Algebraic flux correction for nonconforming finite element discretizations of scalar transport problems

This paper is concerned with the extension of the algebraic flux-correction (AFC) approach (Kuzmin in Computational fluid and solid mechanics, Elsevier, Amsterdam, pp 887–888, 2001; J Comput Phys 219:513–531, 2006; Comput Appl Math 218:79–87, 2008; J Comput Phys 228:2517–2534, 2009; Flux-corrected transport: principles, algorithms, and applications, 2nd edn. Springer, Berlin, pp 145–192, 2012; J Comput Appl Math 236:2317–2337, 2012; Kuzmin et al. in Comput Methods Appl Mech Eng 193:4915–4946, 2004; Int J Numer Methods Fluids 42:265–295, 2003; Kuzmin and Möller in Flux-corrected transport: principles, algorithms, and applications. Springer, Berlin, 2005; Kuzmin and Turek in J Comput Phys 175:525–558, 2002; J Comput Phys 198:131–158, 2004) to nonconforming finite element methods for the linear transport equation. Accurate nonoscillatory approximations to convection-dominated flows are obtained by stabilizing the continuous Galerkin method by solution-dependent artificial diffusion. Its magnitude is controlled by a flux limiter. This concept dates back to flux-corrected transport schemes. The unique feature of AFC is that all information is extracted from the system matrices which are manipulated to satisfy certain mathematical constraints. AFC schemes have been devised with conforming $P_1$ and $Q_1$ finite elements in mind but this is not a prerequisite. Here, we consider their extension to the nonconforming Crouzeix–Raviart element (Crouzeix and Raviart in RAIRO R3 7:33–76, 1973) on triangular meshes and its quadrilateral counterpart, the class of rotated bilinear Rannacher–Turek elements (Rannacher and Turek in Numer Methods PDEs 8:97–111, 1992). The underlying design principles of AFC schemes are shown to hold for (some variant of) both elements. However, numerical tests for a purely convective flow and a convection–diffusion problem demonstrate that flux-corrected solutions are overdiffusive for the Crouzeix–Raviart element. Good resolution of smooth and discontinuous profiles is attested to $Q_1^\mathrm{nc}$ approximations on quadrilateral meshes. A synthetic benchmark is used to quantify the artificial diffusion present in conforming and nonconforming high-resolution schemes of AFC-type. Finally, the implementation of efficient sparse matrix–vector multiplications is addressed.     

Comparison Between Reduced Basis and Stochastic Collocation Methods for Elliptic Problems

The stochastic collocation method (Babu?ka et al. in SIAM J Numer Anal 45(3):1005–1034, 2007; Nobile et al. in SIAM J Numer Anal 46(5):2411–2442, 2008a; SIAM J Numer Anal 46(5):2309–2345, 2008b; Xiu and Hesthaven in SIAM J Sci Comput 27(3):1118–1139, 2005) has recently been applied to stochastic problems that can be transformed into parametric systems. Meanwhile, the reduced basis method (Maday et al. in Comptes Rendus Mathematique 335(3):289–294, 2002; Patera and Rozza in Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations Version 1.0. Copyright MIT, http://augustine.mit.edu, 2007; Rozza et al. in Arch Comput Methods Eng 15(3):229–275, 2008), primarily developed for solving parametric systems, has been recently used to deal with stochastic problems (Boyaval et al. in Comput Methods Appl Mech Eng 198(41–44):3187–3206, 2009; Arch Comput Methods Eng 17:435–454, 2010). In this work, we aim at comparing the performance of the two methods when applied to the solution of linear stochastic elliptic problems. Two important comparison criteria are considered: (1), convergence results of the approximation error; (2), computational costs for both offline construction and online evaluation. Numerical experiments are performed for problems from low dimensions $O(1)$ to moderate dimensions $O(10)$ and to high dimensions $O(100)$ . The main result stemming from our comparison is that the reduced basis method converges better in theory and faster in practice than the stochastic collocation method for smooth problems, and is more suitable for large scale and high dimensional stochastic problems when considering computational costs.     

A linear system form solution to compute the local space average color

In this document, we present an alternative to the method introduced by Ebner (Pattern Recognit 60–67, 2003; J Parallel Distrib Comput 64(1):79–88, 2004; Color constancy using local color shifts, pp 276–287, 2004; Color Constancy, 2007; Mach Vis Appl 20(5):283–301, 2009) for computing the local space average color. We show that when the problem is framed as a linear system and the resulting series is solved, there is a solution based on LU decomposition that reduces the computing time by at least an order of magnitude.     

On Rewriting Rules in Mizar

This paper presents some tentative experiments in using a special case of rewriting rules in Mizar (Mizar homepage: http://www.mizar.org/): rewriting a term as its subterm. A similar technique, but based on another Mizar mechanism called functor identification (Korni?owicz 2009) was used by Caminati, in his paper on basic first-order model theory in Mizar (Caminati, J Form Reason 3(1):49–77, 2010, Form Math 19(3):157–169, 2011). However for this purpose he was obligated to introduce some artificial functors. The mechanism presented in the present paper looks promising and fits the Mizar paradigm.     

A Splitting Algorithm for Image Segmentation on Manifolds Represented by the Grid Based Particle Method

We propose a numerical approach to solve variational problems on manifolds represented by the grid based particle method (GBPM) recently developed in Leung et al. (J. Comput. Phys. 230(7):2540–2561, 2011), Leung and Zhao (J. Comput. Phys. 228:7706–7728, 2009a, J. Comput. Phys. 228:2993–3024, 2009b, Commun. Comput. Phys. 8:758–796, 2010). In particular, we propose a splitting algorithm for image segmentation on manifolds represented by unconnected sampling particles. To develop a fast minimization algorithm, we propose a new splitting method by generalizing the augmented Lagrangian method. To efficiently implement the resulting method, we incorporate with the local polynomial approximations of the manifold in the GBPM. The resulting method is flexible for segmentation on various manifolds including closed or open or even surfaces which are not orientable.     

Fault Estimation for a Quad-Rotor MAV Using a Polynomial Observer

This work addresses the problem of fault detection and diagnosis (FDD) for a quad-rotor mini air vehicle (MAV). Actuator faults are considered on this paper. The basic idea behind the proposed method is to estimate the faults signals using the extended state observers theory. To estimate the faults, a polynomial observer (Aguilar et al. 2011; Mata-Machuca et al., Commun Nonlinear Sci Numer Simul 15(12):4114–4130, 2010, BioSystems 100(1):65–69, 2010) is presented by using the available measurements and know inputs of the system. In order to investigate the diagnosability properties of the system, a differential algebra approach is proposed (Cruz-Victoria et al., J Frankl Inst 345(2):102–118, 2008; and Martinez-Guerra and Diop, IEE P-Contr Theor Ap 151(1):130–135, 2004). The effectiveness of the methodology is illustrated by means of numerical simulations.     

Hermitian Compact Interpolation on the Cubed-Sphere Grid

The cubed-sphere grid is a spherical grid made of six quasi-cartesian square-like patches. It was originally introduced in Sadourny (Mon Weather Rev 100:136–144, 1972). We extend to this grid the design of high-order finite-difference compact operators (Collatz, The numerical treatment of differential equations. Springer, Berlin, 1960; Lele, J Comput Phys 103:16–42, 1992). The present work is limitated to the design of a fourth-order accurate spherical gradient. The treatment at the interface of the six patches relies on a specific interpolation system which is based on using great circles in an essential way. The main interest of the approach is a fully symmetric treatment of the sphere. We numerically demonstrate the accuracy of the approximate gradient on several test problems, including the cosine-bell test-case of Williamson et al. (J Comput Phys 102:211–224, 1992) and a deformational test-case reported in Nair and Lauritzen (J Comput Phys 229:8868–8887, 2010).     

A Comparison of Iterated Optimal Stopping and Local Policy Iteration for American Options Under Regime Switching

A theoretical analysis tool, iterated optimal stopping, has been used as the basis of a numerical algorithm for American options under regime switching (Le and Wang in SIAM J Control Optim 48(8):5193–5213, 2010). Similar methods have also been proposed for American options under jump diffusion (Bayraktar and Xing in Math Methods Oper Res 70:505–525, 2009) and Asian options under jump diffusion (Bayraktar and Xing in Math Fin 21(1):117–143, 2011). An alternative method, local policy iteration, has been suggested in Huang et al. (SIAM J Sci Comput 33(5):2144–2168, 2011), and Salmi and Toivanen (Appl Numer Math 61:821–831, 2011). Worst case upper bounds on the convergence rates of these two methods suggest that local policy iteration should be preferred over iterated optimal stopping (Huang et al. in SIAM J Sci Comput 33(5):2144–2168, 2011). In this article, numerical tests are presented which indicate that the observed performance of these two methods is consistent with the worst case upper bounds. In addition, while these two methods seem quite different, we show that either one can be converted into the other by a simple rearrangement of two loops.     

Some types of filters in residuated lattices

In this paper, inspired by some types of $BL$ -algebra filters (deductive systems) introduced in Haveshki et al. (Soft Comput 10:657–664, 2006), Kondo and Dudek (Soft Comput 12:419–423, 2008) and Turunen (Arch Math Log 40:467–473, 2001), we defined residuated lattice versions of them and study them in connection with Van Gasse et al. (Inf Sci 180(16):3006–3020, 2010), Lianzhen and Kaitai (Inf Sci 177:5725–5738, 2007), Zhu and Xu (Inf Sci 180:3614–3632, 2010). Also we consider some relations between these filters and quotient residuated lattice that are constructed via these filters.     

Insensitizing exact controls for the scalar wave equation and exact controllability of 2-coupled cascade systems of PDE’s by a single control

We study the exact controllability, by a reduced number of controls, of coupled cascade systems of PDE’s and the existence of exact insensitizing controls for the scalar wave equation. We give a necessary and sufficient condition for the observability of abstract-coupled cascade hyperbolic systems by a single observation, the observation operator being either bounded or unbounded. Our proof extends the two-level energy method introduced in Alabau-Boussouira (Siam J Control Opt 42:871–906, 2003) and Alabau-Boussouira and Léautaud (J Math Pures Appl 99:544–576, 2013) for symmetric coupled systems, to cascade systems which are examples of non-symmetric coupled systems. In particular, we prove the observability of two coupled wave equations in cascade if the observation and coupling regions both satisfy the Geometric Control Condition (GCC) of Bardos et al. (SIAM J Control Opt 30:1024–1065, 1992). By duality, this solves the exact controllability, by a single control, of $2$ -coupled abstract cascade hyperbolic systems. Using transmutation, we give null-controllability results for the multidimensional heat and Schrödinger $2$ -coupled cascade systems under GCC and for any positive time. By our method, we can treat cases where the control and coupling coefficients have disjoint supports, partially solving an open question raised by de Teresa (CPDE 25:39–72, 2000). Moreover we answer the question of the existence of exact insensitizing locally distributed as well as boundary controls of scalar multidimensional wave equations, raised by Lions (Actas del Congreso de Ecuaciones Diferenciales y Aplicaciones (CEDYA), Universidad de Málaga, pp 43–54, 1989) and later on by Dáger (Siam J Control Opt 45:1758–1768, 2006) and Tebou (C R Acad Sci Paris 346(Sér I):407–412, 2008).     

Comments and further improvements on “pth moment stability of stochastic neural networks with Markov volatilities”

In a very recent paper, Peng and Liu (Neural Comput Appl 20:543–547, 2011) investigated the pth moment stability of the stochastic Grossberg–Hopfield neural networks with Markov volatilities by Mao et al. (Bernoulli 6:73–90, 2000, Theorem 4.1). We should point out that Mao et al. (Bernoulli 6:73–90, 2000, Theorem 4.1) investigated the pth moment exponentially stable for a class of stochastic dynamical systems with constant delay; however, this theorem cannot apply to the case of variable time delays. It is also worthy to emphasize that Peng and Liu (Neural Comput Appl 20:543–547, 2011) discussed by Mao et al. (Bernoulli 6:73–90, 2000, Theorem 4.1) the pth moment exponentially stable for the Grossberg–Hopfield neural networks with variable delays, and therefore, there are some gaps between Peng and Liu (Neural Comput Appl 20:543–547, 2011, Theorem 1) and Mao et al. (Bernoulli 6:73–90, 2000, Theorem 4.1). In this paper, we fill up this gap. Moreover, a numerical example is also provided to demonstrate the effectiveness and applicability of the theoretical results.     

On sunflowers and matrix multiplication

We present several variants of the sunflower conjecture of Erd?s & Rado (J Lond Math Soc 35:85–90, 1960) and discuss the relations among them. We then show that two of these conjectures (if true) imply negative answers to the questions of Coppersmith & Winograd (J Symb Comput 9:251–280, 1990) and Cohn et al. (2005) regarding possible approaches for obtaining fast matrix-multiplication algorithms. Specifically, we show that the Erd?s–Rado sunflower conjecture (if true) implies a negative answer to the “no three disjoint equivoluminous subsets” question of Coppersmith & Winograd (J Symb Comput 9:251–280, 1990); we also formulate a “multicolored” sunflower conjecture in ${\mathbb{Z}_3^n}$ and show that (if true) it implies a negative answer to the “strong USP” conjecture of Cohn et al. (2005) (although it does not seem to impact a second conjecture in Cohn et al. (2005) or the viability of the general group-theoretic approach). A surprising consequence of our results is that the Coppersmith–Winograd conjecture actually implies the Cohn et al. conjecture. The multicolored sunflower conjecture in ${\mathbb{Z}_3^n}$ is a strengthening of the well-known (ordinary) sunflower conjecture in ${\mathbb{Z}_3^n}$ , and we show via our connection that a construction from Cohn et al. (2005) yields a lower bound of (2.51 . . .) ⁿ on the size of the largest multicolored 3-sunflower-free set, which beats the current best-known lower bound of (2.21 . . . ) ⁿ Edel (2004) on the size of the largest 3-sunflower-free set in ${\mathbb{Z}_3^n}$ .     

Cross-categorization of legal concepts across boundaries of legal systems: in consideration of inferential links

This work contrasts Giovanni Sartor’s view of inferential semantics of legal concepts (Sartor in Artif Intell Law 17:217–251, 2009) with a probabilistic model of theory formation (Kemp et al. in Cognition 114:165–196, 2010). The work further explores possibilities of implementing Kemp’s probabilistic model of theory formation in the context of mapping legal concepts between two individual legal systems. For implementing the legal concept mapping, we propose a cross-categorization approach that combines three mathematical models: the Bayesian Model of Generalization (BMG; Tenenbaum and Griffiths in Behav Brain Sci 4:629–640, 2001), the probabilistic model of theory formation, i.e., the Infinite Relational Model (IRM) first introduced by Kemp et al. (The twenty-first national conference on artificial intelligence, 2006, Cognition 114:165–196, 2010) and its extended model, i.e., the normal-IRM (n-IRM) proposed by Herlau et al. (IEEE International Workshop on Machine Learning for Signal Processing, 2012). We apply our cross-categorization approach to datasets where legal concepts related to educational systems are respectively defined by the Japanese- and the Danish authorities according to the International Standard Classification of Education. The main contribution of this work is the proposal of a conceptual framework of the cross-categorization approach that, inspired by Sartor (Artif Intell Law 17:217–251, 2009), attempts to explain reasoner’s inferential mechanisms.     

Weights Design For Maximal Order WENO Schemes

Weighted essentially non-oscillatory (WENO) finite difference schemes, developed by Liu et al. (Comput Phys 115(1):200–212, 1994) and improved by Jiang and Shu (Comput Phys 126(1):202–228, 1996), are one of the most popular methods to approximate the solutions of hyperbolic equations. But these schemes fail to provide maximal order accuracy near smooth extrema, where the first derivative of the solution becomes zero. Some authors have addressed this problem with different weight designs. In this paper we focus on the weights proposed by Yamaleev and Carpenter (J Comput Phys 228:4248–4272, 2009). They propose new weights to provide faster weight convergence than those presented in Borges et al. (J Comput Phys 227:3191–3211, 2008) and deduce some constraints on the weights parameters to guarantee that the WENO scheme has maximal order for sufficiently smooth solutions with an arbitrary number of vanishing derivatives. We analyze the scheme with the weights proposed in Yamaleev and Carpenter (J Comput Phys 228:4248–4272, 2009) and prove that near discontinuities it achieves worse orders than classical WENO schemes. In order to solve these accuracy problems, we define new weights, based on those proposed in Yamaleev and Carpenter (J Comput Phys 228:4248–4272, 2009), and get some constraints on the weights parameters to guarantee maximal order accuracy for the resulting schemes.     

Testing for Structural Breaks at Unknown Time: A Steeplechase

This paper analyzes the role of common data problems when identifying structural breaks in small samples. Most notably, we survey small sample properties of the most commonly applied endogenous break tests developed by Brown et al. (J R Stat Soc B 37:149–163, 1975) and Zeileis (Stat Pap 45(1):123–131, 2004), Nyblom (J Am Stat Assoc 84(405):223–230, 1989) and Hansen (J Policy Model 14(4):517–533, 1992), and Andrews et al. (J Econ 70(1):9–38, 1996). Power and size properties are derived using Monte Carlo simulations. We find that the Nyblom test is on par with the commonly used F type tests in a small sample in terms of power. While the Nyblom test’s power decreases if the structural break occurs close to the margin of the sample, it proves far more robust to nonnormal distributions of the error term that are found to matter strongly in small samples although being irrelevant asymptotically for all tests that are analyzed in this paper.     

Toward more localized local algorithms: removing assumptions concerning global knowledge

Numerous sophisticated local algorithm were suggested in the literature for various fundamental problems. Notable examples are the MIS and $(\Delta +1)$ -coloring algorithms by Barenboim and Elkin (Distrib Comput 22(5–6):363–379, 2010), by Kuhn (2009), and by Panconesi and Srinivasan (J Algorithms 20(2):356–374, 1996), as well as the $O\mathopen {}(\Delta ^2)$ -coloring algorithm by Linial (J Comput 21:193, 1992). Unfortunately, most known local algorithms (including, in particular, the aforementioned algorithms) are non-uniform, that is, local algorithms generally use good estimations of one or more global parameters of the network, e.g., the maximum degree $\Delta $ or the number of nodes $n$ . This paper provides a method for transforming a non-uniform local algorithm into a uniform one. Furthermore, the resulting algorithm enjoys the same asymptotic running time as the original non-uniform algorithm. Our method applies to a wide family of both deterministic and randomized algorithms. Specifically, it applies to almost all state of the art non-uniform algorithms for MIS and Maximal Matching, as well as to many results concerning the coloring problem (In particular, it applies to all aforementioned algorithms). To obtain our transformations we introduce a new distributed tool called pruning algorithms, which we believe may be of independent interest.     

The Duo-Item Bisection Auction

This paper proposes an iterative sealed-bid auction for selling multiple heterogeneous items to bidders interested in buying at most one item. It generalizes the single item bisection auction (Grigorieva et al. Econ Theory, 30:107–118, 2007) to the environment with multiple heterogeneous items. We focus on the case with two items for sale. We show that the auction elicits a minimal amount of information on preferences required to find the Vickrey–Clark–Groves outcome (Clarke, Public Choice, XI:17–33, 1971; Groves, Econometrica, 61:617–631, 1973; Vickrey, J Finance, 16:8–37, 1961), when there are two items for sale and an arbitrary number of bidders.     

Modeling and optimization of the hot embossing process for micro- and nanocomponent fabrication

Hot embossing, a polymer molding process conceived by Forschungszentrum Karlsruhe, is one of the established replication processes for microstructures The process is especially well suited for manufacturing small and medium series of microcomponents (SPIE Conference 1997; Polymer News 25:224–229, 2000; J Micromech Microeng 14:R1–14, 2004; Sensors Actuators 3:130–135, 2000). However, a wider application of the process currently is seriously hampered by the lack of adequate simulation tools for process optimization and part design. This situation is becoming more critical, as the dimension of the microstructures shrink from micron and submicron levels to the nanoscale and as productivity requirements dictate the enlargement of formats to process larger numbers of devices in parallel. Based on the current scientific work (Forschungszentrum Karlsruhe, FZKA-Bericht 7058 2003; DTIP Conference Montreux 2004; Microsystem Tech 10:432–437 2004), a German–Canadian cooperation has been started. The objective of this cooperation is to fill the gap mentioned above by developing reliable computer models and simulation tools for the hot embossing process and to incorporate these models in a user-friendly computer code. The present paper will give an overview of the activities in the project. The activities related to material characterization, especially the development of a viscoelastic material model, the characterization of friction between polymer and mold during demolding, the development of an 8-in. microstructured mold, and the fabrication of nanostructured molds will be discussed.     

An algebraic axiomatization of the Ewald’s intuitionistic tense logic

Ewald (J Symbolic Logic 51(1):166–179, 1986) considered tense operators \(G\) , \(H\) , \(F\) and \(P\) on intuitionistic propositional calculus and constructed an intuitionistic tense logic system called IKt. The aim of this paper is to give an algebraic axiomatization of the IKt system. We will also show that the algebraic axiomatization given by Chajda (Cent Eur J Math 9(5):1185–1191, 2011) of the tense operators \(P\) and \(F\) in intuitionistic logic is not in accordance with the Halmos definition of existential quantifiers. In this paper, we will study the IKt variety of IKt-algebras. First, we will introduce some examples and we will prove some properties. Next, we will prove that the IKt system has IKt-algebras as algebraic counterpart. We will also describe a discrete duality for IKt-algebras bearing in mind the results indicated by Or?owska and Rewitzky (Fundam Inform 81(1–3):275–295, 2007) for Heyting algebras. We will also get a general construction of tense operators on a complete Heyting algebra, which is a power lattice via the so-called Heyting frame. Finally, we will introduce the notion of tense deductive system which allowed us both to determine the congruence lattice in an IKt-algebra and to characterize simple and subdirectly irreducible algebras of the IKt variety.