An algebraic characterization of the optimum of regularized kernel methods

The representer theorem for kernel methods states that the solution of the associated variational problem can be expressed as the linear combination of a finite number of kernel functions. However, for non-smooth loss functions, the analytic characterization of the coefficients poses nontrivial problems. Standard approaches resort to constrained optimization reformulations which, in general, lack a closed-form solution. Herein, by a proper change of variable, it is shown that, for any convex loss function, the coefficients satisfy a system of algebraic equations in a fixed-point form, which may be directly obtained from the primal formulation. The algebraic characterization is specialized to regression and classification methods and the fixed-point equations are explicitly characterized for many loss functions of practical interest. The consequences of the main result are then investigated along two directions. First, the existence of an unconstrained smooth reformulation of the original non-smooth problem is proven. Second, in the context of SURE (Stein’s Unbiased Risk Estimation), a general formula for the degrees of freedom of kernel regression methods is derived.     

Boolean functions of an odd number of variables with maximum algebraic immunity

In this paper, we study Boolean functions of an odd number of variables with maximum algebraic immunity. We identify three classes of such functions, and give some necessary conditions of such functions, which help to examine whether a Boolean function of an odd number of variables has the maximum algebraic immunity. Further, some necessary conditions for such functions to have also higher nonlinearity are proposed, and a class of these functions are also obtained. Finally, we present a sufficient and necessary condition for Boolean functions of an odd number of variables to achieve maximum algebraic immunity and to be also 1-resilient.     

Convergence of Craig’s method for linear algebraic systems

Craig??s iterative method is designed to solve linear algebraic systems with an asymmetric (or even rectangular) matrix. A simple representation is constructed for the method. Test examples are used to study iterative convergence and compare the method with the conjugate gradient method. Although round-off errors in Craig??s method proved to slow down iterative convergence significantly, they allowed attaining high accuracy (if the matrix was well-conditioned). An efficient iteration-stopping criterion is found.     

Solution of discrete regulation problems using linear algebraic equations†

The solution of linear algebraic equations are used as a method of synthesizing linear, deterministic, multivariable, discrete systems for either stable or unstable open loop plants, Well-defined relationships are established between weights of a quadratic performance index and plant dynamics via the optimal system determinant.

The discrete calculus of variations provides the necessary conditions for minimizing a quadratic cost functional. This gives closed form expressions for U°(z) and Y°(z) and generates the optimal system.     

Partitions of the set of selected unknowns in linear differential–algebraic systems

As was shown earlier, for a linear differential–algebraic system A ₁ y′ + A ₀ y = 0 with a selected part of unknowns (entries of a column vector y), it is possible to construct a differential system ?′ = B ?, where the column vector ? is formed by some entries of y, and a linear algebraic system by means of which the selected entries that are not contained in ? can be expressed in terms of the selected entries included in ?. In the paper, sizes of the differential and algebraic systems obtained are studied. Conditions are established under the fulfillment of which the size of the algebraic system is determined unambiguously and the size of the differential system is minimal.     

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.     

Feedback control of nonlinear differential algebraic systems using Hamiltonian function method

Many complex dynamic systems, such as power systems, robotic systems, etc. can be modeled as the following nonlinear differential algebraic systems (NDAS) [1-4](1) where the vector 1 x1 ∈ X 1 ? Rn represents the state variable, and x2 ∈X2? Rn2 is the al…     

Analysis and control of general logical networks – An algebraic approach

Since Boolean network is a powerful tool in describing the genetic regulatory networks, accompanying the development of systems biology, the analysis and control of Boolean networks have attracted much attention from biologists, physicists, and systems scientists. From mathematical point of view, the dynamics of a Boolean (control) network is a discrete-time logical dynamic process. This paper surveys a recently developed technique, called the algebraic approach, based on semi-tensor product. The new technique can deal with not only Boolean networks, which allow each node to take two values, but also k-valued networks, which allow each node to take k different values, and mix-valued networks, which allow nodes to take different numbers of values.The paper provides a comprehensive introduction to the new technique, including (1) mathematical background of this new technique – semi-tensor product of matrices and the matrix expression of logic; (2) dynamic models of Boolean networks, and general (multi- or mix-valued) logical networks; (3) the topological structure of Boolean networks and general networks; (4) the basic control problems of Boolean/general control networks, which include the controllability, observability, realization, stability and stabilization, disturbance decoupling, identification and optimization, etc.; (5) some other related applications.     

On two versions of the Loomis–Sikorski Theorem for algebraic structures

We present two versions of the Loomis–Sikorski Theorem, one for monotone σ-complete generalized pseudo effect algebras with strong unit satisfying a kind of the Riesz decomposition property. The second one is for Dedekind σ-complete positive pseudo Vitali spaces with strong unit. For any case we can find an appropriate system of nonnegative bounded functions forming an algebra of the given type with the operations defined by points that maps epimorphically onto the algebra. The paper has been supported by the Center of Excellence SAS—Physics of Information—I/2/2005, the grant VEGA No. 2/6088/26 SAV, the Slovak Research and Development Agency under the contract No. APVV-0071-06, Slovak-Italian Project No. 15:“Algebraic and logical systems of soft computing”, and MURST, project “Analisi Reale”.     

Real-time rendering of algebraic B-spline surfaces via Bzier point insertion

This paper presents a GPU-based real-time raycasting algorithm for piecewise algebraic surfaces in terms of tensor product B-splines.3DDDA and depth peeling algorithms are employed to traverse the piecewise surface patches along each ray.The intersection between the ray and the patch is reduced to the root-finding problem of the univariate Bernstein polynomial.The polynomial is obtained via Chebyshev sampling points interpolation.An iterative and unconditionally convergent algorithm called B′ezier point insertion is proposed to find the roots of the univariate polynomials.The B′ezier point insertion is robust and suitable for the SIMD architecture of GPU.Experimental results show that the proposed root-finding algorithm performs better than other root-finding algorithms,such as B′ezier clipping and B-spline knot insertion.Our rendering algorithm can display thousands of piecewise algebraic patches of degrees 6–9 in real time and can achieve the semi-transparent rendering interactively.     

An algebraic variational multiscale–multigrid method for large eddy simulation of turbulent flow

An algebraic variational multiscale–multigrid method is proposed for large eddy simulation of turbulent flow. Level-transfer operators from plain aggregation algebraic multigrid methods are employed for scale separation. In contrast to earlier approaches based on geometric multigrid methods, this purely algebraic strategy for scale separation obviates any coarse discretization besides the basic one. Operators based on plain aggregation algebraic multigrid provide a projective scale separation, enabling an efficient implementation of the proposed method. The application of the algebraic variational multiscale–multigrid method to turbulent flow in a channel produces results notably closer to reference (direct numerical simulation) results than other state-of-the-art methods both for mean streamwise and root-mean-square velocities. For predicting highly sensitive components of the Reynolds-stress tensor in the context of turbulent recirculating flow in a lid-driven cavity, the algebraic variational multiscale–multigrid method also shows a remarkably good performance in predicting reference results from experiment and direct numerical simulation compared to other methods.     

Spectral–spatial classification of hyperspectral images by algebraic multigrid based multiscale information fusion

In this work, we present a novel spectral-spatial classification framework of hyperspectral images (HSIs) by integrating the techniques of algebraic multigrid (AMG), hierarchical segmentation (HSEG) and Markov random field (MRF). The proposed framework manifests two main contributions. First, an effective HSI segmentation method is developed by combining the AMG-based marker selection approach and the conventional HSEG algorithm to construct a set of unsupervised segmentation maps in multiple scales. To improve the computational efficiency, the fast Fish Markov selector (FMS) algorithm is exploited for feature selection before image segmentation. Second, an improved MRF energy function is proposed for multiscale information fusion (MIF) by considering both spatial and inter-scale contextual information. Experiments were performed using two airborne HSIs to evaluate the performance of the proposed framework in comparison with several popular classification methods. The experimental results demonstrated that the proposed framework can provide superior performance in terms of both qualitative and quantitative analysis.     

On the robustness and optimality of algebraic multilevel methods for reaction–diffusion type problems

This paper is on preconditioners for reaction–diffusion problems that are both, uniform with respect to the reaction–diffusion coefficients, and optimal in terms of computational complexity. The considered preconditioners belong to the class of so-called algebraic multilevel iteration (AMLI) methods, which are based on a multilevel block factorization and polynomial stabilization. The main focus of this work is on the construction and on the analysis of a hierarchical splitting of the conforming finite element space of piecewise linear functions that allows to meet the optimality conditions for the related AMLI preconditioner in case of second-order elliptic problems with non-vanishing zero-order term. The finite element method (FEM) then leads to a system of linear equations with a system matrix that is a weighted sum of stiffness and mass matrices. Bounds for the constant \(\gamma \) in the strengthened Cauchy–Bunyakowski–Schwarz inequality are computed for both mass and stiffness matrices in case of a general \(m\) -refinement. Moreover, an additive preconditioner is presented for the pivot blocks that arise in the course of the multilevel block factorization. Its optimality is proven for the case \(m=3\) . Together with the estimates for \(\gamma \) this shows that the construction of a uniformly convergent AMLI method with optimal complexity is possible (for \(m \ge 3\) ). Finally, we discuss the practical application of this preconditioning technique in the context of time-periodic parabolic optimal control problems.     

Application of computational invariant theory to Kobayashi hyperbolicity and to Green–Griffiths algebraic degeneracy

A major unsolved problem (according to Demailly (1997)) towards the Kobayashi hyperbolicity conjecture in optimal degree is to understand jet differentials of germs of holomorphic discs that are invariant under any reparametrization of the source. The underlying group action is not reductive, but we provide a complete algorithm to generate all invariants, in arbitrary dimension n$n$ and for jets of arbitrary order k$k$.     

COOPT — a software package for optimal control of large-scale differential–algebraic equation systems

This paper describes the functionality and implementation of COOPT. This software package implements a direct method with modified multiple shooting type techniques for solving optimal control problems of large-scale differential–algebraic equation (DAE) systems. The basic approach in COOPT is to divide the original time interval into multiple shooting intervals, with the DAEs solved numerically on the subintervals at each optimization iteration. Continuity constraints are imposed across the subintervals. The resulting optimization problem is solved by sparse sequential quadratic programming (SQP) methods. Partial derivative matrices needed for the optimization are generated by DAE sensitivity software. The sensitivity equations to be solved are generated via automatic differentiation.COOPT has been successfully used in solving optimal control problems arising from a wide variety of applications, such as chemical vapor deposition of superconducting thin films, spacecraft trajectory design and contingency/recovery problems, and computation of cell traction forces in tissue engineering.     

The numerical solution of higher index differential–algebraic equations by 4-point spline collocation methods

We study the order, stability, and convergence properties of 4-point spline collocation methods if applied to differential/algebraic systems with index greater than or equal one. These methods do not in general attain the same order of accuracy for higher index differential/algebraic systems as they do for index-1 systems and for purely differential systems. We show that the 4-point spline collocation methods applied to differential/algebraic systems with index-ν are stable and the order of convergence is 8???ν, ν?=?2(1)7. For both index-1 and purely differential systems the order is seven. Finally, some numerical experiments are presented that illustrate the theoretical results.     

An algebraic Riccati equation approach to H∞ optimization

This paper considers the general (so-called four block) H^∞ optimal control problem with the assumption that system states are available for feedback. It is shown that infimization of the H^∞ norm of the closed loop transfer function over all linear constant, i.e., nondynamic, stabilizing state feedback laws can be completely characterized via an algebraic Riccati equation. It is further shown that the optimal norm is not improved by allowing feedback to be dynamic. Thus, the general state-feedback H^∞ optimal control problem can be solved by iteratively solving one ARE and the controller can be chosen to be static gain.