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In the paper (HA14), unfortunately, a computational error occurred in one estimate. Although the wrong estimate does not affect the main results, we want to present the necessary corrections. Essentially, Lemma 5.2 has to be corrected and, since it is used in the proof of Theorem 5.1, the proof of this theorem also requires an adaptation. (i) The corrected formulation of Lemma 5.2 is as follows. Lemma 5.2 For Lagrange finite elements with a shape-regular family of affine meshes { T n h } h>0 there is a constant C > 0 independent of q and h such that for all w ∈ Wh and q = 2m, m ∈N: CΛq−2 p (∇w,∇Ip h (wq−1))T ∫ T ‖∇w‖l2‖w‖ q−2 0,∞,T dx, ∀T ∈ T n h , (5.1) where Λp = ‖ ∑ndof i=1 |φi|‖0,∞,T is the Lebesgue constant.

Olkin & Vaeth (1981) consider a two-way classification model with k rows and p columns in which the residuals (ei1, ..., eip) for row i are independently distributed with a multivariate normal distribution with zero means and common covariance matrix ((ij)). They concentrate their attention particularly on the situation where the row classification corresponds to k subjects, each treated in one of two or more ways; the column classification corresponds to p repetitions, perhaps in time, from each subject, probably resulting in correlated errors. Replication of subjects within treatments enables the elements of the covariance matrix to be estimated from the data, which in turn enables maximum likelihood estimates of the row and column parameters to be computed and likelihood ratio tests to be carried out without having to make assumptions about the form of the covariance matrix. The careful analysis of data of this kind has been the subject of numerous papers, Wishart (1938) perhaps being the first to concentrate on this topic. Despite this, however, the use of erroneous methods in published material is widespread and this provided the stimulus for the publication of our earlier paper (Rowell & Walters, 1976), and also for the present communication. We note here that there is confusion in Olkin & Vaeth's row/column terminology in their worked example: the rows in their tables are referred to as columns in the text, and vice versa. In what follows, when referring to their numerical example, the rows classification will refer to the high/low factor, and the columns classification to the time factor.

The process of grinding and polishing optical surfaces using a Computer Numerically Controlled machine produces a machine material removal profile. The profiles achievable by the machine depend on the nature of the tool used in the process, and the tool center motions. In this paper machine material removal profiles are developed as mollifications of given workpiece profiles for a variety of tool configurations. The form of the mollification, in effect, defines the tool center motion. Convergence of the machine's material removal profile to the given workpiece profile as the support of the tool goes to zero is established under mild assumptions. Numerical examples are included

Differential-algebraic equations (DAEs) are widely used for modeling of dynamical systems. In numerical analysis of DAEs, consistent initialization and index reduction are important preprocessing prior to numerical integration. Existing DAE solvers commonly adopt structural preprocessing methods based on combinatorial optimization. Unfortunately, the structural methods fail if the DAE has numerical or symbolic cancellations. For such DAEs, methods have been proposed to modify them to other DAEs to which the structural methods are applicable, based on the combinatorial relaxation technique. Existing modification methods, however, work only for a class of DAEs that are linear or close to linear. This paper presents two new modification methods for nonlinear DAEs: the substitution method and the augmentation method. Both methods are based on the combinatorial relaxation approach and are applicable to a large class of nonlinear DAEs. The substitution method symbolically solves equations for some derivatives based on the implicit function theorem and substitutes the solution back into the system. Instead of solving equations, the augmentation method modifies DAEs by appending new variables and equations. The augmentation method has advantages that the equation solving is not needed and the sparsity of DAEs is retained. It is shown in numerical experiments that both methods, especially the augmentation method, successfully modify high-index DAEs that the DAE solver in MATLAB cannot handle., Comment: A preliminary version of this paper is to appear in Proceedings of the 44th International Symposium on Symbolic and Algebraic Computation (ISSAC 2019), Beijing, China, July 2019

In 1993, Shub and Smale posed the problem of finding a sequence of univariate polynomials of degree $N$ with condition number bounded above by $N$. In a previous paper by C. Belt\'an, U. Etayo, J. Marzo and J. Ortega-Cerd\`a, it was proved that the optimal value of the condition number is of the form $O(\sqrt{N})$, and the sequence demanded by Shub and Smale was described by a closed formula (for large enough $N\geqslant N_0$ with $N_0$ unknown) and by a search algorithm for the rest of the cases. In this paper we find concrete estimates for the constant hidden in the $O(\sqrt{N})$ term and we describe a simple formula for a sequence of polynomials whose condition number is at most $N$, valid for all $N=4M^2$, with $M$ a positive integer., Comment: 21 pages

The paper focuses on the analysis of the Euler projection Galerkin finite element method (FEM) for the dynamics of magnetization in ferromagnetic materials, described by the Landau–Lifshitz equation with the point-wise constraint $|{\textbf{m}}|=1$. The method is based on a simple sphere projection that projects the numerical solution onto a unit sphere at each time step, and the method has been used in many areas in the past several decades. However, error analysis for the commonly used method has not been done since the classical energy approach cannot be applied directly. In this paper we present an optimal $\textbf{L}^2$ error analysis of the backward Euler sphere projection method by using quadratic or higher order finite elements under a time step condition $\tau =O(\epsilon _0 h)$ with some small $\epsilon _0>0$. The analysis is based on more precise estimates of the extra error caused by the sphere projection in both $\textbf{L}^2$ and $\textbf{H}^1$ norms, and the classical estimate of dual norm. Numerical experiment is provided to confirm our theoretical analysis.

This paper is concerned with optimal error estimates and recovery technique of a classical mixed finite element method for the thermistor problem, which is governed by a parabolic/elliptic system with strong nonlinearity and coupling. The method is based on a popular combination of the lowest-order Raviart–Thomas mixed approximation for the electric potential/field $(\phi , \boldsymbol{\theta })$ and the linear Lagrange approximation for the temperature $u$. A common question is how the first-order approximation influences the accuracy of the second-order approximation to the temperature in such a strongly coupled system, while previous work only showed the first-order accuracy $O(h)$ for all three components in a traditional way. In this paper, we prove that the method produces the optimal second-order accuracy $O(h^2)$ for $u$ in the spatial direction, although the accuracy for the potential/field is in the order of $O(h)$. And more importantly, we propose a simple one-step recovery technique to obtain a new numerical electric potential/field of second-order accuracy. The analysis presented in this paper relies on an $H^{-1}$-norm estimate of the mixed finite element methods and analysis on a nonclassical elliptic map. We provide numerical experiments in both two- and three-dimensional spaces to confirm our theoretical analyses.

In this paper we revisit the problem of performing a $QZ$ step with a so-called ‘perfect shift’, which is an ‘exact’ eigenvalue of a given regular pencil $\lambda B-A$ in unreduced Hessenberg triangular form. In exact arithmetic, the $QZ$ step moves that eigenvalue to the bottom of the pencil, while the rest of the pencil is maintained in Hessenberg triangular form, which then yields a deflation of the given eigenvalue. But in finite precision the $QZ$ step gets ‘blurred’ and precludes the deflation of the given eigenvalue. In this paper we show that when we first compute the corresponding eigenvector to sufficient accuracy, then the $QZ$ step can be constructed using this eigenvector, so that the deflation is also obtained in finite precision. An important application of this technique is the computation of the index of a system of differential algebraic equations, since an exact deflation of the infinite eigenvalues is needed to impose correctly the algebraic constraints of such differential equations.

We propose and analyse numerical algorithms based on weighted least squares for the approximation of a real-valued function on a general bounded domain $\Omega \subset \mathbb{R}^d$. Given any $n$-dimensional approximation space $V_n \subset L^2(\Omega)$, the analysis in [6] shows the existence of stable and optimally converging weighted least-squares estimators, using a number of function evaluations $m$ of the order $n \log n$. When an $L^2(\Omega)$-orthonormal basis of $V_n$ is available in analytic form, such estimators can be constructed using the algorithms described in [6,Section 5]. If the basis also has product form, then these algorithms have computational complexity linear in $d$ and $m$. In this paper we show that, when $\Omega$ is an irregular domain such that the analytic form of an $L^2(\Omega)$-orthonormal basis is not available, stable and quasi-optimally weighted least-squares estimators can still be constructed from $V_n$, again with $m$ of the order $n \log n$, but using a suitable surrogate basis of $V_n$ orthonormal in a discrete sense. The computational cost for the calculation of the surrogate basis depends on the Christoffel function of $\Omega$ and $V_n$. Numerical results validating our analysis are presented., Comment: Version of the paper accepted for publication

In a series of papers published since 2017 the author introduced a simple alternative of the $n$-simplex type, to enhance the accuracy of approximations of second-order boundary value problems subject to Dirichlet boundary conditions, posed on smooth curved domains. This technique is based upon trial functions consisting of piecewise polynomials defined on straight-edged triangular or tetrahedral meshes, interpolating the Dirichlet boundary conditions at points of the true boundary. In contrast, the test functions are defined by the standard degrees of freedom associated with the underlying method for polytopic domains. While the mathematical analysis of the method for Lagrange and Hermite methods for two-dimensional second- and fourth-order problems was carried out in earlier paper by the author this paper is devoted to the study of the three-dimensional case. Well-posedness, uniform stability and optimal a priori error estimates in the energy norm are proved for a tetrahedron-based Lagrange family of finite elements. Novel error estimates in the $L^2$-norm, for the class of problems considered in this work, are also proved. A series of numerical examples illustrates the potential of the new technique. In particular, its superior accuracy at equivalent cost, as compared to the isoparametric technique, is highlighted.

Nonlinear elastic models are widely used to describe the elastic response of crystalline solids, for example, the well-known Cauchy–Born model. While the Cauchy–Born model only depends on the strain, effects of higher-order strain gradients are significant and higher-order continuum models are preferred in various applications such as defect dynamics and modeling of carbon nanotubes. In this paper we rigorously derive a higher-order nonlinear elasticity model for crystals from its atomistic description in one dimension. We show that, compared to the second-order accuracy of the Cauchy–Born model, the higher-order continuum model in this paper is of fourth-order accuracy with respect to the interatomic spacing in the thermal dynamic limit. In addition we discuss the key issues for the derivation of higher-order continuum models in more general cases. The theoretical convergence results are demonstrated by numerical experiments.

In this paper we analyze a class of trace finite element methods for the discretization of vector-Laplace equations. A key issue in the finite element discretization of such problems is the treatment of the constraint that the unknown vector field must be tangential to the surface (‘tangent condition’). We study three different natural techniques for treating the tangent condition, namely a consistent penalty method, a simpler inconsistent penalty method and a Lagrange multiplier method. The main goal of the paper is to present an analysis that reveals important properties of these three different techniques for treating the tangent constraint. A detailed error analysis is presented that takes the approximation of both the geometry of the surface and the solution of the partial differential equation into account. Error bounds in the energy norm are derived that show how the discretization error depends on relevant parameters such as the degree of the polynomials used for the approximation of the solution, the degree of the polynomials used for the approximation of the level set function that characterizes the surface, the penalty parameter and the degree of the polynomials used for the approximation of the Lagrange multiplier.

The Morley finite element method (FEM) is attractive for semilinear problems with the biharmonic operator as a leading term in the stream function vorticity formulation of two-dimensional Navier–Stokes problem and in the von Kármán equations. This paper establishes a best-approximation a priori error analysis and an a posteriori error analysis of discrete solutions close to an arbitrary regular solution on the continuous level to semilinear problems with a trilinear nonlinearity. The analysis avoids any smallness assumptions on the data, and so has to provide discrete stability by a perturbation analysis before the Newton–Kantorovich theorem can provide the existence of discrete solutions. An abstract framework for the stability analysis in terms of discrete operators from the medius analysis leads to new results on the nonconforming Crouzeix–Raviart FEM for second-order linear nonselfadjoint and indefinite elliptic problems with $L^\infty $ coefficients. The paper identifies six parameters and sufficient conditions for the local a priori and a posteriori error control of conforming and nonconforming discretizations of a class of semilinear elliptic problems first in an abstract framework and then in the two semilinear applications. This leads to new best-approximation error estimates and to a posteriori error estimates in terms of explicit residual-based error control for the conforming and Morley FEM.

First, a new convergence analysis is given for the semidiscrete (finite elements in space) numerical method that is used in Le et al. (2016, Numerical solution of the time-fractional Fokker–Planck equation with general forcing. SIAM J. Numer. Anal.,54 1763–1784) to solve the time-fractional Fokker–Planck equation on a domain $\varOmega \times [0,T]$ with general forcing, i.e., where the forcing term is a function of both space and time. Stability and convergence are proved in a fractional norm that is stronger than the $L^2(\varOmega )$ norm used in the above paper. Furthermore, unlike the bounds proved in Le et al., the constant multipliers in our analysis do not blow up as the order of the fractional derivative $\alpha $ approaches the classical value of $1$. Secondly, for the semidiscrete (L1 scheme in time) method for the same Fokker–Planck problem, we present a new $L^2(\varOmega )$ convergence proof that avoids a flaw in the analysis of Le et al.’s paper for the semidiscrete (backward Euler scheme in time) method.

The augmented Lagrangian method (ALM) is fundamental in solving convex programming problems with linear constraints. The proximal version of ALM, which regularizes ALM’s subproblem over the primal variable at each iteration by an additional positive-definite quadratic proximal term, has been well studied in the literature. In this paper we show that it is not necessary to employ a positive-definite quadratic proximal term for the proximal ALM and the convergence can be still ensured if the positive definiteness is relaxed to indefiniteness by reducing the proximal parameter. An indefinite proximal version of the ALM is thus proposed for the generic setting of convex programming problems with linear constraints. We show that our relaxation is optimal in the sense that the proximal parameter cannot be further reduced. The consideration of indefinite proximal regularization is particularly meaningful for generating larger step sizes in solving ALM’s primal subproblems. When the model under discussion is separable in the sense that its objective function consists of finitely many additive function components without coupled variables, it is desired to decompose each ALM’s subproblem over the primal variable in Jacobian manner, replacing the original one by a sequence of easier and smaller decomposed subproblems, so that parallel computation can be applied. This full Jacobian splitting version of the ALM is known to be not necessarily convergent, and it has been studied in the literature that its convergence can be ensured if all the decomposed subproblems are further regularized by sufficiently large proximal terms. But how small the proximal parameter could be is still open. The other purpose of this paper is to show the smallest proximal parameter for the full Jacobian splitting version of ALM for solving multi-block separable convex minimization models.

In this paper we present a derivation of the surface Helmholtz decomposition, discuss its relation to the surface Hodge decomposition and derive a well-posed stream function formulation of a class of surface Stokes problems. We consider a $C^2$ connected (not necessarily simply connected) oriented hypersurface $\varGamma \subset \mathbb{R}^3$ without boundary. The surface gradient, divergence, curl and Laplace operators are defined in terms of the standard differential operators of the ambient Euclidean space $\mathbb{R}^3$. These representations are very convenient for the implementation of numerical methods for surface partial differential equations. We introduce surface $\mathbf H({\mathop{\rm div}}_{\varGamma})$ and $\mathbf H({\mathop{\rm curl}}_{\varGamma})$ spaces and derive useful properties of these spaces. A main result of the paper is the derivation of the Helmholtz decomposition, in terms of these surface differential operators, based on elementary differential calculus. As a corollary of this decomposition we obtain that for a simply connected surface to every tangential divergence-free velocity field there corresponds a unique scalar stream function. Using this result the variational form of the surface Stokes equation can be reformulated as a well-posed variational formulation of a fourth-order equation for the stream function. The latter can be rewritten as two coupled second-order equations, which form the basis for a finite element discretization. A particular finite element method is explained and the results of a numerical experiment with this method are presented.

This paper concerns the adaptive finite element method for elliptic Dirichlet boundary control problems in the energy space. The contribution of this paper is twofold. First, we rigorously derive efficient and reliable a posteriori error estimates for finite element approximations of Dirichlet boundary control problems. As a by-product, a priori error estimates are derived in a simple way by introducing appropriate auxiliary problems and establishing certain norm equivalence. Secondly, for the coupled elliptic partial differential system that resulted from the first-order optimality system, we prove that the sequence of adaptively generated discrete solutions including the control, the state and the adjoint state, guided by our newly derived a posteriori error indicators, converges to the true solution along with the convergence of the error estimators. We give some numerical results to confirm our theoretical findings.

Variants of the coordinate descent approach for minimizing a nonlinear function are distinguished in part by the order in which coordinates are considered for relaxation. Three common orderings are cyclic (CCD), in which we cycle through the components of $x$ in order; randomized (RCD), in which the component to update is selected randomly and independently at each iteration; and random-permutations cyclic (RPCD), which differs from CCD only in that a random permutation is applied to the variables at the start of each cycle. Known convergence guarantees are weaker for CCD and RPCD than for RCD, though in most practical cases, computational performance is similar among all these variants. There is a certain type of quadratic function for which CCD is significantly slower than for RCD; a recent paper by Sun & Ye (2016, Worst-case complexity of cyclic coordinate descent: $O(n^2)$ gap with randomized version. Technical Report. Stanford, CA: Department of Management Science and Engineering, Stanford University. arXiv:1604.07130) has explored the poor behavior of CCD on functions of this type. The RPCD approach performs well on these functions, even better than RCD in a certain regime. This paper explains the good behavior of RPCD with a tight analysis.

This paper provides a unified error analysis for nonconforming space discretizations of linear wave equations in the time domain. We propose a framework that studies wave equations as first-order evolution equations in Hilbert spaces and their space discretizations as differential equations in finite-dimensional Hilbert spaces. A lift operator maps the semidiscrete solution from the approximation space to the continuous space. Our main results are a priori error bounds in terms of interpolation, data and conformity errors of the method. Such error bounds are the key to the systematic derivation of convergence rates for a large class of problems. To show that this approach significantly eases the proof of new convergence rates, we apply it to an isoparametric finite element discretization of the wave equation with acoustic boundary conditions in a smooth domain. Moreover, our results reproduce known convergence rates for already investigated conforming and nonconforming space discretizations in a concise and unified way. The examples discussed in this paper comprise discontinuous Galerkin discretizations of Maxwell’s equations and finite elements with mass lumping for the acoustic wave equation.

In this paper we focus on the analysis of energy-preserving schemes for solving high-dimensional nonlinear Klein–Gordon equations. A novel energy-preserving scheme is developed based on the discrete gradient method and the Duhamel principle. The local error, global convergence and nonlinear stability of the new scheme are analysed in detail. Numerical experiments are implemented to compare with existing numerical methods in the literature, and the numerical results show the remarkable efficiency of the new energy-preserving scheme presented in this paper.

Weighting methods are widely used to adjust for covariates in observational studies, sample surveys, and regression settings. In this paper, we study a class of recently proposed weighting methods which find the weights of minimum dispersion that approximately balance the covariates. We call these weights "minimal weights" and study them under a common optimization framework. The key observation is the connection between approximate covariate balance and shrinkage estimation of the propensity score. This connection leads to both theoretical and practical developments. From a theoretical standpoint, we characterize the asymptotic properties of minimal weights and show that, under standard smoothness conditions on the propensity score function, minimal weights are consistent estimates of the true inverse probability weights. Also, we show that the resulting weighting estimator is consistent, asymptotically normal, and semiparametrically efficient. From a practical standpoint, we present a finite sample oracle inequality that bounds the loss incurred by balancing more functions of the covariates than strictly needed. This inequality shows that minimal weights implicitly bound the number of active covariate balance constraints. We finally provide a tuning algorithm for choosing the degree of approximate balance in minimal weights. We conclude the paper with four empirical studies that suggest approximate balance is preferable to exact balance, especially when there is limited overlap in covariate distributions. In these studies, we show that the root mean squared error of the weighting estimator can be reduced by as much as a half with approximate balance., 41 pages

The energy-dependent form of the radiative transfer equation (RTE) is important in a variety of applications. In a previous paper the well-posedness and an energy discretization for the energy-dependent RTE were investigated. In this paper the fully discrete scheme is introduced and analysed. Optimal-order error estimates and a well-posedness analysis of the discrete system are provided. The theoretical results are validated through numerical examples.

In this paper, we propose two low-order nonconforming finite element methods (FEMs) for the three-dimensional Stokes flow that generalize the nonconforming FEM of Kouhia & Stenberg (1995, A linear nonconforming finite element method for nearly incompressible elasticity and Stokes flow. Comput. Methods Appl. Mech. Eng, 124, 195–212). The finite element spaces proposed in this paper consist of two globally continuous components (one piecewise affine and one enriched component) and one component that is continuous at the midpoints of interior faces. We prove that the discrete Korn inequality and a discrete inf–sup condition hold uniformly in the mesh size and also for a nonempty Neumann boundary. Based on these two results, we show the well-posedness of the discrete problem. Two counterexamples prove that there is no direct generalization of the Kouhia–Stenberg FEM to three space dimensions: the finite element space with one nonconforming and two conforming piecewise affine components does not satisfy a discrete inf–sup condition with piecewise constant pressure approximations, while finite element functions with two nonconforming and one conforming component do not satisfy a discrete Korn inequality.

In this paper we suggest a method for transforming a vector subdivision scheme generating $C^{\ell}$ limits to another such scheme of the same dimension, generating $C^{\ell+1}$ limits. In scalar subdivision, it is well known that a scheme generating $C^{\ell}$ limit curves can be transformed to a new scheme producing $C^{\ell+1}$ limit curves by multiplying the scheme's symbol with the smoothing factor $\tfrac{z+1}{2}$. We extend this approach to vector and Hermite subdivision schemes, by manipulating symbols. The algorithms presented in this paper allow to construct vector (Hermite) subdivision schemes of arbitrarily high regularity from a convergent vector scheme (from a Hermite scheme whose Taylor scheme is convergent with limit functions of vanishing first component)., 28 pages, 4 figures. Corrected typos, updated contact information

n this paper, optimal control problems subject to a nonlinear scalar conservation law are studied. Such optimal control problems are challenging both at the continuous and at the discrete level since the control-to-state operator poses difficulties as it is, e.g., not differentiable. Therefore discretization of the underlying optimal control problem should be designed with care. Here the discretize-then-optimize approach is employed where first the full discretization of the objective function as well as the underlying PDE is considered. Then, the derivative of the reduced objective is obtained by using an adjoint calculus. In this paper total variation diminishing Runge-Kutta (TVD-RK) methods for the time discretization of such problems are studied. TVD-RK methods, also called strong stability preserving (SSP), are originally designed to preserve total variation of the discrete solution. It is proven in this paper that providing an SSP state scheme, is enough to ensure stability of the discrete adjoint. However requiring SSP for both discrete state and adjoint is too strong. Also approximation properties that the discrete adjoint inherits from the discretization of the state equation are studied. Moreover order conditions are derived. In addition, optimal choices with respect to CFL constant are discussed and numerical experiments are presented.

The envelope model allows efficient estimation in multivariate linear regression. In this paper, we propose the sparse envelope model, which is motivated by applications where some response variables are invariant with respect to changes of the predictors and have zero regression coefficients. The envelope estimator is consistent but not sparse, and in many situations it is important to identify the response variables for which the regression coefficients are zero. The sparse envelope model performs variable selection on the responses and preserves the efficiency gains offered by the envelope model. Response variable selection arises naturally in many applications, but has not been studied as thoroughly as predictor variable selection. In this paper, we discuss response variable selection in both the standard multivariate linear regression and the envelope contexts. In response variable selection, even if a response has zero coefficients, it should still be retained to improve the estimation efficiency of the nonzero coefficients. This is different from the practice in predictor variable selection. We establish consistency and the oracle property and obtain the asymptotic distribution of the sparse envelope estimator.

In a recent paper Dette et al. (2014) introduced optimal design problems for dose finding studies with an active control. These authors concentrated on regression models with normal distributed errors (with known variance) and the problem of determining optimal designs for estimating the smallest dose, which achieves the same treatment effect as the active control. This paper discusses the problem of designing active-controlled dose finding studies from a broader perspective. In particular, we consider a general class of optimality criteria and models arising from an exponential family, which are frequently used analyzing count data. We investigate under which circumstances optimal designs for dose finding studies including a placebo can be used to obtain optimal designs for studies with an active control. Optimal designs are constructed for several situations and the differences arising from different distributional assumptions are investigated in detail. In particular, our results are applicable for constructing optimal experimental designs to analyze active-controlled dose finding studies with discrete data, and we illustrate the efficiency of the new optimal designs with two recent examples from our consulting projects.

This paper analyses discontinuous Galerkin finite element methods (DGFEM) to approximate a regular solution to the von Karman equations defined on a polygonal domain. A discrete inf–sup condition sufficient for the stability of the discontinuous Galerkin discretization of a well-posed linear problem is established, and this allows the proof of local existence and uniqueness of a discrete solution to the nonlinear problem with a Banach fixed point theorem. The Newton scheme is locally second-order convergent and appears to be a robust solution strategy up to machine precision. A comprehensive a priori and a posteriori energy-norm error analysis relies on one sufficiently large stabilization parameter and sufficiently fine triangulations. In case the other stabilization parameter degenerates towards infinity, the DGFEM reduces to a novel C0-interior penalty method (IPDG). In contrast to the known C0-IPDG dueto Brenner et al., (2016, A C0 interior penalty method for a von Karman plate. Numer. Math., 1–30), the overall discrete formulation maintains symmetry of the trilinear form in the first two components—despite the general nonsymmetry of the discrete nonlinear problems. Moreover, a reliable and efficient a posteriori error analysis immediately follows for the DGFEM of this paper, while the different norms in the known C0-IPDG lead to complications with some nonresidual-type remaining terms. Numerical experiments confirm the best-approximation results and the equivalence of the error and the error estimators. A related adaptive mesh-refining algorithm leads to optimal empirical convergence rates for a nonconvex domain.

This paper proposes a unified framework to characterize the rate function of a recurrent event process through shape and size parameters. In contrast to the intensity function, which is the event occurrence rate conditional on the event history, the rate function is the occurrence rate unconditional on the event history, and thus it can be interpreted as a population-averaged count of events in unit time. In this paper, shape and size parameters are introduced and used to characterize the association between the rate function λ(⋅) and a random variable X. Measures of association between X and λ(⋅) are defined via shape- and size-based coefficients. Rate-independence of X and λ(⋅) is studied through tests of shape-independence and size-independence, where the shape- and size-based test statistics can be used separately or in combination. These tests can be applied when X is a covariable possibly correlated with the recurrent event process through λ(⋅) or, in the one-sample setting, when X is the censoring time at which the observation of N(⋅) is terminated. The proposed tests are shape- and size-based, so when a null hypothesis is rejected, the test results can serve to distinguish the source of violation.

This paper deals with the design of the likelihood continual reassessment method, which is an increasingly widely used model-based method for dose-finding studies. It is common to implement the method in a two-stage approach, whereby the model-based stage is activated after an initial sequence of patients has been treated. While this two-stage approach is practically appealing, it lacks a theoretical framework, and it is often unclear how the design components should be specified. This paper develops a general framework based on the coherence principle, from which we derive a design calibration process. A real clinical-trial example is used to demonstrate that the proposed process can be implemented in a timely and reproducible manner, while offering competitive operating characteristics. We explore the operating characteristics of different models within this framework and show the performance to be insensitive to the choice of dose-toxicity model.

In an earlier paper Kume & Wood (2005) showed how the normalizing constant of the Fisher– Bingham distribution on a sphere can be approximated with high accuracy using a univariate saddlepoint density approximation. In this sequel, we extend the approach to a more general setting and derive saddlepoint approximations for the normalizing constants of multicomponent Fisher– Bingham distributions on Cartesian products of spheres, and Fisher–Bingham distributions on Stiefel manifolds. In each case, the approximation for the normalizing constant is essentially a multivariate saddlepoint density approximation for the joint distribution of a set of quadratic forms in normal variables. Both first-order and second-order saddlepoint approximations are considered. Computational algorithms, numerical results and theoretical properties of the approximations are presented. In the challenging high-dimensional settings considered in this paper the saddlepoint approximations perform very well in all examples considered. Some key words: Directional data; Fisher matrix distribution; Kent distribution; Orientation statistics.