Authors:Philip Greengard; Vladimir Rokhlin Pages: 23 - 49 Abstract: We introduce an algorithm for the evaluation of the Incomplete Gamma Function, P(m, x), for all m, x > 0. For small m, a classical recursive scheme is used to evaluate P(m, x), whereas for large m a newly derived asymptotic expansion is used. The number of operations required for evaluation is O(1) for all x and m. Nearly full double and extended precision accuracies are achieved in their respective environments. The performance of the scheme is illustrated via several numerical examples. PubDate: 2019-02-01 DOI: 10.1007/s10444-018-9604-x Issue No:Vol. 45, No. 1 (2019)

Authors:Xudong Yao Pages: 269 - 310 Abstract: In Yao (J. Sci. Comput. 66, 19–40 2016), two Ljusternik-Schnirelman minimax algorithms for capturing multiple free saddle points are developed from well-known Ljusternik-Schnirelman critical point theory, numerical experiment is carried out and global convergence is established. In this paper, a Ljusternik-Schnirelman minimax algorithm for calculating multiple equality constrained saddle points is presented. The algorithm is applied to numerically solve eigen problems. Finally, global convergence for the algorithm is verified. PubDate: 2019-02-01 DOI: 10.1007/s10444-018-9616-6 Issue No:Vol. 45, No. 1 (2019)

Authors:Scott Congreve; Paul Houston; Ilaria Perugia Pages: 361 - 393 Abstract: In this article, we develop an hp-adaptive refinement procedure for Trefftz discontinuous Galerkin methods applied to the homogeneous Helmholtz problem. Our approach combines not only mesh subdivision (h–refinement) and local basis enrichment (p–refinement), but also incorporates local directional adaptivity, whereby the elementwise plane wave basis is aligned with the dominant scattering direction. Numerical experiments based on employing an empirical a posteriori error indicator clearly highlight the efficiency of the proposed approach for various examples. PubDate: 2019-02-01 DOI: 10.1007/s10444-018-9621-9 Issue No:Vol. 45, No. 1 (2019)

Authors:Burak Aksoylu; Fatih Celiker; Orsan Kilicer Pages: 493 - 493 Abstract: In the original publication, Figure 4 image should be Figure 5, Figure 5 image was a repetition of Figure 6 and the correct image of Figure 4 was not shown. The original article was updated by correcting the images of figures 4, 5, and 6. PubDate: 2019-02-01 DOI: 10.1007/s10444-018-9632-6 Issue No:Vol. 45, No. 1 (2019)

Abstract: This paper describes the usage of the finite element library CFEM for solution of boundary value problems for partial differential equations. The application of the finite element method is shown based on the weak formulation of a boundary value problem. A unified approach for solution of linear scalar, linear vector, and nonlinear vector problems is presented. A direct link between the mathematical formulation and the design of the computer code is shown. Several examples and results are shown. PubDate: 2019-03-18

Abstract: We show that the sparse polynomial interpolation problem reduces to a discrete super-resolution problem on the n-dimensional torus. Therefore, the semidefinite programming approach initiated by Candès and Fernandez-Granda (Commun. Pure Appl. Math. 67(6) 906–956, 2014) in the univariate case can be applied. We extend their result to the multivariate case, i.e., we show that exact recovery is guaranteed provided that a geometric spacing condition on the supports holds and evaluations are sufficiently many (but not many). It also turns out that the sparse recovery LP-formulation of ℓ1-norm minimization is also guaranteed to provide exact recovery provided that the evaluations are made in a certain manner and even though the restricted isometry property for exact recovery is not satisfied. (A naive sparse recovery LP approach does not offer such a guarantee.) Finally, we also describe the algebraic Prony method for sparse interpolation, which also recovers the exact decomposition but from less point evaluations and with no geometric spacing condition. We provide two sets of numerical experiments, one in which the super-resolution technique and Prony’s method seem to cope equally well with noise, and another in which the super-resolution technique seems to cope with noise better than Prony’s method, at the cost of an extra computational burden (i.e., a semidefinite optimization). PubDate: 2019-03-15

Abstract: Image colorization aims to recover the whole color image based on a known grayscale image (luminance or brightness) and some known color pixel values. In this paper, we generalize the graph Laplacian to its second-order variant called graph bi-Laplacian, and then propose an image colorization method by using graph bi-Laplacian. The eigenvalue analysis of graph bi-Laplacian matrix and its corresponding normalized bi-Laplacian matrix is given to show their properties. We apply graph bi-Laplacian approach to image colorization by formulating it as an optimization problem and solving the resulting linear system efficiently. Numerical results show that the proposed method can perform quite well for image colorization problem, and its performance in terms of efficiency and colorization quality for test images can be better than that by the state-of-the-art colorization methods when the randomly given color pixels ratio attains some level. PubDate: 2019-03-12

Abstract: In this paper, we study a newly developed hybrid shearlet-wavelet system on bounded domains which yields frames for Hs(Ω) for some \(s\in \mathbb {N}\) , \({\Omega } \subset \mathbb {R}^{2}\) . We will derive approximation rates with respect to Hs(Ω) norms for functions whose derivatives admit smooth jumps along curves and demonstrate superior rates to those provided by pure wavelet systems. These improved approximation rates demonstrate the potential of the novel shearlet system for the discretization of partial differential equations. Therefore, we implement an adaptive shearlet-wavelet-based algorithm for the solution of an elliptic PDE and analyze its computational complexity and convergence properties. PubDate: 2019-03-11

Abstract: We illustrate the potential applications in machine learning of the Christoffel function, or, more precisely, its empirical counterpart associated with a counting measure uniformly supported on a finite set of points. Firstly, we provide a thresholding scheme which allows approximating the support of a measure from a finite subset of its moments with strong asymptotic guaranties. Secondly, we provide a consistency result which relates the empirical Christoffel function and its population counterpart in the limit of large samples. Finally, we illustrate the relevance of our results on simulated and real-world datasets for several applications in statistics and machine learning: (a) density and support estimation from finite samples, (b) outlier and novelty detection, and (c) affine matching. PubDate: 2019-03-07

Abstract: A fully discrete Galerkin scheme for a thermodynamically consistent transient Maxwell–Stefan system for the mass particle densities, coupled to the Poisson equation for the electric potential, is investigated. The system models the diffusive dynamics of an isothermal ionized fluid mixture with vanishing barycentric velocity. The equations are studied in a bounded domain, and different molar masses are allowed. The Galerkin scheme preserves the total mass, the nonnegativity of the particle densities, their boundedness and satisfies the second law of thermodynamics in the sense that the discrete entropy production is nonnegative. The existence of solutions to the Galerkin scheme and the convergence of a subsequence to a solution to the continuous system is proved. Compared to previous works, the novelty consists in the treatment of the drift terms involving the electric field. Numerical experiments show the sensitive dependence of the particle densities and the equilibration rate on the molar masses. PubDate: 2019-03-07

Abstract: The paper develops the essentially optimal sparse tensor product finite element method for a parabolic equation in a domain in \(\mathbb {R}^{d}\) which depends on a microscopic scale in space and a microscopic scale in time. We consider the critical self similar case which has the most interesting homogenization limit. We solve the high dimensional time-space multiscale homogenized equation, which provides the solution to the homogenized equation which describes the multiscale equation macroscopically, and the corrector which encodes the microscopic information. For obtaining an approximation within a prescribed accuracy, the method requires an essentially optimal number of degrees of freedom that is essentially equal to that for solving a macroscopic parabolic equation in a domain in \(\mathbb {R}^{d}\) . A numerical corrector is deduced from the finite element solution. Numerical examples for one and two dimensional problems confirm the theoretical results. Although the theory is developed for problems with one spatial microscopic scale, we show numerically that the method is capable of solving problems with more than one spatial microscopic scale. PubDate: 2019-03-06

Abstract: We consider numerical approximations for the modified phase field crystal equation in this paper. The model is a nonlinear damped wave equation that includes both diffusive dynamics and elastic interactions. To develop easy-to-implement time-stepping schemes with unconditional energy stabilities, we adopt the “Invariant Energy Quadratization” approach. By using the first-order backward Euler, the second-order Crank–Nicolson, and the second-order BDF2 formulas, we obtain three linear and symmetric positive definite schemes. We rigorously prove their unconditional energy stabilities and implement a number of 2D and 3D numerical experiments to demonstrate the accuracy, stability, and efficiency. PubDate: 2019-03-02

Abstract: In this paper, we propose and analyse a new unbiased stochastic approach for solving a class of integral equations. We study and compare the proposed unbiased approach against the known biased Monte Carlo method based on evaluation of truncated Liouville-Neumann series. We also compare the proposed algorithm against the deterministic Nystrom method. Extensions of the unbiased method for the weak and global solutions are described. Extensive numerical experiments have been performed to support the theoretical studies regarding the convergence of the unbiased algorithms. The results are compared to the best known biased Monte Carlo algorithms for numerical integration done in our previous studies. Conclusions about the applicability and efficiency of the proposed algorithms have been drawn. PubDate: 2019-03-02

Abstract: In this contribution, we are concerned with tight a posteriori error estimation for projection-based model order reduction of \(\inf \) - \(\sup \) stable parameterized variational problems. In particular, we consider the Reduced Basis Method in a Petrov-Galerkin framework, where the reduced approximation spaces are constructed by the (weak) greedy algorithm. We propose and analyze a hierarchical a posteriori error estimator which evaluates the difference of two reduced approximations of different accuracy. Based on the a priori error analysis of the (weak) greedy algorithm, it is expected that the hierarchical error estimator is sharp with efficiency index close to one, if the Kolmogorov N-with decays fast for the underlying problem and if a suitable saturation assumption for the reduced approximation is satisfied. We investigate the tightness of the hierarchical a posteriori estimator both from a theoretical and numerical perspective. For the respective approximation with higher accuracy, we study and compare basis enrichment of Lagrange- and Taylor-type reduced bases. Numerical experiments indicate the efficiency for both, the construction of a reduced basis using the hierarchical error estimator in a greedy algorithm, and for tight online certification of reduced approximations. This is particularly relevant in cases where the \(\inf \) - \(\sup \) constant may become small depending on the parameter. In such cases, a standard residual-based error estimator—complemented by the successive constrained method to compute a lower bound of the parameter dependent \(\inf \) - \(\sup \) constant—may become infeasible. PubDate: 2019-02-27

Authors:Sinem Arslan; Münevver Tezer-Sezgin Abstract: In this study, the steady, laminar, and fully developed magnetohydrodynamic (MHD) flow is considered in a long channel along with the z-axis under an external magnetic field which is perpendicular to the channel axis. The fluid velocity u and the induced magnetic field b depend on the plane coordinates x and y on the cross-section of the channel. When the lateral channel walls are extended to infinity, the problem turns out to be MHD flow between two parallel plates (Hartmann flow). Now, the variations of u and b are only with respect to y-coordinate. The finite difference method (FDM) is used to solve the governing MHD equations with the wall conditions which include both the slip and the conductivity of the plates. The numerical results obtained from FDM discretized equations are compared with the exact solution derived here for the 1D MHD flow with Robin’s type boundary conditions. The fluid velocity and the induced magnetic field are simulated for each special case of boundary conditions on the plates including no-slip to highly slipping and insulated to perfectly conducting plates. The well-known characteristics of the MHD flow are observed. It is found that the increase in the slip length weakens the formation of boundary layers. Thus, the FDM which is simple to implement, enables one to depict the effects of Hartmann number, conductivity parameter, and the slip length on the behavior of both the velocity and the induced magnetic field at a small expense. PubDate: 2019-02-21 DOI: 10.1007/s10444-019-09669-x

Authors:Jan Nikl; Milan Kuchařík; Jiří Limpouch; Richard Liska; Stefan Weber Abstract: Models of the laser propagation and absorption are a crucial part of the laser–plasma interaction models. Hydrodynamic codes are afflicted by usage of the simplified, not self-consistent, models of the geometrical optics, limiting their physical accuracy. A robust and efficient method is presented for computing the stationary wave solution, not restricted to this field of application exclusively. The method combines the semi-analytic and high-order differential approaches to benefit from both. Flexibility of the discretization is maintained, including the discontinuous methods. Performance of the model is evaluated for the problem of a transition layer by comparison with the analytic solution. Reliable results on coarse computational meshes and high convergence rates on fine meshes are obtained. The relevance to the current fusion research and non-local energy transport is pointed out. PubDate: 2019-02-18 DOI: 10.1007/s10444-019-09671-3

Authors:José L. Galán-García; Gabriel Aguilera-Venegas; María Á. Galán-García; Pedro Rodríguez-Cielos; Iván Atencia-McKillop; Yolanda Padilla-Domínguez; Ricardo Rodríguez-Cielos Abstract: In a previous paper, the authors developed new rules for computing improper integrals which allow computer algebra systems (Cas) to deal with a wider range of improper integrals. The theory used in order to develop such rules where Laplace and Fourier transforms and the residue theorem. In this paper, we describe new rules for computing symbolic improper integrals using extensions of the residue theorem and analyze how some of the most important Cas could improve their improper integral computations using these rules. To achieve this goal, different tests are developed. The Cas considered have been evaluated using these tests. The obtained results show that all Cas involved, considering the new developed rules, could improve their capabilities for computing improper integrals. The results of the evaluations of the Cas are described providing a sorted list of the Cas depending on their scores. PubDate: 2019-02-07 DOI: 10.1007/s10444-018-09660-y

Authors:Zhifang Liu; Chunlin Wu; Yanan Zhao Abstract: We consider the non-Lipschitz ℓp-ℓq (0 < p < 1 ≤ q < ∞) minimization problem, which has many applications and is a great challenge for optimization. The problem contains a non-Lipschitz regularization term and a possibly nonsmooth fidelity. In this paper, we present a new globally convergent algorithm, which gradually shrinks the variable support and uses linearization and proximal approximations. The subproblem at each iteration is then convex with increasingly fewer unknowns. By showing a lower bound theory for the sequence generated by our algorithm, we prove that the sequence globally converges to a stationary point of the ℓp-ℓq objective function. Our method can be extended to the ℓp-regularized elastic net model. Numerical experiments demonstrate the performances and flexibilities of the proposed algorithm, such as the applicability to measurements with either Gaussian or heavy-tailed noise. PubDate: 2019-02-06 DOI: 10.1007/s10444-019-09668-y

Authors:Jiří Fürst; Zdeněk Žák Abstract: The article deals with the numerical simulation of unsteady flows through the turbine part of the turbocharger. The main focus of the article is the extension of the in-house CFD finite volume solver for the case of unsteady flows in radial turbines and the coupling to an external zero-dimensional model of the inlet and outlet parts. In the second part, brief description of a simplified one-dimensional model of the turbine is given. The final part presents a comparison of the results of numerical simulations using both the 3D CFD method and the 1D simplified model with the experimental data. The comparison shows that the properly calibrated 1D model gives accurate predictions of mass flow rate and turbine performance at much less computational time than the full 3D CFD method. On the other hand, the more expensive 3D CFD method does not need any specific calibration and allows detailed inspections of the flow fields. PubDate: 2019-02-06 DOI: 10.1007/s10444-019-09670-4

Authors:Tomasz Talaśka Abstract: Fuzzy systems play an important role in many industrial applications. Depending on the application, they can be implemented using different techniques and technologies. Software implementations are the most popular, which results from the ease of such implementations. This approach facilitates modifications and testing. On the other hand, such realizations are usually not convenient when high data rate, low cost per unit, and large miniaturization are required. For this reason, we propose efficient, fully digital, parallel, and asynchronous (clock-less) fuzzy logic (FL) systems suitable for the implementation as ultra low-power-specific integrated circuits (ASICs). On the basis of our former work, in which single FL operators were proposed, here we demonstrate how to build larger structures, composed of many operators of this type. As an example, we consider Lukasiewicz neural networks (LNN) that are fully composed of selected FL operators. In this work, we propose FL OR, and AND Lukasiewicz neurons, which are based on bounded sum and bounded product FL operators. In the comparison with former analog implementations of such LNNs, digital realization, presented in this work, offers important advantages. The neurons have been designed in the CMOS 130nm technology and thoroughly verified by means of the corner analysis in the HSpice environment. The only observed influence of particular combinations on the process, voltage, and temperature parameters was on delays and power dissipation, while from the logical point of view, the system always worked properly. This shows that even larger FL systems may be implemented in this way. PubDate: 2019-02-04 DOI: 10.1007/s10444-018-09659-5