisted fracture reduction (RAFR) system for pelvic fracture.MethodsThe presented method is divided into three parts. The first part is the preparation of 20 pelvic fracture models. In the second part, we offer an automatic reduction algorithm of our robotic reduction system, including Intraoperative real-time 3D navigation, reduction path planning, control and fixation, and robotic-assisted fracture reduction. In the third part, image registration accuracy and fracture reduction accuracy were calculated and analyzed.ResultsAll 20 pelvic fracture bone models were reduced by the RAFR system; the mean registration error E1 of the 20 models was 1.29 ± 0.57 mm. The mean reduction error E2 of the 20 models was 2.72 ± 0.82 mm. The global error analysis of registration and reduction results showed that higher errors are mainly located at the edge of the pelvis, such as the iliac wing.ConclusionThe accuracy of image registration error and fracture reduction error in our study was excellent, which could reach the requirements of the clinical environment. Our study demonstrated the precision and effectiveness of our RAFR system and its applicability and usability in clinical practice, thus paving the way toward robot minimally invasive pelvic fracture surgeries.

A stability preserved time-integration method for nonlinear advection–diffusion-reaction processes

A new implicit-explicit local differential transform method (IELDTM) is derived here for time integration of the nonlinear (2 + 1)-dimensional advection–diffusion-reaction (ADR) equations. The IELDTM is adaptively constructed as a stability preserved and high order time integrator for spatially discretized ADR equations. For spatial discretization of the model equation, the Chebyshev spectral collocation method (ChCM) is utilized. A robust stability analysis and global error analysis of the IELDTM are presented with respect to the direction parameter \(\theta \). With the help of the global error analysis, adaptivity equations are derived to minimize the computational costs of the algorithms. The produced method is shown to eliminate the accuracy disadvantage of the classical \(\theta \)-method and the stability disadvantages of the existing differential transform-based methods. Two examples of the Burgers equation in one and two space dimensions and the Chapman oxygen-ozone ADR model are solved via the ChCM-IELDTM hybridization. The present time integrator is proven to provide more efficient numerical characteristics than the various multi-step and multi-stage time integration methods. The IELDTM is extensively compared with the widely used MATLAB solvers, ode45 and ode15s. The adaptive IELDTM has been proven to integrate the stiff Chapman ADR equations with optimum costs over relatively long-time intervals.

CLOSED NEWTON-COTES CUBATURE SCHEMES FOR TRIPLE INTEGRALS WITH ERROR ANALYSIS

Most of the problems in applied sciences in engineering contain integrals, not only in one dimension but also in higher dimensions. The complexity of integrands of functions in one variable or higher variables motivates the quadrature and cubature approximations. Much of the work is focused on the literature on single integral quadrature approximations and double integral cubature schemes. On the other hand, the work on triple integral schemes has been quite rarely focused. In this work, we propose the closed Newton-Cotes-type cubature schemes for triple integrals and discuss consequent error analysis of these schemes in terms of the degree of precision and local error terms for the basic form approximations. The results obtained for the proposed triple integral schemes are in line with the patterns observed in single and double integral schemes. The theorems proved in this work on the local error analysis will be a great aid in extending the work towards global error analysis of the schemes in the future.

Global error analysis of the Chebyshev rational approximation method

The Chebyshev rational approximation method (CRAM) has become a widely adopted method for solving nuclear depletion problems. Therefore, understanding CRAM’s accuracy is important for the safe operation of nuclear power plants. This article performs a global error analysis of CRAM and finds that, as the length of the time step approaches zero, the relative error measured between the exact and CRAM solutions at a fixed end time approaches one and infinity for even and odd orders, respectively; for intermediate time step sizes, a minimum in relative error is observed. We show that the reason for CRAM’s behavior is that the method is inconsistent. Two best practices for using CRAM, derived from these results, are: (1) use CRAM order 16 or higher, (2) if necessary, increase the CRAM order when multiphysics coupling requires smaller time steps.

Moving quantile regression

Quantile regression is a technique to estimate the conditional quantile. In this paper we propose a localized method for quantile regression, the regularized moving quantile regression, which can be used to analyze scattered data efficiently. We present a rigorous global error analysis in the learning theory framework. The main results include an inequality that bridges the gap between the global risk and local risk, a characterization of the approximation that shows the moving technique allows to approximate very complicated functions by simple function classes, and a learning rate analysis. These results indicate that the moving quantile regression method converges fast under mild conditions.

Global error analysis of two-dimensional panel methods for Neumann formulation

There are two main formulations for panel methods based on Green’s formula: Dirichlet and Neumann methods. In a previous work a rigorous analytical study of the global error of Dirichlet method was performed. In this work an analytical study of the global error is performed for the other formulation: Neumann method. The analysis is performed for a wide variety of body shapes and different panel geometries to fully understand their effect on the convergence of the method. In particular, we study the global error associated with panel methods applied to thin or thick bodies with purely convex parts or with both convex and concave parts, and with smooth or non-smooth boundaries. In order to validate the analytical results, both numerical and analytical solutions for different body geometries have been considered to compare the actual and predicted errors in each case. Finally, a comparison of the error order between both methods, Dirichlet and Neumann, has also been performed for different configurations.

Extended explicit pseudo two-step RKN methods for oscillatory systems y ″ + M y = f(y)

In this paper, extended explicit pseudo two-step Runge-Kutta-Nystrom (EEPTSRKN) methods for the numerical integration of oscillatory system y ″ + M y = f(y) are proposed and studied. These methods inherit the framework of explicit pseudo two-step Runge-Kutta-Nystrom (EPTSRKN) methods (Cong et al. Comput. Math. Appl. 38, 17–39, 1999) and they are parallel methods. Furthermore, these new methods take into account the special feature of the oscillatory problem so that they integrate exactly unperturbed problem y ″ + M y = 0. The study of corresponding order conditions shows that an s-stage EEPTSRKN method can obtain maximum step order s + 2 and maximum stage order s + 1. We make a global error analysis and present the global error bound for the new method, which is proved to be independent of ∥M∥ under suitable assumptions. Four two-stage methods with step order two, three, three, and four, respectively, are constructed and the corresponding stability regions are given. Numerical results show that our new methods are more efficient in comparison with other well-known high quality methods proposed in the scientific literature.

Global error analysis of two-dimensional panel methods for dirichlet formulation

A rigorous analytical study of the global error of panel methods is presented. The analysis is performed for a wide variety of body shapes and different panel geometries to fully understand their effect on the convergence of the method. In particular, we study the global error associated with panel methods applied to thin or thick bodies with purely convex parts or with both convex and concave parts, and with smooth or non-smooth boundaries. Most previous studies focused on the analysis of local error, considering only the influence of the nearest panels and excluding the rest. The difference is shown to be appreciable in many configurations. Generally, there is a lack of consensus concerning the order of magnitude of the error for panel methods even in the simplest case with flat panels and a constant distribution of doublets along them. This paper clarifies apparently different or inconsistent results obtained by other authors.

Generalized polynomial chaos for nonlinear random pantograph equations

This paper is concerned with the application of generalized polynomial chaos (gPC) method to nonlinear random pantograph equations. An error estimation of gPC method is derived. The global error analysis is given for the error arising from finite-dimensional noise (FDN) assumption, projection error, aliasing error and discretization error. In the end, with several numerical experiments, the theoretical results are further illustrated.

Global error analysis and inertial manifold reduction

Four types of global error for initial value problems are considered in a common framework. They include classical forward error analysis and shadowing error analysis together with extensions of both to include rescaling of time. To determine the amplification of the local error that bounds the global error we present a linear analysis similar in spirit to condition number estimation for linear systems of equations. We combine these ideas with techniques for dimension reduction of differential equations via a boundary value formulation of numerical inertial manifold reduction. These global error concepts are exercised to illustrate their utility on the Lorenz equations and inertial manifold reductions of the Kuramoto–Sivashinsky equation.

Quantitative Analysis Tools and Digital Phantoms for Deformable Image Registration Quality Assurance.

This article proposes quantitative analysis tools and digital phantoms to quantify intrinsic errors of deformable image registration (DIR) systems and establish quality assurance (QA) procedures for clinical use of DIR systems utilizing local and global error analysis methods with clinically realistic digital image phantoms. Landmark-based image registration verifications are suitable only for images with significant feature points. To address this shortfall, we adapted a deformation vector field (DVF) comparison approach with new analysis techniques to quantify the results. Digital image phantoms are derived from data sets of actual patient images (a reference image set, R, a test image set, T). Image sets from the same patient taken at different times are registered with deformable methods producing a reference DVFref. Applying DVFref to the original reference image deforms T into a new image R'. The data set, R', T, and DVFref, is from a realistic truth set and therefore can be used to analyze any DIR system and expose intrinsic errors by comparing DVFref and DVFtest. For quantitative error analysis, calculating and delineating differences between DVFs, 2 methods were used, (1) a local error analysis tool that displays deformation error magnitudes with color mapping on each image slice and (2) a global error analysis tool that calculates a deformation error histogram, which describes a cumulative probability function of errors for each anatomical structure. Three digital image phantoms were generated from three patients with a head and neck, a lung and a liver cancer. The DIR QA was evaluated using the case with head and neck.

Weighted Reduced Basis Method for Stochastic Optimal Control Problems with Elliptic PDE Constraint

In this paper we develop and analyze an efficient computational method for solving stochastic optimal control problems constrained by an elliptic partial differential equation (PDE) with random input data. We first prove both existence and uniqueness of the optimal solution. Regularity of the optimal solution in the stochastic space is studied in view of the analysis of stochastic approximation error. For numerical approximation, we employ a finite element method for the discretization of physical variables, and a stochastic collocation method for the discretization of random variables. In order to alleviate the computational effort, we develop a model order reduction strategy based on a weighted reduced basis method. A global error analysis of the numerical approximation is carried out, and several numerical tests are performed to verify our analysis.

Error analysis of explicit TSERKN methods for highly oscillatory systems

In this paper, we are concerned with the error analysis for the two-step extended Runge-Kutta-Nystrom-type (TSERKN) methods [Comput. Phys. Comm. 182 (2011) 2486–2507] for multi-frequency and multidimensional oscillatory systems y″(t) + My(t) = f(t, y(t)), where high-frequency oscillations in the solutions are generated by the linear part My(t). TSERKN methods extend the two-step hybrid methods [IMA J. Numer. Anal. 23 (2003) 197–220] by reforming both the internal stages and the updates so that they are adapted to the oscillatory properties of the exact solutions. However, the global error analysis for the TSERKN methods has not been investigated. In this paper we construct a new three-stage explicit TSERKN method of order four and present the global error bound for the new method, which is proved to be independent of ∥M∥ under suitable assumptions. This property of our new method is very important for solving highly oscillatory systems (1), where ∥M∥ may be arbitrarily large. We also analyze the stability and phase properties for the new method. Numerical experiments are included and the numerical results show that the new method is very competitive and promising compared with the well-known high quality methods proposed in the scientific literature.

Verification of a three-dimensional unstructured finite element method using analytic and manufactured solutions

We report on the verification of a three-dimensional unstructured finite element method applicable to compressible fluid dynamics and diffusion problems. Our verification methodology uses a combination of analytic and manufactured solutions to formally measure convergence rates in global error for both shock-dominated flows and smooth problems. In addition we measure the global error in vorticity, which should converge at reduced-order relative to the velocity solution. The numerical method under investigation is an edge-based Finite Element formulation on linear tetrahedra with a parabolic MUSCL reconstruction for the advective fluxes. The scheme is nominally second-order accurate on smooth flows. For diffusion problems the formulation also is nominally second-order accurate.Using global error analysis we measure convergence rates of 0.8–1.0 for shock-dominated problems and 1.5–2.4 for smooth problems. Calculations with Adaptive Mesh Refinement (AMR) are observed to produce errors comparable to finer mesh simulations but at significantly reduced computational cost. A convergence rate of 2.2 also is observed for a simplified diffusion problem. Examples of how these studies can inform simulation practices are provided.

Rational approximation method for delay differential equations with proportional delay

Consider the pantograph differential equation y′( t) = ay( t) + by( qt) + f( t), 0 ⩽ t ⩽ T, y(0) = y 0, with proportional delay qt, 0 < q < 1. In this paper, we propose a piecewise (2 m, m)-rational approximation with “quasi-uniform meshes” which corresponds to mth collocation method, and establish the global error analysis of O( h 2 m ) on successive mesh points. This method is useful than the known collocation method in case that a long time integration is needed, that is, if T is large, then the number of steps in our method is less than that of the collocation method. Numerical experiments are also contained.

Local uniform grid refinement and systems of coupled partial differential equations

In this paper we consider an adaptive grid method based on local uniform grid refinement applied to systems of coupled time-dependent PDEs. Local uniform grid refinement means that the PDEs are solved on a series of nested, uniform, increasingly finer subgrids which cover only a part of the domain. These subgrids are created up to a level of refinement where sufficient spatial accuracy is obtained and their location and shape is adjusted after each time step in order to follow the moving steep fronts. When a system of coupled PDEs is solved, the behavior of the local and global error associated with each separate PDE can be very different from one PDE to another. A refinement strategy based on a global error analysis has been developed which takes these differences into account. This refinement strategy aims at the domination of the global space error by the space discretization error at the finest subgrid.

Optimal extravascular dosing intervals

An explicit formula is presented for simple calculations of the dosing time, tau, that results in a steady-state peak-to-trough ratio of 2 in extravascular dosings. Contrary to other formulae presented, the calculations are guaranteed to be well bound in the percentage error (less than 1%) for any parameter value combination. It is shown that the biexponential dosing interval problem can be transformed into a general, dimensionless problem enabling a global error analysis in the approximation. The proposed formula is demonstrated in the calculation of an "optimal" dosing interval for quinidine. An algorithm and FORTRAN computer program OPTAU for exact calculation of tau and dosing simulations is also demonstrated in the quinidine example.

Condition Estimates

Condition Estimates