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We describe a subdivision algorithm for isolating the complex roots of a polynomial F ∈ C [ x ] . Given an oracle that provides approximations of each of the coefficients of F to any absolute error bound and given an arbitrary square B in the complex plane containing only simple roots of F , our algorithm returns disjoint isolating disks for the roots of F in B . Our complexity analysis bounds the absolute error to which the coefficients of F have to be provided, the total number of iterations, and the overall bit complexity. It further shows that the complexity of our algorithm is controlled by the geometry of the roots in a near neighborhood of the input square B , namely, the number of roots, their absolute values and pairwise distances. The number of subdivision steps is near-optimal. For the benchmark problem , namely, to isolate all the roots of a polynomial of degree n with integer coefficients of bit size less than τ , our algorithm needs O ˜ ( n 3 + n 2 τ ) bit operations, which is comparable to the record bound of Pan (2002) . It is the first time that such a bound has been achieved using subdivision methods, and independent of divide-and-conquer techniques such as Schönhage's splitting circle technique. Our algorithm uses the quadtree construction of Weyl (1924) with two key ingredients: using Pellet's Theorem (1881) combined with Graeffe iteration, we derive a “soft-test” to count the number of roots in a disk. Using Schröder's modified Newton operator combined with bisection, in a form inspired by the quadratic interval method from Abbot (2006), we achieve quadratic convergence towards root clusters. Relative to the divide-conquer algorithms, our algorithm is quite simple with the potential of being practical. This paper is self-contained: we provide pseudo-code for all subroutines used by our algorithm. [ABSTRACT FROM AUTHOR]

Hybrid polynomial correlated function expansion (H-PCFE) is a novel metamodel formulated by coupling polynomial correlated function expansion (PCFE) and Kriging. Unlike commonly available metamodels, H-PCFE performs a bi-level approximation and hence, yields more accurate results. However, till date, it is only applicable to medium scaled problems. In order to address this apparent void, this paper presents an improved H-PCFE, referred to as locally refined hp – adaptive H-PCFE. The proposed framework computes the optimal polynomial order and important component functions of PCFE, which is an integral part of H-PCFE, by using global variance based sensitivity analysis. Optimal number of training points are selected by using distribution adaptive sequential experimental design. Additionally, the formulated model is locally refined by utilizing the prediction error, which is inherently obtained in H-PCFE. Applicability of the proposed approach has been illustrated with two academic and two industrial problems. To illustrate the superior performance of the proposed approach, results obtained have been compared with those obtained using hp – adaptive PCFE. It is observed that the proposed approach yields highly accurate results. Furthermore, as compared to hp – adaptive PCFE, significantly less number of actual function evaluations are required for obtaining results of similar accuracy. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a development of the direction splitting algorithm for flows in complex geometries proposed in [2] to the case of flows containing rigid particles. The main novelty of this method is that the grid is fit to the time-dependent domain at each time step which allows for an optimal spatial approximation of the Navier–Stokes equations. In general, this is a very hard task for multidimensional problems. However, it is significantly simplified if the momentum equations are discretized by a direction splitting scheme. Such a scheme requires only solving a set of one-dimensional problems, and the grid can be easily fit to the boundary in the direction of each of these problems. Here we use a MAC discretization stencil but the same idea can be applied to other discretizations. The equations of motion of each particle are discretized explicitly and the computed particle velocities are imposed as Dirichlet boundary conditions for the Navier–Stokes equations on the adapted grid. The pressure is extended within the particles in a fictitious domain fashion, but the divergence stencil is fit to the particle boundaries. In this paper, the direction splitting algorithm with boundary fitting is compared to pure fictitious domain methods. [Copyright &y& Elsevier]

In this paper we introduce a general framework that exponentially improves the space, degree of independence, and time needed by min-wise-based algorithms. The authors, in SODA '11 [1] , introduced an exponential time improvement for min-wise-based algorithms. Here we develop an alternative approach that achieves both exponential time and exponential space improvement. The new approach relaxes the need for approximately min-wise hash functions, hence getting around the Ω ( log ⁡ 1 ϵ ) independence lower bound, by defining and constructing a d-k-min-wise independent family of hash functions; surprisingly, for most cases, only 8-wise independence is needed for the additional improvement. Furthermore, we discuss how this construction can be used to improve many min-wise-based algorithms. To our knowledge such definitions, for hash functions, were never previously studied or constructed. Finally, we show how to apply it for similarity and rarity estimation over data streams; other min-wise-based algorithms can be adjusted in the same way. [ABSTRACT FROM AUTHOR]

The blocking probability to the path requests is an important issue in flexible bandwidth optical communications. In this paper, we propose a blocking probability approximation method of path requests in flexi-grid networks. It models the bundled neighboring carrier allocation with a group of birth–death processes and provides a theoretical analysis to the blocking probability under variable bandwidth traffic. The numerical results show the effect of traffic parameters to the blocking probability of path requests. We use the first fit algorithm in network nodes to allocate neighboring carriers to path requests in simulations, and verify approximation results. [ABSTRACT FROM AUTHOR]

In this paper we establish an algorithmic framework between bin packing and strip packing, with which strip packing can be very well approximated by applying some bin packing algorithms. More precisely we obtain the following results: (1) Any off-line bin packing algorithm can be applied to strip packing maintaining almost the same asymptotic worst-case ratio. (2) A class of Harmonic-based algorithms for bin packing, such as Refined Harmonic, Modified Harmonic, Harmonic++, can be applied to online strip packing. In particular, we show that online strip packing admits an upper bound of 1.58889 + ϵ on the asymptotic competitive ratio, for any arbitrarily small ϵ > 0 . This significantly improves the previously best bound of 1.6910 and affirmatively answers an open question posed by Csirik and Woeginger (1997). Moreover, the time complexity mainly depends on a sorting procedure and the bin packing algorithms employed. [ABSTRACT FROM AUTHOR]

Abstract: Simulation of charged-particle beam devices using particle-in-cell methods in (r, z) cylindrical coordinates can be extremely efficient, incorporating a majority of the physics and yielding high resolution with minimal computational costs. However, accuracy requires adjustment of solution coefficients and particle/current weights with radius, especially near the cylindrical axis. Prior algorithms have been based on special models and approximations with no consistent method for computing coefficients in general cases. The Local Taylor Polynomial (LTP) technique presented in this paper provides a general framework for determination of coefficients and development of PDE solution algorithms. The LTP method employs a local (polynomial) solution of the continuum PDE to construct numerical algorithms. As a result, LTP systematically generates accurate coefficients and incorporates fields and sources on an equal basis. This paper focuses on the scalar Poisson PDE in cylindrical coordinates to illustrate the LTP technique, coefficient derivation, electric field calculation and handling of space charge effects. Application of LTP to non-conformal boundaries and non-uniform grids is presented. Although focused on cylindrical coordinates in this paper, the LTP technique has general applicability to other PDE’s and geometries, and is broadly applicable to problems in beam/field simulation. [Copyright &y& Elsevier]

Abstract: Trigonometric non-Fourier moment problems arise as a result of various control problem study. In current paper, the extremal solution, i.e. the one with the least -norm is searched for. Proposed is an algorithm that allows to change an infinite system of equations into the linear one with only a finite number of equations. The mentioned algorithm is based on the fact, that in the case of a Fourier moment problem, the extremal solution is periodic and easy to construct. The extremal solution of a non-Fourier moment problem close to a Fourier one is approximated by a sequence of solutions with periodicity disturbed in a finite number of equations. It is proved that this sequence of approximations converges to the desired extremal solution. The paper is concluded with the particular example whose consideration leads to a moment problem elaborated in the first part of the article. [Copyright &y& Elsevier]

Abstract: This is a follow-up paper of “Liberating the dimension for function approximation”, where we studied approximation of infinitely variate functions by algorithms that use linear information consisting of finitely many linear functionals. In this paper, we study similar approximation problems, however, now the algorithms can only use standard information consisting of finitely many function values. We assume that the cost of one function value depends on the number of active variables. We focus on polynomial tractability, and occasionally also study weak tractability. We present non-constructive and constructive results. Non-constructive results are based on known relations between linear and standard information for finitely variate functions, whereas constructive results are based on Smolyak’s construction generalized to the case of infinitely variate functions. Surprisingly, for many cases, the results for standard information are roughly the same as for linear information. [Copyright &y& Elsevier]

Abstract: Recovering missing data from its partial samples is a fundamental problem in mathematics and it has wide range of applications in image and signal processing. While many such algorithms have been developed recently, there are very few papers available on their error estimations. This paper is to analyze the error of a frame based data recovery approach from random samples. In particular, we estimate the error between the underlying original data and the approximate solution that interpolates (or approximates with an error bound depending on the noise level) the given data that has the minimal norm of the canonical frame coefficients among all the possible solutions. [Copyright &y& Elsevier]

Abstract: This paper works on joint approximate diagonalization of simplified fourth order cumulant matrices for very fast and large scale blind separation of instantaneous mixing model sources. The JADE algorithm is widely accepted but only limited to small scale separation tasks. The SHIBBS algorithm calculates a fraction of the fourth order cumulant set and avoids eigenmatrix decomposition to reduce calculation cost. However, it was seen to be slower than JADE at the time of its first publication and is hence less known. On the other hand, the SJAD algorithm using the same approach is shown to be very fast. This paper studies the iteration convergence criterion and proposes to use a signal to noise ratio based iteration stopping threshold approach. The improved SHIBBS/SJAD algorithm is very fast, and capable of large scale separation. Experimental separation comparisons between the SHIBBS/SJAD and FastICA are presented. [Copyright &y& Elsevier]

Abstract: This paper describes a statistically motivated framework for performing real-time dialogue state updates and policy learning in a spoken dialogue system. The framework is based on the partially observable Markov decision process (POMDP), which provides a well-founded, statistical model of spoken dialogue management. However, exact belief state updates in a POMDP model are computationally intractable so approximate methods must be used. This paper presents a tractable method based on the loopy belief propagation algorithm. Various simplifications are made, which improve the efficiency significantly compared to the original algorithm as well as compared to other POMDP-based dialogue state updating approaches. A second contribution of this paper is a method for learning in spoken dialogue systems which uses a component-based policy with the episodic Natural Actor Critic algorithm. The framework proposed in this paper was tested on both simulations and in a user trial. Both indicated that using Bayesian updates of the dialogue state significantly outperforms traditional definitions of the dialogue state. Policy learning worked effectively and the learned policy outperformed all others on simulations. In user trials the learned policy was also competitive, although its optimality was less conclusive. Overall, the Bayesian update of dialogue state framework was shown to be a feasible and effective approach to building real-world POMDP-based dialogue systems. [Copyright &y& Elsevier]

Abstract: In this paper, we focus on the edit distance between two given strings where block-edit operations are allowed and better fitting to the human natural edit behaviors. Previous results showed that this problem is NP-hard when block moves are allowed. Various approximations to this problem have been proposed in recent years. However, this problem can be solved by the polynomial-time optimization algorithms if some reasonable restrictions are applied. The restricted variations which we consider involve character insertions, character deletions, block copies and block deletions. In this paper, three problems are defined with different measuring functions, which are , and . Then we show that with some preprocessing, the minimum block edit distances of these three problems can be obtained by dynamic programming in , and time, respectively, where n and m are the lengths of the two input strings. [Copyright &y& Elsevier]

Abstract: The cyclic projections algorithm is an important method for determining a point in the intersection of a finite number of closed convex sets in a Hilbert space. That is, for determining a solution to the “convex feasibility” problem. This is the third paper in a series on a study of the rate of convergence for the cyclic projections algorithm. In the first of these papers, we showed that the rate could be described in terms of the “angles” between the convex sets involved. In the second, we showed that these angles often had a more tractable formulation in terms of the “norm” of the product of the (nonlinear) metric projections onto related convex sets. In this paper, we show that the rate of convergence of the cyclic projections algorithm is also intimately related to the “linear regularity property” of Bauschke and Borwein, the “normal property” of Jameson (as well as Bakan, Deutsch, and Li’s generalization of Jameson’s normal property), the “strong conical hull intersection property” of Deutsch, Li, and Ward, and the rate of convergence of iterated parallel projections. Such properties have already been shown to be important in various other contexts as well. [Copyright &y& Elsevier]

Abstract: In a previous paper, the author introduced a class of multivariate rational interpolants, which are called optimal Padé-type approximants (OPTA). The main goal of this paper is to extend classical results on convergence both in measure and in capacity of sequences of Padé approximants to the multivariate case using OPTA. To this end, we obtain some estimations of the size of multivariate polynomial lemniscates in terms of the Hausdorff content, which we also think are of some interest. [Copyright &y& Elsevier]

This paper considers a modification of a PAC learning theory problem in which each instance of the training data is supplemented with side information. In this case, a transformation, given by a side-information map, of the training instance is also classified. However, the learning algorithm needs only to classify a new instance, not the instance and its value under the side information map. Side information can improve general learning rates, but not always. This paper shows that side information leads to the improvement of standard PAC learning theory rate bounds, under restrictions on the probable overlap between concepts and their images under the side information map. [Copyright &y& Elsevier]

Kernelization algorithms in the context of Parameterized Complexity are often based on a combination of data reduction rules and combinatorial insights. We will expose in this paper a similar strategy for obtaining polynomial-time approximation algorithms. Our method features the use of approximation-preserving reductions, akin to the notion of parameterized reductions. We exemplify this method to obtain the currently best approximation algorithms for Harmless Set , Differential and Multiple Nonblocker , all of them can be considered in the context of securing networks or information propagation. [ABSTRACT FROM AUTHOR]

Computing the roots of a univariate polynomial is a fundamental and long-studied problem of computational algebra with applications in mathematics, engineering, computer science, and the natural sciences. For isolating as well as for approximating all complex roots, the best algorithm known is based on an almost optimal method for approximate polynomial factorization, introduced by Pan in 2002. Pan's factorization algorithm goes back to the splitting circle method from Schönhage in 1982. The main drawbacks of Pan's method are that it is quite involved 2 2 In Victor Pan's own words: “Our algorithms are quite involved, and their implementation would require a non-trivial work, incorporating numerous known implementation techniques and tricks”. In fact, we are not aware of any implementation of Pan's method. and that all roots have to be computed at the same time. For the important special case, where only the real roots have to be computed, much simpler methods are used in practice; however, they considerably lag behind Pan's method with respect to complexity. In this paper, we resolve this discrepancy by introducing a hybrid of the Descartes method and Newton iteration, denoted A New Dsc , which is simpler than Pan's method, but achieves a run-time comparable to it. Our algorithm computes isolating intervals for the real roots of any real square-free polynomial, given by an oracle that provides arbitrary good approximations of the polynomial's coefficients. A New Dsc can also be used to only isolate the roots in a given interval and to refine the isolating intervals to an arbitrary small size; it achieves near optimal complexity for the latter task. [ABSTRACT FROM AUTHOR]

This paper presents an in-depth analysis on the accuracy and performance of the windowed multipole Doppler broadening method. The basic theory behind cross section data is described, along with the basic multipole formalism followed by the approximations leading to windowed multipole method and the algorithm used to efficiently evaluate Doppler broadened cross sections. The method is tested by simulating the BEAVRS benchmark with a windowed multipole library composed of 70 nuclides. Accuracy of the method is demonstrated on a single assembly case where total neutron production rates and U 238 capture rates compare within 0.1% to ACE format files at the same temperature. With regards to performance, clock cycle counts and cache misses were measured for single temperature ACE table lookup and for windowed multipole. The windowed multipole method was found to require 39.6% more clock cycles to evaluate, translating to a 7.9% performance loss overall. However, the algorithm has significantly better last-level cache performance, with 3 fewer misses per evaluation, or a 65% reduction in last-level misses. This is due to the small memory footprint of the windowed multipole method and better memory access pattern of the algorithm. [ABSTRACT FROM AUTHOR]

The most common method to compute wedgelet transform is based on exhaustive search which requires assessment of all wedgelet atoms and hence is prohibitively slow. In this paper, the discontinuous edge model of the wedgelet is replaced by a continuous differentiable model to make gradient based estimation of the edge parameters possible. The proposed estimation based approach contributes to a surprisingly fast algorithm called Iterative Wedgelet Transform , which leads to a better tradeoff between quality and computation speed compared to the existing approaches. Performance of the proposed method is studied in the image approximation task and the results are compared with two other major algorithms, namely classical Wedgelet Transform and Moments-based Wedgelet Transform . The iterative algorithm is shown to be much faster than classical wedgelet transform, O ( N 2 log 2 N ) given an N × N image; and compared to the moments-based method exhibits higher precision. [ABSTRACT FROM AUTHOR]

In this paper, we present algorithms for the approximation of multivariate periodic functions by trigonometric polynomials. The approximation is based on sampling of multivariate functions on rank-1 lattices. To this end, we study the approximation of periodic functions of a certain smoothness. Our considerations include functions from periodic Sobolev spaces of generalized mixed smoothness. Recently an algorithm for the trigonometric interpolation on generalized sparse grids for this class of functions was investigated by Griebel and Hamaekers (2014). The main advantage of our method is that the algorithm is based mainly on a single one-dimensional fast Fourier transform, and that the arithmetic complexity of the algorithm depends only on the cardinality of the support of the trigonometric polynomial in the frequency domain. Therefore, we investigate trigonometric polynomials with frequencies supported on hyperbolic crosses and energy norm based hyperbolic crosses in more detail. Furthermore, we present an algorithm for sampling multivariate functions on perturbed rank-1 lattices and show the numerical stability of the suggested method. Numerical results are presented up to dimension d = 10 , which confirm the theoretical findings. [ABSTRACT FROM AUTHOR]

This paper presents a new non-overlapping domain decomposition method for the time harmonic Maxwell's equations, whose effective convergence is quasi-optimal. These improved properties result from a combination of an appropriate choice of transmission conditions and a suitable approximation of the Magnetic-to-Electric operator. A convergence theorem of the algorithm is established and numerical results validating the new approach are presented. [ABSTRACT FROM AUTHOR]

In this paper, we firstly develop two high-order approximate formulas for the Riesz fractional derivative. Secondly, we propose a temporal second order numerical method for a fractional reaction-dispersion equation, where we discretize the Riesz fractional derivative by using two numerical schemes. We prove that the numerical methods for a spatial Riesz fractional reaction dispersion equation are both unconditionally stable and convergent, and the orders of convergence are O ( τ 2 + h 6 ) and O ( τ 2 + h 8 ) , in which τ and h are spatial and temporal step sizes, respectively. Finally, we test our numerical schemes and observe that the numerical results are in good agreement with the theoretical analysis. [ABSTRACT FROM AUTHOR]

Fractional calculus allows variable-order of fractional operators, which can be exploited in diverse physical and biological applications where rates of change of the quantity of interest may depend on space and/or time. In this paper, we derive two second-order approximation formulas for the variable-order fractional time derivatives involved in anomalous diffusion and wave propagation. We then present numerical tests that verify the theoretical estimates of convergence rate and also simulations of anomalous sub-diffusion and super-diffusion that demonstrate new localized diffusion rates that depend on the curvature of the variable-order function. Finally, we perform simulations of wave propagation in a truncated domain to demonstrate how erroneous wave reflections at the boundaries can be eliminated by super-diffusion, and also simulations of the Burgers equation that serve as a testbed for studying the loss and recovery of monotonicity using again variable rate diffusion as a function of space and/or time. Taken together, our results demonstrate that variable-order fractional derivatives can be used to model the physics of anomalous transport with spatiotemporal variability but also as new effective numerical tools that can deal with the long-standing issues of outflow boundary conditions and monotonicity of integer-order PDEs. [ABSTRACT FROM AUTHOR]

Fourier extensions have been shown to be an effective means for the approximation of smooth, nonperiodic functions on bounded intervals given their values on an equispaced, or in general, scattered grid. Related to this method are two parameters. These are the extension parameter T (the ratio of the size of the extended domain to the physical domain) and the oversampling ratio η (the number of sampling nodes per Fourier mode). The purpose of this paper is to investigate how the choice of these parameters affects the accuracy and stability of the approximation. Our main contribution is to document the following interesting phenomenon: namely, if the desired condition number of the algorithm is fixed in advance, then the particular choice of such parameters makes little difference to the algorithm's accuracy. As a result, one is free to choose T without concern that it is suboptimal. In particular, one may use the value T=2 - which corresponds to the case where the extended domain is precisely twice the size of the physical domain - for which there is known to be a fast algorithm for computing the approximation. In addition, we determine the resolution power (points-per-wavelength) of the approximation to be equal to Tη, and address the trade-off between resolution power and stability. [ABSTRACT FROM AUTHOR]

The Douglas-Rachford splitting algorithm is a classical optimization method that has found many applications. When specialized to two normal cone operators, it yields an algorithm for finding a point in the intersection of two convex sets. This method for solving feasibility problems has attracted a lot of attention due to its good performance even in nonconvex settings. In this paper, we consider the Douglas-Rachford algorithm for finding a point in the intersection of two subspaces. We prove that the method converges strongly to the projection of the starting point onto the intersection. Moreover, if the sum of the two subspaces is closed, then the convergence is linear with the rate being the cosine of the Friedrichs angle between the subspaces. Our results improve upon existing results in three ways: First, we identify the location of the limit and thus reveal the method as a best approximation algorithm; second, we quantify the rate of convergence, and third, we carry out our analysis in general (possibly infinite-dimensional) Hilbert space. We also provide various examples as well as a comparison with the classical method of alternating projections. [ABSTRACT FROM AUTHOR]

Abstract: This article considers recovery of signals that are sparse or approximately sparse in terms of a (possibly) highly overcomplete and coherent tight frame from undersampled data corrupted with additive noise. We show that the properly constrained -analysis optimization problem, called analysis Dantzig selector, stably recovers a signal which is nearly sparse in terms of a tight frame provided that the measurement matrix satisfies a restricted isometry property adapted to the tight frame. As a special case, we consider the Gaussian noise. Further, under a sparsity scenario, with high probability, the recovery error from noisy data is within a log-like factor of the minimax risk over the class of vectors which are at most s sparse in terms of the tight frame. Similar results for the analysis LASSO are shown. The above two algorithms provide guarantees only for noise that is bounded or bounded with high probability (for example, Gaussian noise). However, when the underlying measurements are corrupted by sparse noise, these algorithms perform suboptimally. We propose new algorithms for reconstructing signals that are nearly sparse in terms of a tight frame in the presence of bounded noise combined with sparse noise, and present corresponding recovery guarantees. The analysis in this paper is based on the restricted isometry property adapted to a tight frame, which is a natural extension to the standard restricted isometry property. [Copyright &y& Elsevier]

Abstract: Recently, a fast alternating minimization algorithm for total variation image deblurring (FTVd) has been presented by Wang, Yang, Yin, and Zhang (2008) [32]. The method in a nutshell consists of a discrete Fourier transform-based alternating minimization algorithm with periodic boundary conditions and in which two fast Fourier transforms (FFTs) are required per iteration. In this paper, we propose an alternating minimization algorithm for the continuous version of the total variation image deblurring problem. We establish convergence of the proposed continuous alternating minimization algorithm. The continuous setting is very useful to have a unifying representation of the algorithm, independently of the discrete approximation of the deconvolution problem, in particular concerning the strategies for dealing with boundary artifacts. Indeed, an accurate restoration of blurred and noisy images requires a proper treatment of the boundary. A discrete version of our continuous alternating minimization algorithm is obtained following two different strategies: the imposition of appropriate boundary conditions and the enlargement of the domain. The first one is computationally useful in the case of a symmetric blur, while the second one can be efficiently applied for a nonsymmetric blur. Numerical tests show that our algorithm generates higher quality images in comparable running times with respect to the Fast Total Variation deconvolution algorithm. [Copyright &y& Elsevier]

Abstract: This paper presents a preconditioner for non-overlapping Schwarz methods applied to the Helmholtz problem. Starting from a simple analytic example, we show how such a preconditioner can be designed by approximating the inverse of the iteration operator for a layered partitioning of the domain. The preconditioner works by propagating information globally by concurrently sweeping in both directions over the subdomains, and can be interpreted as a coarse grid for the domain decomposition method. The resulting algorithm is shown to converge very fast, independently of the number of subdomains and frequency. The preconditioner has the advantage that, like the original Schwarz algorithm, it can be implemented as a matrix-free routine, with no additional preprocessing. [Copyright &y& Elsevier]

Abstract: In this paper, we present an algebraic perspective of the de Rham transform of a binary subdivision scheme and propose an elegant strategy for constructing dual -ary approximating subdivision schemes of de Rham-type, starting from two primal schemes of arity and 2, respectively. On the one hand, this new strategy allows us to show that several existing dual corner-cutting subdivision schemes fit into a unified framework. On the other hand, the proposed strategy provides a straightforward algorithm for constructing new dual subdivision schemes having higher smoothness and higher polynomial reproduction capabilities with respect to the two given primal schemes. [Copyright &y& Elsevier]

Abstract: In this paper, an algorithm is given for determining the topology of an algebraic space curve and to compute a certified rational parametric approximation of the algebraic space curve. The algorithm works by extending to dimension one the local generic position method for solving zero-dimensional polynomial equation systems. Here, certified means that the approximation curve and the original curve have the same topology and their Hausdorff distance is smaller than a given precision. The main advantage of the algorithm, inherited from the local generic position method, is that the topology computation and approximation for a space curve are directly reduced to the same tasks for two plane curves. In particular, an error bound of the approximation space curve is deduced explicitly from the error bounds of the approximation plane curves. The complexity of the algorithm is also analyzed. Its effectivity is shown on some non-trivial examples. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate a non-iterative imaging algorithm based on the topological derivative in order to retrieve the shape of penetrable electromagnetic inclusions when their dielectric permittivity and/or magnetic permeability differ from those in the embedding (homogeneous) space. The main objective is the imaging of crack-like thin inclusions, but the algorithm can be applied to arbitrarily shaped inclusions. For this purpose, we apply multiple time-harmonic frequencies and normalize the topological derivative imaging function by its maximum value. In order to verify its validity, we apply it for the imaging of two-dimensional crack-like thin electromagnetic inhomogeneities completely hidden in a homogeneous material. Corresponding numerical simulations with noisy data are performed for showing the efficacy of the proposed algorithm. [Copyright &y& Elsevier]

Abstract: In this paper we propose a variation of the soft-thresholding algorithm for finding sparse approximate solutions of the equation , where as the sparsity of the iterate increases the penalty function changes. In this approach, sufficiently large entries in a sparse iterate are left untouched. The advantage of this approach is that a higher regularization constant can be used, leading to a significant reduction of the total number of iterations. Numerical experiments for sparse recovery problems, also with noisy data, are included. [Copyright &y& Elsevier]

Abstract: The computation of approximate Gröbner bases is reported to be highly unstable in the literature. Selecting a suitable length of floats is helpful for stabilizing this computation. In this paper, we present a method to compute such a suitable length of floats. We concentrate on a family of polynomial systems sharing the same support and study the relation between the lengths of floats and the coefficients of relevant polynomials. Then we give a reference length of floats for all the polynomial systems in the family. One feature of our method is that it need not utilize numerical algorithms of Gröbner bases. Hence, our method can avoid the influence of the instabilities of the existing numerical algorithms and thus will be helpful for designing stable ones (e.g., stable Shirayanagiʼs algorithm). Experiments show that our method can work out reliable and reasonably large lengths of floats for most of the tested benchmarks. [Copyright &y& Elsevier]

Abstract: In this paper we present a technique for shape reconstruction based on the topological and shape gradients. The shape in consideration is a solution of an inverse conductivity problem. To solve such a problem numerically, we compute the topological gradient of a Kohn–Vogelius-type cost function when the domain under consideration is perturbed by the introduction of a small inclusion instead of a hole. The reconstruction is done by considering the shape as a superposition of very thin elliptic inclusions to get a first approximation. Then, we use a gradient-type algorithm to perform a good reconstruction. Various numerical experiments of single and multiple inclusions demonstrate the viability of the designed algorithm. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate -average sampling numbers of a periodic Sobolev space with a Gaussian measure in the metric for and , and obtain their asymptotic orders. Moreover, we show that in the average case setting, the operators , which are the Lagrange interpolating operators, are asymptotically optimal in the metric for all . It is interesting to note that in the worst case setting, are not asymptotically optimal algorithms in the metric for or . [Copyright &y& Elsevier]

Abstract: Regularization schemes with an -regularizer often produce sparse representations for objects in approximation theory, image processing, statistics and learning theory. In this paper, we study a kernel-based learning algorithm for regression generated by regularization schemes associated with the -regularizer. We show that convergence rates of the learning algorithm can be independent of the dimension of the input space of the regression problem when the kernel is smooth enough. This confirms the effectiveness of the learning algorithm. Our error analysis is carried out by means of an approximation theory approach using a local polynomial reproduction formula and the norming set condition. [Copyright &y& Elsevier]

Abstract: In this paper modified variants of the sparse Fourier transform algorithms from Iwen (2010) [32] are presented which improve on the approximation error bounds of the original algorithms. In addition, simple methods for extending the improved sparse Fourier transforms to higher dimensional settings are developed. As a consequence, approximate Fourier transforms are obtained which will identify a near-optimal k-term Fourier series for any given input function, , in time (neglecting logarithmic factors). Faster randomized Fourier algorithm variants with runtime complexities that scale linearly in the sparsity parameter k are also presented. [Copyright &y& Elsevier]

Abstract: The solution of time-dependent PDE-constrained optimization problems subject to unsteady flow equations presents a challenge to both algorithms and computing power. In this paper we present an all-at-once approach where we solve for all time-steps of the discretized unsteady Stokes problem at once. The most desirable feature of this approach is that for all steps of an iterative scheme we only need approximate solutions of the discretized Stokes operator. This leads to an efficient scheme which exhibits mesh-independent behaviour. [Copyright &y& Elsevier]

Abstract: In this paper the two-dimensional -fraction with independent variables which is the generalization of the continued -fraction is considered. The correspondence between the formal double power series and the above mentioned fraction is studied. As a result the algorithm for the expansion of the formal double power series into the corresponding two-dimensional -fraction with independent variables has been constructed and the conditions for the existence of such an algorithm have been established. The expansion of some functions into the corresponding two-dimensional -fraction with independent variables is constructed and the efficiency of the obtained expansion by using approximants is shown. [Copyright &y& Elsevier]

Abstract: Let be a separable polynomial over a local field. The Montes algorithm computes certain approximations to the different irreducible factors of , with strong arithmetic properties. In this paper, we develop an algorithm to improve any one of these approximations, till a prescribed precision is attained. The most natural application of this “single-factor lifting” routine is to combine it with the Montes algorithm to provide a fast polynomial factorization algorithm. Moreover, the single-factor lifting algorithm may be applied as well to accelerate the computational resolution of several global arithmetic problems in which the improvement of an approximation to a single local irreducible factor of a polynomial is required. [Copyright &y& Elsevier]

Abstract: A three-body potential function can account for interactions among triples of particles which are uncaptured by pairwise interaction functions such as Coulombic or Lennard-Jones potentials. Likewise, a multibody potential of order n can account for interactions among n-tuples of particles uncaptured by interaction functions of lower orders. To date, the computation of multibody potential functions for a large number of particles has not been possible due to its scaling cost. In this paper we describe a fast tree-code for efficiently approximating multibody potentials that can be factorized as products of functions of pairwise distances. For the first time, we show how to derive a Barnes–Hut type algorithm for handling interactions among more than two particles. Our algorithm uses two approximation schemes: (1) a deterministic series expansion-based method; (2) a Monte Carlo-based approximation based on the central limit theorem. Our approach guarantees a user-specified bound on the absolute or relative error in the computed potential with an asymptotic probability guarantee. We provide speedup results on a three-body dispersion potential, the Axilrod–Teller potential. [Copyright &y& Elsevier]

Abstract: The error function erf is a special function. It is widely used in statistical computations for instance, where it is also known as the standard normal cumulative probability. The complementary error function is defined as . In this paper, the computation of and in arbitrary precision is detailed: our algorithms take as input a target precision and deliver approximate values of or with a relative error guaranteed to be bounded by . We study three different algorithms for evaluating erf and erfc. These algorithms are completely detailed. In particular, the determination of the order of truncation, the analysis of roundoff errors and the way of choosing the working precision are presented. The scheme used for implementing erf and erfc and the proofs are expressed in a general setting, so they can directly be reused for the implementation of other functions. We have implemented the three algorithms and studied experimentally what is the best algorithm to use in function of the point x and the target precision . [Copyright &y& Elsevier]

Abstract: This paper addresses the problem of scheduling non-preemptive moldable tasks to minimize the stretch of the tasks in an online non-clairvoyant setting. To the best of the authors’ knowledge, this problem has never been studied before. To tackle this problem, first the sequential subproblem is studied through the lens of the approximation theory. An algorithm, called DASEDF, is proposed and, through simulations, it is shown to outperform the first-come, first-served scheme. Furthermore, it is observed that machine availability is the key to getting good stretch values. Then, the moldable task scheduling problem is considered, and, by leveraging the results from the sequential case, another algorithm, DBOS, is proposed to optimize the stretch while scheduling moldable tasks. This work is motivated by a task scheduling problem in the context of parallel short sequence mapping which has important applications in biology and genetics. The proposed DBOS algorithm is evaluated both on synthetic data sets that represent short sequence mapping requests and on data sets generated using log files of real production clusters. The results show that the DBOS algorithm significantly outperforms the two state-of-the-art task scheduling algorithms on stretch optimization. [Copyright &y& Elsevier]

Abstract: High-order compact finite difference scheme with operator splitting technique for solving two-dimensional time fractional diffusion equation is considered in this paper. A Grünwald–Letnikov approximation is used for the Riemann–Liouville time derivative, and the second order spatial derivatives are approximated by the compact finite differences to obtain a fully discrete implicit scheme. Alternating direction implicit (ADI) method is used to split the original problem into two separate one-dimensional problems. The local truncation error is analyzed and the stability is discussed by the Fourier method. The proposed scheme is suitable when the order of the time fractional derivative γ lies in the interval . A correction term is added to maintain high accuracy when . Numerical results are provided to verify the accuracy and efficiency of the proposed algorithm. [Copyright &y& Elsevier]

Abstract: A number of people have suggested that there is a link between information integration and consciousness, and a number of algorithms for calculating information integration have been put forward. The most recent of these is Balduzzi and Tononi’s state-based Φ algorithm, which has factorial dependencies that severely limit the number of neurons that can be analyzed. To address this issue an alternative state-based measure known as liveliness has been developed, which uses the causal relationships between neurons to identify the areas of maximum information integration. This paper outlines the state-based Φ and liveliness algorithms and sets out a number of test networks that were used to compare their accuracy and performance. The results show that liveliness is a reasonable approximation to state-based Φ for some network topologies, and it has a much more scalable performance than state-based Φ. [Copyright &y& Elsevier]

Abstract: Symbolic numeric algorithms for polynomials are very important, especially for practical computations since we have to operate with empirical polynomials having numerical errors on their coefficients. Recently, for those polynomials, a number of algorithms have been introduced, such as approximate univariate GCD and approximate multivariate factorization for example. However, for polynomials over integers having coefficients rounded from empirical data, changing their coefficients over reals does not remain them in the polynomial ring over integers; hence we need several approximate operations over integers. In this paper, we discuss computing a polynomial GCD of univariate or multivariate polynomials over integers approximately. Here, “approximately” means that we compute a polynomial GCD over integers by changing their coefficients slightly over integers so that the input polynomials still remain over integers. [Copyright &y& Elsevier]

Abstract: The aim of the paper is the construction and the analysis of nonlinear and non-separable multiscale representations for multivariate functions defined using a non-diagonal dilation matrix . We show that a function in or Besov spaces can be characterized by means of its multiscale representation. We also study the stability of these representations, a key issue to design adaptive algorithms. [Copyright &y& Elsevier]

Abstract: This paper shows that the FPTAS for the min–max disjoint paths problem on directed acyclic graphs by Yu et al. (2010) can be improved by a rounding and searching technique. [Copyright &y& Elsevier]

Abstract: In this paper, we present iteration schemes to weakly and strongly approximate common fixed points of a finite family of a class of strict pseudocontractions. [Copyright &y& Elsevier]