An a posteriori error estimator for model adaptivity in electrocardiology

We introduce an a posteriori modeling error estimator for the effective computation of electric potential propagation in the heart. Starting from the Bidomain problem and an extended formulation of the simplified Monodomain system, we build a hybrid model, called Hybridomain, which is dynamically adapted to be either Bi- or Monodomain ones in different regions of the computational domain according to the error estimator. We show that accurate results can be obtained with the adaptive Hybridomain model with a reduced computational cost compared to the full Bidomain model. We discuss the effectivity of the estimator and the reliability of the results on simulations performed on real human left ventricle geometries retrieved from healthy subjects.     

Optimal control in non-convex domains: a priori discretization error estimates

Abstract An optimal control problem for a two-dimensional elliptic equation with pointwise control constraints is investigated. The domain is assumed to be polygonal but non-convex. The corner singularities are treated by a priori mesh grading. Approximations of the optimal solution of the continuous optimal control problem are constructed by a projection of the discrete adjoint state. It is proved that these approximations have convergence order h². Keywords Linear-quadratic optimal control problems, error estimates, elliptic equations, non-convex domains, corner singularities, control constraints, superconvergence. Mathematics Subject Classification (2000): 49K20, 49M25, 65N30, 65N50     

An upper bound for the recursive least squares estimation error

A new upper bound for the convergence rate of recursive least squares (RLS) errors is presented. The bound is free of some deficiencies of a cell-known RLS upper bound and allows a realistic assessment of factors influencing convergence rate, such as input-output data scaling, disturbances, signal-to-noise ratio, number of estimated parameters, data discounting, and excitation properties of plant inputs. Some of the properties of the new bound are discussed     

Guaranteed and robust a posteriori error estimates and balancing discretization and linearization errors for monotone nonlinear problems

We derive a posteriori error estimates for a class of second-order monotone quasi-linear diffusion-type problems approximated by piecewise affine, continuous finite elements. Our estimates yield a guaranteed and fully computable upper bound on the error measured by the dual norm of the residual, as well as a global error lower bound, up to a generic constant independent of the nonlinear operator. They are thus fully robust with respect to the nonlinearity, thanks to the choice of the error measure. They are also locally efficient, albeit in a different norm, and hence suitable for adaptive mesh refinement. Moreover, they allow to distinguish, estimate separately, and compare the discretization and linearization errors. Hence, the iterative (Newton–Raphson, fixed point) linearization can be stopped whenever the linearization error drops to the level at which it does not affect significantly the overall error. This can lead to important computational savings, as performing an excessive number of unnecessary linearization iterations can be avoided. A strategy combining the linearization stopping criterion and adaptive mesh refinement is proposed and numerically tested for the p-Laplacian.     

Implicit Runge-Kutta formulae with built-in estimates of the accumulated truncation error

Explicit Runge-Kutta formulae with built-in estimates of the accumulated truncation error are well known. A method is presented for developingA-stable and stronglyA-stable Runge-Kutta algorithms with built-in estimates of the accumulated truncation error.     

Goal-oriented error estimation and adaptivity for the finite element method

In this paper, we study a new approach in a posteriori error estimation, in which the numerical error of finite element approximations is estimated in terms of quantities of interest rather than the classical energy norm. These so-called quantities of interest are characterized by linear functionals on the space of functions to where the solution belongs. We present here the theory with respect to a class of elliptic boundary-value problems, and in particular, show how to obtain accurate estimates as well as upper and lower bounds on the error. We also study the new concept of goal-oriented adaptivity, which embodies mesh adaptation procedures designed to control error in specific quantities. Numerical experiments confirm that such procedures greatly accelerate the attainment of local features of the solution to preset accuracies as compared to traditional adaptive schemes based on energy norm error estimates.     

A tight upper bound on the Bayesian probability of error

In this paper, we present a new upper bound on the minimum probability of error of Bayesian decision systems for statistical pattern recognition. This new bound is continuous everywhere and is shown to be tighter than several existing bounds such as the Bhattacharyya and the Bayesian bounds. Numerical results are also presented     

Comments on "A least upper bound on quantization error"

A summary of methods proposed by Cavin and Bertram for bounding quantization errors in hybrid control systems is presented. Their methods are compared and similarities in the methods are noted. An apparent error by Cavin is demonstrated.     

An upper bound on the mean-square error in the sampling expansion for non-band-limited processes

An upper bound is obtained on the mean-square error involved when a real-valued non-band-limited nonstationary random process x(t) is approximated by the sampling expansion

$\text{∑}\text{n=?∞}\text{∞}\text{x(nT)}\text{sin}\text{π(t?nT)/T}\text{π(t?nT)/T}$

for some T > 0. When the process x(t) is band-limited over [$\text{?}\text{1}\text{2T}\text{,}\text{1}\text{2T}$], this error bound reduces to zero.     

A priori error estimates for the finite element discretization of optimal distributed control problems governed by the biharmonic operator

In this article a priori error estimates are derived for the finite element discretization of optimal distributed control problems governed by the biharmonic operator. The state equation is discretized in primal mixed form using continuous piecewise biquadratic finite elements, while piecewise constant approximations are used for the control. The error estimates derived for the state variable as well as that for the control are order-optimal on general unstructured meshes. However, on uniform meshes not all error estimates are optimal due to the low-order control approximation. All theoretical results are confirmed by numerical tests.     

A distribution-free geometric upper bound for the probability of error of a minimum distance classifier

An upperbound to the probability of error per class in a multivariate pattern classification is derived. The bound, given by

$\text{P(E|class w}{}_{\mathrm{i}}\text{)≤}\text{N}\text{R}{}^2{}_{\mathrm{i}}$

is derived with minimal assumptions; specifically the mean vectors exist and are distinct and the covariance matrices exist and are non-singular. No other assumptions are made about the nature of the distributions of the classes. In equation (i) N is the number of features in the feature (vector) space and R_i is a measure of the “radial neighbourhood” of a class. An expression for R_i is developed. A comparison to the multivariate Gaussian hypothesis is presented.     

Toward a tight upper bound for the error probability of the binary Gaussian classification problem

It is well known that the error probability, of the binary Gaussian classification problem with different class covariance matrices, cannot be generally evaluated exactly because of the lack of closed-form expression. This fact pointed out the need to find a tight upper bound for the error probability. This issue has been for more than 50 years ago and is still of interest. All derived upper-bounds are not free of flaws. They might be loose, computationally inefficient particularly in highly dimensional situations, or excessively time consuming if high degree of accuracy is desired. In this paper, a new technique is developed to estimate a tight upper bound for the error probability of the well-known binary Gaussian classification problem with different covariance matrices. The basic idea of the proposed technique is to replace the optimal Bayes decision boundary with suboptimal boundaries which provide an easy-to-calculate upper bound for the error probability. In particular, three types of decision boundaries are investigated: planes, elliptic cylinders, and cones. The new decision boundaries are selected in such a way as to provide the tightest possible upper bound. The proposed technique is found to provide an upper bound, tighter than many of the often used bounds such as the Chernoff bound and the Bayesian-distance bound. In addition, the computation time of the proposed bound is much less than that required by the Monte-Carlo simulation technique. When applied to real world classification problems, obtained from the UCI repository [H. Chernoff, A measure for asymptotic efficiency of a hypothesis based on a sum of observations, Ann. Math. Statist. 23 (1952) 493–507.], the proposed bound was found to provide a tight bound for the analytical error probability of the quadratic discriminant analysis (QDA) classifier and a good approximation to its empirical error probability.     

Improving feature extraction by replacing the Fisher criterion by an upper error bound

A lot of alternatives and constraints have been proposed in order to improve the Fisher criterion. But most of them are not linked to the error rate, the primary interest in many applications of classification. By introducing an upper bound for the error rate a criterion is developed which can improve the classification performance.     

On componentwise error estimates for inverse matrices

LetA be an×n-matrix with the property I–A<1. LetY be an approximation of the inverse ofA. This paper shows how to get a componentwise error estimate forY, that does not require too much numerical effort but generally presents better results than global error estimates do. Although proved by means of interval mathematics, the given error estimate can also be calculated in absence of any implementation of interval arithmetic.

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Über komponentenweise Fehlerabschätzungen für inverse Matrizen

Zusammenfassung SeiA einen×n-Matrix mit der Eigenschaft I–A<1. SeiY eine Approximation der Inversen vonA. In dieser Arbeit wird gezeigt, wie man eine komponentenweise Fehlerabschätzung fürY erhalten kann, deren Berechnung nicht sehr aufwendig ist, die aber im allgemeinen schärfer ist als globale Fehlerabschätzungen. Obwohl mit intervallmathematischen Mitteln bewiesen, kann die angegebene Fehlerabschätzung auch berechnet werden, wenn keine Intervallarithmetik implementiert ist.

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This research was supported in part by Sonderforschungsbereich 72-Approximation und Optimierung, University of Bonn.     

An upper bound model for TCP and UDP throughput in IPv4 and IPv6

Due to the shortage of public IPv4 addresses, the IETF has developed a new version of the Internet Protocol called IPv6. Many institutions all over the world had already started the migration to IPv6. Since this migration has to be done slowly, the first step is the coexistence of the two protocols (IPv4 and IPv6) for some years. One important issue for IPv6 to gain acceptance, is its performance in end-user applications. Hence, due to the availability of a variety of IPv6 implementations on different operating systems, it is important to evaluate the performance of the different IPv6 stacks, and compare it to the one shown by IPv4. In this paper, we present an upper bound model to compute TCP and UDP throughput for IPv4 and IPv6, in a full-duplex point-to-point connection. Our model can be used for any variant of Ethernet technology (10, 100, and 1000 Mbps). To validate this model, we did experiments and compared the maximum theoretical throughput with the experimental ones. Experiments were done with Windows XP SP2, Solaris 10, and Debian 3.1, which are very popular operating systems. The results show that 10 Mbps Ethernet technology is already very mature, since it gave performance very close to the maximum theoretical throughput. Experiments with FastEthernet (100 Mbps) show a TCP and UDP throughput close to the maximum theoretical throughput, especially for large payload. In the case of GigaEthernet (1000 Mbps), experimental results are not far from the maximum throughput for large TCP and UDP payload. However, for small TCP and UDP payload, the differences between our model (the maximum throughput) and the experiments are important. These differences should significantly decrease with the release of faster technology (processors and RAM).     

An error bound for a discrete reduced order model of a linear multivariable system

The design of feasible controllers for high dimension multivariable systems can be greatly aided by a method of model reduction. In order for the design based on the order reduction to include a guarantee of stability, it is sufficient to have a bound on the model error. Previous work has provided such a bound for continuous-time systems for algorithms based on balancing. In this note an L^inftybound is derived for model error for a method of order reduction of discrete linear multivariable systems based on balancing.     

A relative error bound for balanced stochastic truncation

Recently, several methods have appeared for the approximation of (power) spectra, notably balanced stochastic truncation (BST). It is shown that BST satisfies a relative error bound approximately twice the bound for the relative error method (REM) proper. This offers a quantitative basis for the observation that BST and REM produce similar reduced-order models. Balanced stochastic truncation can therefore be interpreted as providing a computationally simple algorithm for relative error approximation     

Improved error bound for multivariate Chebyshev polynomial interpolation

ABSTRACT

Chebyshev interpolation is a highly effective, intensively studied method and enjoys excellent numerical properties which provides tremendous application potential in mathematical finance. The interpolation nodes are known beforehand, implementation is straightforward and the method is numerically stable. For efficiency, a sharp error bound is essential, in particular for high-dimensional applications. For tensorized Chebyshev interpolation, we present an error bound that improves existing results significantly.