Error Bound

This paper presents a comprehensive analysis of uniform convergence with explicit error bounds, a fundamental result in operator theory that establishes precise convergence rates for sequences of operators. The main theorem provides explicit error bounds combining exponential and algebraic decay terms, offering a powerful framework for understanding convergence behavior in infinite-dimensional spaces. Through rigorous mathematical analysis, we explore the master equation ∥T-lim_{n→∞} T_n∥_{op} = 0, its derivative relations, and the spectral gap properties that determine the convergence parameters α, β, C₁, and C₂. This work contributes to the theoretical foundations of functional analysis and provides practical insights for applications in numerical analysis, quantum mechanics, and mathematical physics.

The describing function method is widely used without much attention being paid to the error analysis so vital in any approximate method. One reason for this is the lack of a straightforward, user-oriented method for checking error bounds except when the nonlinear element characteristic is single value and of bounded slope. This paper attempts to eliminate that defect. A far wider range of nonlinear elements is now amenable to straightforward, usually graphical treatment; discontinuities, hysteresis and backlash characteristics are included. In addition, previous results for slope-bounded single-valued nonlinear characteristics may be substantially improved with a small additional computational effort.

The existing image and data compression tech- niques try to minimize the mean square deviation between the original data f(x;y;z) and the compressed-decompressed data e f(x;y;z). In many practical situations, reconstruction that only guaranteed mean square error over the data set is unacceptable: for example, if we use the meteorological data to plan a best trajectory for a plane, what we really want to know are the meteorological parameters such as wind, tem- perature, and pressure along the trajectory. If along this line, the values are not reconstructed accurately enough, the plane may crash - and the fact that on average, we get a good reconstruction, does not help. What we need is a com- pression that guarantees that for each (x;y), the difference jf(x;y;z) ¡ e f(x;y;z)j is bounded by a given value ¢ - i.e., that the actual value f(x;y;z) belongs to the interval (e f(x;y;z) ¡ ¢; e f(x;y;z) + ¢). In this paper, we describe new efficient techniques for data compression under such...

Efficient Kernel Machines Using the Improved Fast Gauss Transform Changjiang Yang, Ramani Duraiswami and Larry Davis Department of Computer Science, Perceptual Interfaces and Reality Laboratory University of Maryland, College Park, MD 20742 {yangcj,ramani, lsd}@umiacs.umd.edu Abstract The computation and memory required for kernel machines with N training samples is at least O(N 2). Such a complexity is significant even for moderate size problems and is prohibitive for large datasets. ...

is essential in the analysis of the slow viscous fluid flow in the neighbourhood of a sharp corner which subtends an angle a E (0, 2~r] to the fluid. Existing methods for finding all roots A essentially require an a priori knowledge of the solution structure; given that {A1, A2,..., Am} are known, Am+l is determined via iterations, and/or a solution procedure initiated by Am. We present herein a general interval analysis method which exhaustively finds all roots A via only the original eigenvalue equation: no other information is required. The interval-analysis method automatically guarantees root existence and uniqueness while simultaneously providing error bounds.

This paper presents a volumetric approach to reconstructing a smooth surface from a sparse set of parallel binary contours, e.g. those produced via histologic imaging. It creates a volume dataset by interpolating 2D filtered distance fields. The zero isosurface embedded in the computed volume provides the desired result. MPU implicit functions are fit to the input contours, defined as binary images, to produce smooth curves with controllable error bounds. Full 2D Euclidean distance fields are then calculated from the implicit curves. A distance-dependent Gaussian filter is applied to the distance fields to smooth their medial axis discontinuities. Monotonicity-constraining cubic splines are used to construct smooth, blending slices between the input slices. A mesh that approximates the zero isosurface is then extracted from the resulting volume. The effectiveness of the approach is demonstrated on a number of complex, multi-component contour datasets.

This paper presents a volumetric approach to reconstructing a smooth surface from a sparse set of parallel binary contours, e.g. those produced via histologic imaging. It creates a volume dataset by interpolating 2D filtered distance fields. The zero isosurface embedded in the computed volume provides the desired result. MPU implicit functions are fit to the input contours, defined as binary images, to produce smooth curves with controllable error bounds. Full 2D Euclidean distance fields are then calculated from the implicit curves. A distance-dependent Gaussian filter is applied to the distance fields to smooth their medial axis discontinuities. Monotonicity-constraining cubic splines are used to construct smooth, blending slices between the input slices. A mesh that approximates the zero isosurface is then extracted from the resulting volume. The effectiveness of the approach is demonstrated on a number of complex, multi-component contour datasets.

This paper describes performance bounds for compressed sensing (CS) where the underlying sparse or compressible (sparsely approximable) signal is a vector of nonnegative intensities whose measurements are corrupted by Poisson noise. In this setting, standard CS techniques cannot be applied directly for several reasons. First, the usual signal-independent and/or bounded noise models do not apply to Poisson noise, which is non-additive and signal-dependent. Second, the CS matrices typically considered are not feasible in real optical systems because they do not adhere to important constraints, such as nonnegativity and photon flux preservation. Third, the typical 2-1 minimization leads to overfitting in the high-intensity regions and oversmoothing in the low-intensity areas. In this paper, we describe how a feasible positivity-and flux-preserving sensing matrix can be constructed, and then analyze the performance of a CS reconstruction approach for Poisson data that minimizes an objective function consisting of a negative Poisson log likelihood term and a penalty term which measures signal sparsity. We show that, as the overall intensity of the underlying signal increases, an upper bound on the reconstruction error decays at an appropriate rate (depending on the compressibility of the signal), but that for a fixed signal intensity, the signal-dependent part of the error bound actually grows with the number of measurements or sensors. This surprising fact is both proved theoretically and justified based on physical intuition.

We propose a new family of multilevel methods for unconstrained minimization. The resulting strategies are multilevel extensions of high-order optimization methods based on q-order Taylor models (with q ≥ 1) that have been recently proposed in the literature. The use of high-order models, while decreasing the worst-case complexity bound, makes these methods computationally more expensive. Hence, to counteract this effect, we propose a multilevel strategy that exploits a hierarchy of problems of decreasing dimension, still approximating the original one, to reduce the global cost of the step computation. A theoretical analysis of the family of methods is proposed. Specifically, local and global convergence results are proved and a complexity bound to reach first order stationary points is also derived. A multilevel version of the well known adaptive method based on cubic regularization (ARC, corresponding to q = 2 in our setting) has been implemented. Numerical experiments clearly highlight the relevance of the new multilevel approach leading to considerable computational savings in terms of floating point operations compared to the classical one-level strategy.

We propose a new family of multilevel methods for unconstrained minimization. The resulting strategies are multilevel extensions of high-order optimization methods based on q-order Taylor models (with q ≥ 1) that have been recently proposed in the literature. The use of high-order models, while decreasing the worst-case complexity bound, makes these methods computationally more expensive. Hence, to counteract this effect, we propose a multilevel strategy that exploits a hierarchy of problems of decreasing dimension, still approximating the original one, to reduce the global cost of the step computation. A theoretical analysis of the family of methods is proposed. Specifically, local and global convergence results are proved and a complexity bound to reach first order stationary points is also derived. A multilevel version of the well known adaptive method based on cubic regularization (ARC, corresponding to q = 2 in our setting) has been implemented. Numerical experiments clearly highlight the relevance of the new multilevel approach leading to considerable computational savings in terms of floating point operations compared to the classical one-level strategy.

In this paper, we study the problem of recovering a low-rank matrix (the principal components) from a highdimensional data matrix despite both small entry-wise noise and gross sparse errors. Recently, it has been shown that a convex program, named Principal Component Pursuit (PCP), can recover the low-rank matrix when the data matrix is corrupted by gross sparse errors. We further prove that the solution to a related convex program (a relaxed PCP) gives an estimate of the low-rank matrix that is simultaneously stable to small entrywise noise and robust to gross sparse errors. More precisely, our result shows that the proposed convex program recovers the low-rank matrix even though a positive fraction of its entries are arbitrarily corrupted, with an error bound proportional to the noise level. We present simulation results to support our result and demonstrate that the new convex program accurately recovers the principal components (the low-rank matrix) under quite broad conditions. To our knowledge, this is the first result that shows the classical Principal Component Analysis (PCA), optimal for small i.i.d. noise, can be made robust to gross sparse errors; or the first that shows the newly proposed PCP can be made stable to small entry-wise perturbations.

Distributed classification fusion using error-correcting codes (DCFECC) has recently been proposed for wireless sensor networks operating in a harsh environment. It has been shown to have a considerably better capability against unexpected sensor faults than the optimal likelihood fusion. In this paper, we analyze the performance of a DCFECC code with minimum Hamming distance fusion. No assumption on identical distribution for local observations, as well as common marginal distribution for the additive noises of the wireless links, is made. In addition, sensors are allowed to employ their own local classification rules. Upper bounds on the probability of error that are valid for any finite number of sensors are derived based on large deviations technique. A necessary and sufficient condition under which the minimum Hamming distance fusion error vanishes as the number of sensors tends to infinity is also established. With the necessary and sufficient condition and the upper error bounds, the relation between the fault-tolerance capability of a DCFECC code and its pair-wise Hamming distances is characterized, and can be used together with any code search criterion in finding the code with the desired fault-tolerance capability. Based on the above results, we further propose a code search criterion of much less complexity than the minimum Hamming distance fusion error criterion adopted earlier by the authors. This makes the code construction with acceptable fault-tolerance capability for a network with over a hundred of sensors practical. Simulation results show that the code determined based on the new criterion of much less complexity performs almost identically to the best code that minimizes the minimum Hamming distance fusion error. Also simulated and discussed are the performance trends of the codes Manuscript

We present an algorithm, called BioLab, for verifying temporal properties of rule-based models of cellular signalling networks. Bi-oLab models are encoded in the BioNetGen language, and properties are expressed as formulae in probabilistic bounded linear temporal logic. Temporal logic is a formalism for representing and reasoning about propositions qualified in terms of time. Properties are then verified using sequential hypothesis testing on executions generated using stochastic simulation. BioLab is optimal, in the sense that it generates the minimum number of executions necessary to verify the given property. Bio-Lab also provides guarantees on the probability of it generating Type-I (i.e., false-positive) and Type-II (i.e., false-negative) errors. Moreover, these error bounds are pre-specified by the user. We demonstrate Bio-Lab by verifying stochastic effects and bistability in the dynamics of the T-cell receptor signaling network.

The stabilized version of the sequential quadratic programming algorithm (sSQP) had been developed in order to achieve fast convergence despite possible degeneracy of constraints of optimization problems, when the Lagrange multipliers associated to a solution are not unique. Superlinear convergence of sSQP had been previously established under the strong second-order sufficient condition for optimality (without any constraint qualification assumptions). We prove a stronger superlinear convergence result than the above, assuming the usual second-order sufficient condition only. In addition, our analysis is carried out in the more general setting of variational problems, for which we introduce a natural extension of sSQP techniques. In the process, we also obtain a new error bound for Karush-Kuhn-Tucker systems for variational problems that holds under an appropriate second-order condition.

Figure 1: Closeup view of an isosurface feature in the mixing interface of two gases showing the texture mapped surface, underlying triangle mesh, and the adaptively refined tetrahedral mesh around the region of interest. Time step = 273, Isovalue = 206, Isosurface error = 1.5, 50K Triangles, rendered at 7 frames per second.

The task of estimating the number of distinct values (DVs) in a large dataset arises in a wide variety of settings in computer science and elsewhere. We provide DV estimation techniques that are designed for use within a flexible and scalable "synopsis warehouse" architecture. In this setting, incoming data is split into partitions and a synopsis is created for each partition; each synopsis can then be used to quickly estimate the number of DVs in its corresponding partition. By combining and extending a number of results in the literature, we obtain both appropriate synopses and novel DV estimators to use in conjunction with these synopses. Our synopses can be created in parallel, and can then be easily combined to yield synopses and DV estimates for arbitrary unions, intersections or differences of partitions. Our synopses can also handle deletions of individual partition elements. We use the theory of order statistics to show that our DV estimators are unbiased, and to establish moment formulas and sharp error bounds. Based on a novel limit theorem, we can exploit results due to Cohen in order to select synopsis sizes when initially designing the warehouse. Experiments and theory indicate that our synopses and estimators lead to lower computational costs and more accurate DV estimates than previous approaches.

We propose a similarity index for set-valued features and study algorithms for executing various set similarity queries on it. Such queries are fundamental for many application areas, including data integration and cleaning, data profiling as well as near duplicate document detection. In this paper, we focus on Jaccard similarity and present estimators that work for arbitrary similarity thresholds based on a single similarity index. We show how to build this similarity index a-priori, without knowledge about query similarity thresholds, based on recently proposed synopses for multiset operations. The index is deployed using existing disk-based inverted indexing implementations and our algorithms exploit available techniques, like skip-lists, to further optimize the query performance. The index has provably small space footprints, is orders of magnitude smaller and faster to create/incrementally maintain than exact solutions, and the algorithms provide approximate answers, with an error that is controlled by a user-specified parameter. We prove the error bounds of our algorithms analytically, and, finally, we demonstrate the performance of the algorithms and verify their accuracy experimentally.

The task of estimating the number of distinct values (DVs) in a large dataset arises in a wide variety of settings in computer science and elsewhere. We provide DV estimation techniques for the case in which the dataset of interest is split into partitions. We create for each partition a synopsis that can be used to estimate the number of DVs in the partition. By combining and extending a number of results in the literature, we obtain both suitable synopses and DV estimators. The synopses can be created in parallel, and can be easily combined to yield synopses and DV estimates for "compound" partitions that are created from the base partitions via arbitrary multiset union, intersection, or difference operations. Our synopses can also handle deletions of individual partition elements. We prove that our DV estimators are unbiased, provide error bounds, and show how to select synopsis sizes in order to achieve a desired estimation accuracy. Experiments and theory indicate tha...

Global Minimization of Indefinite Quadratic Problems. A branch and bound algorithm is proposed for finding the global optimum of large-scale indefinite quadratic problems over a polytope. The algorithm uses separable programming and techniques from concave optimization to obtain approximate solutions. Results on error bounding are given and preliminary computational results using the Cray 1 S supercomputer as reported.

Recently, a new iterative method, called Newton-Lavrentiev regularization (NLR) method, was considered by George ( ) for regularizing a nonlinear ill-posed Hammerstein-type operator equation in Hilbert spaces. In this paper we introduce a modified form of the NLR method and derive order optimal error bounds by choosing the regularization parameter according to the adaptive scheme considered by .

In this paper we consider a collocation method for solving Fredholm integral equations of the first kind, which is known to be an ill-posed problem. An "unregularized" use of this method can give reliable results in the case when the rate at which smallest singular values of the collocation matrices decrease is known a priori. In this case the number of collocation points plays the role of a regularization parameter. If the a priori information mentioned above is not available, then a combination of collocation with Tikhonov regularization can be the method of choice. We analyze such regularized collocation in a rather general setting, when a solution smoothness is given as a source condition with an operator monotone index function. This setting covers all types of smoothness studied so far in the theory of Tikhonov regularization. One more issue discussed in this paper is an a posteriori choice of the regularization parameter, which allows us to reach an optimal order of accuracy for deterministic noise model without any knowledge of solution smoothness.

For solving linear ill-posed problems regularization methods are required when the available data include some noise. In the present paper regularized approximations are obtained by a general regularization scheme in Hilbert scales which include well-known regularization methods such as the method of Tikhonov regularization and its higher-order forms, spectral methods, asymptotical regularization and iterative regularization methods. For both the cases of high-and low-order regularization, we study a priori and a posteriori rules for choosing the regularization parameter and provide order optimal error bounds that characterize the accuracy of the regularized approximations. These error bounds have been obtained under general smoothing conditions. The results extend earlier results and cover the case of finitely and infinitely smoothing operators. The theory is illustrated by a special ill-posed deconvolution problem arising in geoscience.

For solving linear ill-posed problems with noisy data regularization methods are required. We analyze a simplified regularization scheme in Hilbert scales for operator equations with nonnegative self-adjoint operators. By exploiting the operator monotonicity of certain functions, order-optimal error bounds are derived that characterize the accuracy of the regularized approximations. These error bounds have been obtained under general smoothness conditions.

We examine the long time error behavior of discontinuous Galerkin spectral element approximations to hyperbolic equations. We show that the choice of numerical flux at interior element boundaries affects the growth rate and asymptotic value of the error. Using the upwind flux, the error reaches the asymptotic value faster, and to a lower value than a central flux gives, especially for low resolution computations. The differences in the error caused by the numerical flux choice decrease as the solution becomes better resolved.

The performance enhancements achievable under signal masking such as urban canyons, indoor and under the forestry canopy with High Sensivity GPS (HSGPS) technology are first described using sample field results. HSGPS uses a longer integration time in order to utilize signals that are 25-30 dB weaker than the nominal line-of-sight GPS signals. A self-aided implementation of HSGPS that does not require external aiding from an existing network was tested. Depending on the type of signal masking, availability increases substantially. However, this occurs at the cost of increased susceptibility to interference which leads to very large measurement errors in some cases. Many tests obtained under a variety of conditions and summarized in the paper demontrate this clearly. In order to improve HSGPS overall performance (availability, accuracy, and reliability), integration with self-contained portable, preferably low cost sensors, is described. These sensors include miniature accelerometers, gyros, and six degrees of freedom IMUs. A performance analysis of vehicular and pedestrian applications of HSGPS conducted under a variety of conditions shows the advantages and limitations of self-contained sensor augmented HSGPS.

In this work we propose to assess the relation between HRV and QTV measured by an automatic delineator. A low order linear autoregressive model on RR versus QT interactions was used to explore short term relations and quantify the fractions of QTV correlated and not correlated with HRV. Power spectral density measures were estimated from the total QTV and from the two separated fractions using the proposed model. Simulated RR and QT series were used to quantify the error bounds associated to the method performance. ECG records of young normal subjects were processed to obtain the RR and QT series. The high QTV fraction not correlated with RR found i n these records (over 40% in 98% of the segments) indicates that an important part of QTV can be driven by other factors rather than by heart rate, and may contain complementary information.

A straightforward semi-implicit nite di erence method approximating a system of conservation laws including a sti relaxation term is analyzed. We show that the error, measured in L 1 , is bounded by O( p t) independent of the sti ness, where the time step t represents the mesh size. As a simple corollary we obtain that solutions of the sti system converge towards the solution of an equilibrium model at a rate of O( 1=3 ) in L 1 as the relaxation time tends to zero.

This paper presents a spectrally-weighted balanced truncation technique for RLC interconnects, when the interconnect circuit parameters change as a result of variations in the manufacturing process. The salient features of this algorithm are the inclusion of the parameter variation in the RLC interconnect, the guaranteed stability of the reduced transfer function, and the availability of provable frequency-weighted error bounds for the reduced-order system. This paper shows that the balanced truncation technique is an effective model-order reduction technique when variations in the circuit parameters are taken into consideration. Experimental results show that the new variational spectrally-weighted balanced truncation attains, on average, 20% more accuracy than the variational Krylovsubspace-based model-order reduction techniques while the run-time is also, on average, 5% faster.

This paper presents a spectrally-weighted balanced truncation technique for tightly coupled integated circuit interconnects, when the interconnect circuit parameters change as a result of statistical variations in the manufacturing process. The salient features of this algorithm are the inclusion of the parameter variation in the RLC interconnect, the guaranteed passivity of the reduced transfer function, and the availability of provable spectrally-weighted error bounds for the reduced-order system. This paper shows that the variational balanced truncation technique produces reduced systems that accurately follow the time-and frequency-domain responses of the original system when variations in the circuit parameters are taken into consideration. Experimental results show that the new variational spectrally-weighted balanced truncation attains, in average, 30% more accuracy than the variational Krylov-subspacebased model-order reduction techniques.

A thesis submitted in conformity wit h the requirements for the degree of Master of Science Graduate Department of Computer Science University of Toronto Copyright @ 2000 by Kit Sun Ng The author has gcanted a nonexclusive licence dîowing the National L i i of Canada to copies of diis thesis in microform, paper or electronic formats. The author retains ownership of the copyright in this thesis. Neither the may be printed or othawise reproduced without the author's L'auteur a accord6 me licence non exclusive parnettant A la Bibliothdque nationale du Canada de reproduire, pr&r, distribuer ou vendre des copies de cette Wse sous la forme de microfiche/iiim, de reproduction sur papier ou siu fonnat électronique. L'auteur conserve la propriété du droit d'auteur qui protège cette bitse. Ni la these ni des extraits substantiels de celle-ci ne doivent &ûe imprimes ou autrement reproduits sans son autorisation.

We develop optimal quadratic and cubic spline collocation methods for solving linear secondorder two-point boundary value problems on non-uniform partitions. To develop optimal non-uniform partition methods, we use a mapping function from uniform to non-uniform partitions and develop expansions of the error at the non-uniform collocation points of some appropriately defined spline interpolants. The existence and uniqueness of the spline collocation approximations are shown, under some conditions. Optimal global and local orders of convergence of the spline collocation approximations and derivatives are derived, similar to those of the respective methods for uniform partitions. Numerical results on a variety of problems, including a boundary layer problem, and a non-linear problem, verify the optimal convergence of the methods, even under more relaxed conditions than those assumed by theory.

We integrate optimal quadratic and cubic spline collocation methods for second-order two-point boundary value problems with adaptive grid techniques, and grid size and error estimators. Some adaptive grid techniques are based on the construction of a mapping function that maps uniform to non-uniform points, placed appropriately to minimize a certain norm of the error. One adaptive grid technique for cubic spline collocation is mapping-free and resembles the technique used in COLSYS (COLNEW) [2,. Numerical results on a variety of problems, including problems with boundary or interior layers, and singular perturbation problems indicate that, for most problems, the cubic spline collocation method requires less computational effort for the same error tolerance, and has equally reliable error estimators, when compared to Hermite piecewise cubic collocation. Comparison results with quadratic spline collocation are also presented.

This paper explores Tandem feature extraction used in a large-vocabulary speech recognition system. In this framework a multi-layer perceptron estimates phone probabilities which are treated as acoustic observations in a traditional HMM-GMM system. To determine a lower error bound, we simulated an idealized classifier based on alignment of reference transcriptions. This cheating experiment demonstrated a best-case scenario for Tandem feature extraction, highlighting the potential for dramatic system improvement. More importantly, we discovered a way to exploit the result without cheating: using the simulated classifier during training and a MLP classifier at test, the performance improved despite the mismatched Tandem features.

An integral equation representation is given for parabolic partial differential equations. When the equations are defined in unbounded domains, as in the initial value (Cauchy) problem, the solution of the integral equation by the method of successive approximation has inherent advantages over other methods. Error bounds for the method are of order h ~/2 and h ~/2 (h is the increment size) depending on the finite difference approximations involved.

Surface fluxes and soil moisture were mea&red in situ during the Echival Field Experiment in Desertification-threatened Areas (EFEDA), executed in Castilla la Mancha, Central Spain. Although the observation network had a high density (20 flux towers and 46 soil moisture plots), the area-average values at a scale of 100 km for a Mediterranean agricultural landscape were difficult to deduce. An attempt with remote sensing data has been made to study the possibility to extrapolate in situ measurements to a coarser scale. A remote sensing flux algorithm was applied to estimate (1) actual evaporation, (2) evaporative fraction, (3) bulk surface resistance and (4) top soil moisture from spectral radiances at different spatial scales. The spatial variability and area-average values were computed for two different golden days (June 12 and 29). The evaporative fraction (latent heat flux/net available energy) estimated from remote sensing was within the error bounds of the values derived from instrumented flux towers in 85% of the cases compared. The bulk surface resistance derived from the remote sensing flux algorithm could be related successfully to in situ near-surface soil moisture measurements conducted with Time Domain Reflectometers (TDR). Unfortunately, the relationship between resistance and soil moisture is shown to be space and time dependent. Nevertheless, after calibration with field scale measurements, instantaneous relationships between resistance and top soil moisture could be applied to estimate soil moisture at unsampled locations. Thereafter, the area-average top soil moisture could be assessed. It was concluded that an arithmetic means of the distributed field measurements of evaporation (n = 13) and soil moisture (n = 46) gives a wrong indication of the area-average values at a scale of 100 km, and that weighting factors for the area1 integration can be derived from remote sensing data. During a 17-day drying period in the Special Observation Period, top soil moisture in Castilla la Mancha at a scale of 100 km decreased from 0.16 to 0.10 cm3 cm-3 resulting in an increase of the area1 bulk surface resistance from 661 to 1166 s m-' . The associated daily evaporation at this scale decreased from 2.0 to 1.3 mm d-'. 0 1997 Elsevier Science B.V.

In this paper we consider the scattering of a plane acoustic or electromagnetic wave by a one-dimensional, periodic rough surface. We restrict the discussion to the case when the boundary is sound soft in the acoustic case, perfectly reflecting with TE polarization in the EM case, so that the total field vanishes on the boundary. We propose a uniquely solvable first kind integral equation formulation of the problem, which amounts to a requirement that the normal derivative of the Green's representation formula for the total field vanish on a horizontal line below the scattering surface. We then discuss the numerical solution by Galerkin's method of this (ill-posed) integral equation. We point out that, with two particular choices of the trial and test spaces, we recover the so-called SC (spectral-coordinate) and SS (spectral-spectral) numerical schemes of DeSanto et al., Waves Random Media, 8, 315-414, 1998. We next propose a new Galerkin scheme, a modification of the SS method that we term the SS * method, which is an instance of the well-known dual least squares Galerkin method. We show that the SS * method is always well-defined and is optimally convergent as the size of the approximation space increases. Moreover, we make a connection with the classical least squares method, in which the coefficients in the Rayleigh expansion of the solution are determined by enforcing the boundary condition in a least squares sense, pointing out that the linear system to be solved in the SS * method is identical to that in the least squares method. Using this connection we show that (reflecting the ill-posed nature of the integral equation solved) the condition number of the linear system in the SS * and least squares methods approaches infinity as the approximation space increases in size. We also provide theoretical error bounds on the condition number and on the errors induced in the numerical solution computed as a result of ill-conditioning. Numerical results confirm the convergence of the SS * method and illustrate the ill-conditioning that arises.

The stabilized version of the sequential quadratic programming algorithm (sSQP) had been developed in order to achieve fast convergence despite possible degeneracy of constraints of optimization problems, when the Lagrange multipliers associated to a solution are not unique. Superlinear convergence of sSQP had been previously established under the strong second-order sufficient condition for optimality (without any constraint qualification assumptions). We prove a stronger superlinear convergence result than the above, assuming the usual second-order sufficient condition only. In addition, our analysis is carried out in the more general setting of variational problems, for which we introduce a natural extension of sSQP techniques. In the process, we also obtain a new error bound for Karush-Kuhn-Tucker systems for variational problems that holds under an appropriate second-order condition.

This note is concerned with estimates for the remainder term of the Gauss-Turán quadrature formula, where w(t) = (U n-1 (t)/n) 2 1t 2 is the Gori-Michelli weight function, with U n-1 (t) denoting the (n -1)th degree Chebyshev polynomial of the second kind, and f is a function analytic in the interior of and continuous on the boundary of an ellipse with foci at the points ±1 and sum of semiaxes > 1. The present paper generalizes the results in [G.V. Milovanović, M.M. Spalević, Bounds of the error of Gauss-Turán-type quadratures, J. Comput. Appl. Math. 178 (2005) 333-346], which is concerned with the same problem when s = 1.

For analytic functions the remainder term of Gaussian quadrature formula and its Kronrod extension can be represented as a contour integral with a complex kernel. We study these kernels on elliptic contours with foci at the points ±1 and the sum of semi-axes > 1 for the Chebyshev weight functions of the first, second and third kind, and derive representation of their difference. Using this representation and following Kronrod's method of obtaining a practical error estimate in numerical integration, we derive new error estimates for Gaussian quadratures.

For analytic functions the remainder term of Gauss-Radau quadrature formulae can be represented as a contour integral with a complex kernel. We study the kernel on elliptic contours with foci at the points ±1 and a sum of semi-axes > 1 for the Chebyshev weight function of the second kind. Starting from explicit expressions of the corresponding kernels the location of their maximum modulus on ellipses is determined. The corresponding Gautschi's conjecture from [On the remainder term for analytic functions of Gauss-Lobatto and Gauss-Radau quadratures, Rocky Mountain J. Math. 21 (1991), 209-226] is proved.

We analyze the error introduced by approximately calculating the s-dimensional Lebesgue measure of a Jordan-measurable subset of I s = [0, 1) s . We give an upper bound for the error of a method using a (t, m, s)-net, which is a set with a very regular distribution behavior. When the subset of I s is defined by some function of bounded variation on Īs-1 , the error is estimated by means of the variation of the function and the discrepancy of the point set which is used. A sharper error bound is established when a (t, m, s)-net is used. Finally a lower bound of the error is given, for a method using a (0, m, s)-net. The special case of the 2-dimensional Hammersley point set is discussed.

We propose a family of numerical schemes to solve the initial value problem for a system of differential equations y (t) = f (t, y(t)) in which f is smooth in space (y), but only of bounded variation in time (t). The family is akin to the Runge-Kutta family. However, the discretization with respect to time only retains mean properties. The means are estimated with Monte Carlo simulation. We analyze first and second order methods which use quasi-random point sets for the simulation. Error bounds are derived which involve powers of the step size as well as the discrepancy of the point set used for quasi-Monte Carlo approximation. The numerical results indicate that by using quasi-random points in place of pseudo-random points we are able to obtain smaller errors.

We propose a family of numerical schemes to solve the initial value problem for a system of differential equations y (t) = f (t, y(t)) in which f is smooth in space (y), but only of bounded variation in time (t). The family is akin to the Runge-Kutta family. However, the discretization with respect to time only retains mean properties. The means are estimated with Monte Carlo simulation. We analyze first and second order methods which use quasi-random point sets for the simulation. Error bounds are derived which involve powers of the step size as well as the discrepancy of the point set used for quasi-Monte Carlo approximation. The numerical results indicate that by using quasi-random points in place of pseudo-random points we are able to obtain smaller errors.

The most important asset of a Massively Multiplayer Online Game is its world state, as it represents the combined efforts and progress of all its participants. Thus, it is extremely important that this state is not lost in case of server failures. Survival of the world state is typically achieved by making it persistent, e.g., by storing it in a relational database. The main challenge of this approach is to track the large volume of modifications applied to the world in real time. This paper compares a variety of strategies to persist changes of the game world. While critical events must be written synchronously to the persistent storage, a set of approximation strategies are discussed and compared that are suitable for events with low consistency requirements, such as player movements. An analysis to better understand the possible limitations and bottlenecks of these strategies is presented using experimental data from an MMOG research framework. Our analysis shows that a distance-based solution offers the scalability and efficiency required for large-scale games as well as offering error bounds and eliminating unnecessary updates associated with localized movement.