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 Advances in Computational MathematicsJournal Prestige (SJR): 0.812 Citation Impact (citeScore): 1Number of Followers: 19      Hybrid journal (It can contain Open Access articles) ISSN (Print) 1572-9044 - ISSN (Online) 1019-7168 Published by Springer-Verlag  [2352 journals]
• Enhancing Cas improper integrals computations using extensions of the
residue theorem
• 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

• A new globally convergent algorithm for non-Lipschitz ℓ p - ℓ
q minimization
• 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

• Parallel, asynchronous, fuzzy logic systems realized in CMOS technology
• 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

• On error indicators for optimizing parameters in stabilized methods
• Authors: Petr Knobloch; Petr Lukáš; Pavel Solin
Abstract: Numerical solution of convection-dominated problems requires special techniques to suppress spurious oscillations in approximate solutions. Often, stabilized methods are applied which involve user-chosen parameters. These parameters significantly influence the quality of the solution but their optimal choice is usually not known. One possibility is to define them in an adaptive way by minimizing an error indicator characterizing the quality of the approximate solution. A non-trivial requirement on the error indicator is that its minimization with respect to the stabilization parameters should suppress spurious oscillations without smearing layers. In this paper, a new error indicator is introduced and its suitability is tested on two newly proposed benchmark problems for which previously proposed indicators do not provide satisfactory results.
PubDate: 2019-01-29
DOI: 10.1007/s10444-019-09662-4

• A prototype of a functional approach to personalized menus generation
using set operations
• Authors: Eugenio Roanes–Lozano; José Luis Galán–García; Gabriel Aguilera–Venegas
Abstract: The authors developed some time ago a RBES devoted to preparing personalized menus at restaurants according to the allergies, religious constraints, likes, and other diet requirements as well as products availability. This can be specially important when traveling abroad and facing unknown dishes in a menu. Some restaurants include icons in their menu regarding their adequateness for celiacs or vegetarians and vegans, but this is not always a complete information, as it doesn’t consider, for instance, personal dislikes, or uncommon allergies. The tool previously developed uses logic deduction to obtain a personalized menu for each customer, according to the precise recipes of the restaurant and taking into account the data provided by the customer and the ingredients out of stock (if any). That previous work had an impact in Spanish society: news about it were disseminated by different news agencies and appeared in some newspapers. The authors were also interviewed in radio networks and television channels. Now a new approach that uses functions and set operations has been followed and the speed has been increased by three orders of magnitude, allowing to deal with huge menus instantly. Both approaches have been implemented in the computer algebra system Maple and are exemplified using the same recipes in order to compare their performances.
PubDate: 2019-01-28
DOI: 10.1007/s10444-019-09665-1

• An arriving decision problem in a discrete-time queueing system
• Authors: Ivan Atencia; José L. Galán-García; Gabriel Aguilera-Venegas; Pedro Rodríguez-Cielos; M. Ángeles Galán-García
Abstract: This paper discusses a discrete-time queueing system in which an arriving customer may adopt four different strategies; two of them correspond to a LCFS discipline where displacements or expulsions occur, and in the other two, the arriving customer decides to follow a FCFS discipline or to become a negative customer eliminating the customer in the server, if any. The different choices of the involved parameters make this model to enjoy a great versatility, having several special cases of interest. We carry out a thorough analysis of the system, and using a generating function approach, we derive analytical results for the stationary distributions obtaining performance measures for the number of customers in the queue and in the system. Also, recursive formulae for calculating the steady-state distributions of the queue and system size has been developed. Making use of the busy period of an auxiliary system, the sojourn times of a customer in the queue and in the system have also been obtained. Finally, some numerical examples are given.
PubDate: 2019-01-21
DOI: 10.1007/s10444-019-09663-3

• A computational study of the gene expression in the tryptophan operon with
two types of cooperativity
• Authors: José Roberto Cantú-González; O. Díaz-Hernández; Elizeth Ramírez-Álvarez; C. I. Enríquez Flores; A. Flores Rosas; Gerardo J. Escalera Santos
Abstract: Intrinsic noise is inherent to many biological processes and provokes variation in gene expression in a population of isogenic cells leading to phenotypic diversity. Intrinsic noise is generated by different sources of noise such as the number of molecules, the stochastic binding and unbinding of transcription factor and/or the number and strength of transcription factor binding sites. In this work, we use numerical simulations to study the effects of the number of operators and different types of cooperativity on the Fano factor of three different molecules of the tryptophan (trp) operon of E. Coli. We analyze the Fano factor for the mRNA, anthranilate synthase and tryptophan molecules, because it represents the effects of the noise in the variation or variability of the gene expression, a larger Fano factor implies a larger variation. Our model takes into consideration the presence of intrinsic noise and all the known mechanisms of regulation. In particular, we consider hypothetical promoters in the repression mechanism with different numbers of operators and three cases of cooperativity: positive, negative, and no-cooperativity.
PubDate: 2019-01-19
DOI: 10.1007/s10444-018-09661-x

• Randomized model order reduction
• Authors: Alessandro Alla; J. Nathan Kutz
Abstract: The singular value decomposition (SVD) has a crucial role in model order reduction. It is often utilized in the offline stage to compute basis functions that project the high-dimensional nonlinear problem into a low-dimensional model which is then evaluated cheaply. It constitutes a building block for many techniques such as the proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD). The aim of this work is to provide an efficient computation of low-rank POD and/or DMD modes via randomized matrix decompositions. This is possible due to the randomized singular value decomposition (rSVD) which is a fast and accurate alternative of the SVD. Although this is considered an offline stage, this computation may be extremely expensive; therefore, the use of compressed techniques drastically reduce its cost. Numerical examples show the effectiveness of the method for both POD and DMD.
PubDate: 2019-01-17
DOI: 10.1007/s10444-018-09655-9

• A certified model reduction approach for robust parameter optimization
with PDE constraints
• Authors: Alessandro Alla; Michael Hinze; Philip Kolvenbach; Oliver Lass; Stefan Ulbrich
Abstract: We investigate an optimization problem governed by an elliptic partial differential equation with uncertain parameters. We introduce a robust optimization framework that accounts for uncertain model parameters. The resulting nonlinear optimization problem has a bilevel structure due to the min-max formulation. To approximate the worst case in the optimization problem, we propose linear and quadratic approximations. However, this approach still turns out to be very expensive; therefore, we propose an adaptive model order reduction technique which avoids long offline stages and provides a certified reduced order surrogate model for the parametrized PDE which is then utilized in the numerical optimization. Numerical results are presented to validate the presented approach.
PubDate: 2019-01-16
DOI: 10.1007/s10444-018-9653-1

• A substructuring preconditioner with vertex-related interface solvers for
elliptic-type equations in three dimensions
• Authors: Qiya Hu; Shaoliang Hu
Abstract: In this paper, we propose a variant of the substructuring preconditioner for solving three-dimensional elliptic-type equations with strongly discontinuous coefficients. In the proposed preconditioner, we use the simplest coarse solver associated with the finite element space induced by the coarse partition and construct inexact interface solvers based on overlapping domain decomposition with small overlaps. This new preconditioner has an important merit: its construction and efficiency do not depend on the concrete form of the considered elliptic-type equations. We apply the proposed preconditioner to solve the linear elasticity problems and Maxwell’s equations in three dimensions. Numerical results show that the convergence rate of PCG method with the preconditioner is nearly optimal, and also robust with respect to the (possibly large) jumps of the coefficients in the considered equations.
PubDate: 2019-01-07
DOI: 10.1007/s10444-018-9648-y

• Regularization theory in the study of generalization ability of a
biological neural network model
• Authors: Aleksandra Świetlicka
Abstract: This paper focuses on the generalization ability of a dendritic neuron model (a model of a simple neural network). The considered model is an extension of the Hodgkin-Huxley model. The Markov kinetic schemes have been used in the mathematical description of the model, while the Lagrange multipliers method has been applied to train the model. The generalization ability of the model is studied using a method known from the regularization theory, in which a regularizer is added to the neural network error function. The regularizers in the form of the sum of squared weights of the model (the penalty function), a linear differential operator related to the input-output mapping (the Tikhonov functional), and the square norm of the network curvature are applied in the study. The influence of the regularizers on the training process and its results are illustrated with the problem of noise reduction in images of electronic components. Several metrics are used to compare results obtained for different regularizers.
PubDate: 2019-01-04
DOI: 10.1007/s10444-018-09658-6

• A new computable sufficient condition for the convergence of subdivision
• Authors: Li Cheng; Xinlong Zhou
Abstract: We are interested in nontrivial conditions on the nonnegative masks that guarantee the convergence of the correspondent subdivision schemes. Roughly speaking, a certain convexity of the support of the given mask implies the convergence of the subdivision scheme. Moreover, those conditions are computable. The key of proving our main theorem is to find out an irreducible or primitive mapping on some multi-integer set and to show the uniqueness of this mapping.
PubDate: 2019-01-03
DOI: 10.1007/s10444-018-09656-8

• Damping optimization of parameter dependent mechanical systems by rational
interpolation
• Authors: Zoran Tomljanović; Christopher Beattie; Serkan Gugercin
Pages: 1797 - 1820
Abstract: We consider an optimization problem related to semi-active damping of vibrating systems. The main problem is to determine the best damping matrix able to minimize influence of the input on the output of the system. We use a minimization criteria based on the $$\mathcal {H}_{2}$$ system norm. The objective function is non-convex and the associated optimization problem typically requires a large number of objective function evaluations. We propose an optimization approach that calculates ‘interpolatory’ reduced order models, allowing for significant acceleration of the optimization process. In our approach, we use parametric model reduction (PMOR) based on the Iterative Rational Krylov Algorithm, which ensures good approximations relative to the $$\mathcal {H}_{2}$$ system norm, aligning well with the underlying damping design objectives. For the parameter sampling that occurs within each PMOR cycle, we consider approaches with predetermined sampling and approaches using adaptive sampling, and each of these approaches may be combined with three possible strategies for internal reduction. In order to preserve important system properties, we maintain second-order structure, which through the use of modal coordinates, allows for very efficient implementation. The methodology proposed here provides a significant acceleration of the optimization process; the gain in efficiency is illustrated in numerical experiments.
PubDate: 2018-12-01
DOI: 10.1007/s10444-018-9605-9
Issue No: Vol. 44, No. 6 (2018)

• Clustering approach to model order reduction of power networks with
distributed controllers
• Authors: Xiaodong Cheng; Jacquelien M. A. Scherpen
Pages: 1917 - 1939
Abstract: This paper considers the network structure preserving model reduction of power networks with distributed controllers. The studied system and controller are modeled as second-order and first-order ordinary differential equations, which are coupled to a closed-loop model for analyzing the dissimilarities of the power units. By transfer functions, we characterize the behavior of each node (generator or load) in the power network and define a novel notion of dissimilarity between two nodes by the $$\mathcal {H}_{2}$$ -norm of the transfer function deviation. Then, the reduction methodology is developed based on separately clustering the generators and loads according to their behavior dissimilarities. The characteristic matrix of the resulting clustering is adopted for the Galerkin projection to derive explicit reduced-order power models and controllers. Finally, we illustrate the proposed method by the IEEE 30-bus system example.
PubDate: 2018-12-01
DOI: 10.1007/s10444-018-9617-5
Issue No: Vol. 44, No. 6 (2018)

• POD reduced-order modeling for evolution equations utilizing arbitrary
finite element discretizations
• Authors: Carmen Gräßle; Michael Hinze
Pages: 1941 - 1978
Abstract: The main focus of the present work is the inclusion of spatial adaptivity for the snapshot computation in the offline phase of model order reduction utilizing proper orthogonal decomposition (POD-MOR) for nonlinear parabolic evolution problems. We consider snapshots which live in different finite element spaces, which means in a fully discrete setting that the snapshots are vectors of different length. From a numerical point of view, this leads to the problem that the usual POD procedure which utilizes a singular value decomposition of the snapshot matrix, cannot be carried out. In order to overcome this problem, we here construct the POD model/basis using the eigensystem of the correlation matrix (snapshot Gramian), which is motivated from a continuous perspective and is set up explicitly, e.g., without the necessity of interpolating snapshots into a common finite element space. It is an advantage of this approach that the assembly of the matrix only requires the evaluation of inner products of snapshots in a common Hilbert space. This allows a great flexibility concerning the spatial discretization of the snapshots. The analysis for the error between the resulting POD solution and the true solution reveals that the accuracy of the reduced-order solution can be estimated by the spatial and temporal discretization error as well as the POD error. Finally, to illustrate the feasibility of our approach, we present a test case of the Cahn–Hilliard system utilizing h-adapted hierarchical meshes and two settings of a linear heat equation using nested and non-nested grids.
PubDate: 2018-12-01
DOI: 10.1007/s10444-018-9620-x
Issue No: Vol. 44, No. 6 (2018)

• A nonlinearity lagging method for non-steady diffusion equations with
nonlinear convection terms
• Authors: Francesco Mezzadri; Emanuele Galligani
Abstract: We analyze an iterative procedure for solving nonlinear algebraic systems arising from the discretization of nonlinear, non-steady reaction-convection-diffusion equations with non-constant (and, in general, nonlinear) velocity terms. The basic idea underlying the procedure consists in lagging the diffusion and the velocity terms of the discretized system, which is thus partly linearized. After analyzing the discretized system and proving some results on the monotonicity of the operators and on the uniqueness of the solution, we prove sufficient conditions that ensure the convergence of this lagged method. We also describe the inner iteration and show how the weakly nonlinear systems arising at each lagged iteration can be solved efficiently. Finally, we analyze numerically the entire solution process by several numerical experiments.
PubDate: 2018-12-17
DOI: 10.1007/s10444-018-9652-2

• A computational algebraic geometry approach to analyze pseudo-random
sequences based on Latin squares
• Authors: Raúl M. Falcón; Víctor Álvarez; Félix Gudiel
Abstract: Latin squares are used as scramblers on symmetric-key algorithms that generate pseudo-random sequences of the same length. The robustness and effectiveness of these algorithms are respectively based on the extremely large key space and the appropriate choice of the Latin square under consideration. It is also known the importance that isomorphism classes of Latin squares have to design an effective algorithm. In order to delve into this last aspect, we improve in this paper the efficiency of the known methods on computational algebraic geometry to enumerate and classify partial Latin squares. Particularly, we introduce the notion of affine algebraic set of a partial Latin square L = (lij) of order n over a field $$\mathbb {K}$$ as the set of zeros of the binomial ideal $$\langle x_{i}x_{j}-x_{l_{ij}}\colon (i,j) \text { is a non-empty cell in} L \rangle \subseteq \mathbb {K}[x_{1},\ldots ,x_{n}]$$ . Since isomorphic partial Latin squares give rise to isomorphic affine algebraic sets, every isomorphism invariant of the latter constitutes an isomorphism invariant of the former. In particular, we deal computationally with the problem of deciding whether two given partial Latin squares have either the same or isomorphic affine algebraic sets. To this end, we introduce a new pair of equivalence relations among partial Latin squares: being partial transpose and being partial isotopic.
PubDate: 2018-12-12
DOI: 10.1007/s10444-018-9654-0

• Two numerical methods for the Zakharov-Rubenchik equations
• Authors: Xuanxuan Zhou; Tingchun Wang; Luming Zhang
Abstract: Two numerical methods are presented for the approximation of the Zakharov-Rubenchik equations (ZRE). The first one is the finite difference integrator Fourier pseudospectral method (FFP), which is implicit and of the optimal convergent rate at the order of O(N−r + τ2) in the discrete L2 norm without any restrictions on the grid ratio. The second one is to use the Fourier pseudospectral approach for spatial discretization and exponential wave integrator for temporal integration. Fast Fourier transform is applied to the discrete nonlinear system to speed up the numerical computation. Numerical examples are given to show the efficiency and accuracy of the new methods.
PubDate: 2018-12-11
DOI: 10.1007/s10444-018-9651-3

• Model reduction of complex dynamical systems
• Authors: Peter Benner; Heike Faßbender; Michael Hinze; Tatjana Stykel; Ralf Zimmermann
PubDate: 2018-10-30
DOI: 10.1007/s10444-018-9640-6

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