Monotonic Problem Research Articles

Adaptive fixed point iterations for semilinear elliptic partial differential equations

In this paper we study the behaviour of finite dimensional fixed point iterations, induced by discretization of a continuous fixed point iteration defined within a Banach space setting. We show that the difference between the discrete sequence and its continuous analogue can be bounded in terms depending on the discretization of the infinite dimensional space and the contraction factor, defined by the continuous iteration. Furthermore, we show that the comparison between the finite dimensional and the continuous fixed point iteration naturally paves the way towards a general a posteriori error analysis that can be used within the framework of a fully adaptive solution procedure. In order to demonstrate our approach, we use the Galerkin approximation of singularly perturbed semilinear monotone problems. Our scheme combines the fixed point iteration with an adaptive finite element discretization procedure (based on a robust a posteriori error analysis), thereby leading to a fully adaptive fixed-point-Galerkin scheme. Numerical experiments underline the robustness and reliability of the proposed approach.

Monotonic classification: An overview on algorithms, performance measures and data sets

Currently, knowledge discovery in databases is an essential first step when identifying valid, novel and useful patterns for decision making. There are many real-world scenarios, such as bankruptcy prediction, option pricing or medical diagnosis, where the classification models to be learned need to fulfill restrictions of monotonicity (i.e. the target class label should not decrease when input attributes values increase). For instance, it is rational to assume that a higher debt ratio of a company should never result in a lower level of bankruptcy risk. Consequently, there is a growing interest from the data mining research community concerning monotonic predictive models. This paper aims to present an overview of the literature in the field, analyzing existing techniques and proposing a taxonomy of the algorithms based on the type of model generated. For each method, we review the quality metrics considered in the evaluation and the different data sets and monotonic problems used in the analysis. In this way, this paper serves as an overview of monotonic classification research in specialized literature and can be used as a functional guide for the field.

The Ordering of Shannon Entropies for the Multivariate Distributions and Distributions of Eigenvalues.

In this paper, we prove the Shannon entropy inequalities for the multivariate distributions via the notion of convex ordering of two multivariate distributions. We further characterize the multivariate totally positive of order 2 () property of the distribution functions of eigenvalues of both central Wishart and central MANOVA models, and of both noncentral Wishart and noncentral MANOVA models under the general population covariance matrix set-up. These results can be directly applied to both the comparisons of two Shannon entropy measures and the power monotonicity problem for the MANOVA problem.

Acceleration and Global Convergence of a First-Order Primal-Dual Method for Nonconvex Problems

The primal--dual hybrid gradient method (PDHGM, also known as the Chambolle--Pock method) has proved very successful for convex optimization problems involving linear operators arising in image processing and inverse problems. In this paper, we analyze an extension to nonconvex problems that arise if the operator is nonlinear. Based on the idea of testing, we derive new step length parameter conditions for the convergence in infinite-dimensional Hilbert spaces and provide acceleration rules for suitably (locally and/or partially) monotone problems. Importantly, we prove linear convergence rates as well as global convergence in certain cases. We demonstrate the efficacy of these step length rules for PDE-constrained optimization problems.

Homogenization of a nonlinear monotone problem with nonlinear Signorini boundary conditions in a domain with highly rough boundary

In this paper, we consider a domain Ωε⊂RN, N≥2, with a very rough boundary depending on ε. For instance, if N=3Ωε has the form of a brush with an ε-periodic distribution of thin cylindrical teeth with fixed height and a small diameter of order ε. In Ωε we consider a nonlinear monotone problem with nonlinear Signorini boundary conditions, depending on ε, on the lateral boundary of the teeth. We study the asymptotic behavior of this problem, as ε vanishes, i.e. when the number of thin attached cylinders increases unboundedly, while their cross sections tend to zero. We identify the limit problem which is a nonstandard homogenized problem. Namely, in the region filled up by the thin cylinders the limit problem is given by a variational inequality coupled to an algebraic system.

Transmit-Power Minimization for NOMA-Enabled Traffic Offloading With Security Provisioning

Non-orthogonal multiple access (NOMA) has been proposed as an effective technique to cope with the enormous mobile traffic in wireless communication networks in a spectrum-efficient manner. Meanwhile, to further relieve the traffic pressure, 3GPP has proposed dual-connectivity (DC), through which each mobile user can simultaneously communicate with the macro cell and small cell. In this letter, we investigate the transmit-power minimization problem for DC-based traffic offloading in downlink NOMA systems while taking the security provisioning into account. The proposed power allocation problem is difficult to solve due to natural non-convexity. In spite of this, we transform it into an equivalent traffic scheduling problem by exploiting the feature of DC. The transformed traffic scheduling problem is proved to be a monotonic optimization problem which can be solved by the polyblock approximation method. Numerical results are provided to demonstrate the performance and advantages of our proposed traffic offloading scheme with security consideration.

Drawbacks of Uncertainty Measures Based on the Pignistic Transformation

Recently, a measure of “total uncertainty” (TU) in Dempster–Shafer theory, based on the pignistic distribution called ambiguity measure (AM), have been modified. The resulting new measure has been simply referred as modified AM (MAM). In the literature, it has been shown that AM, in addition to showing some undesirable behaviors, has important drawbacks related to two essential properties for such measures: 1) subadditivity and 2) monotonicity. The MAM measure has been developed to solve the AM subadditivity problem, but this paper demonstrates that MAM suffers the same drawback as AM with respect to monotonicity. A measure of uncertainty that cannot meet the monotonicity requirement has an important drawback for its exploitation in operational contexts such as in analytics, information fusion, and decision support. This paper aims at identifying and discussing drawbacks of this type of measures (AM, MAM). Our main motivation is to insist upon the important requirement of monotonicity that a TU measure should possess to improve its potential of being used and trusted in applications. This discussion is due time since the monotonicity problem needs first to be solved to avoid building too high expectations for usefulness and potential exploitation of such measures in operational communities.

Switching-iterative learning control method for discrete-time switching system

In this paper, we propose a new monotonically convergent switching iterative learning control for a class of linear discrete time switched system. It is assumed that the considered switched systems are operated during a finite time interval repetitively, and then the iterative learning control scheme can be introduced. After the switched system is transformed into a 2D repetitive system, sufficient conditions in terms of linear matrix inequalities (LMIs) are derived by using a Lyapunov functional approach and a quadratic performance function. It is shown that if certain LMIs are met, the tracking error $$l_2$$ norm converges monotonically to zero between (subsystem/iteration), while the switching learning gains could be determined directly by solving the LMIs.The integrated design of this SILC scheme is transformed into a robust monotonic stabilizability problem (RMS) of an uncertain switched system. A numerical simulation example is established shown the effectiveness of the proposed method .

Partial coherence with application to the monotonicity problem of coherence involving skew information

Quantifications of coherence are intensively studied in the context of completely decoherent operations (i.e., von Neuamnn measurements, or equivalently, orthonormal bases) in recent years. Here we investigate partial coherence (i.e., coherence in the context of partially decoherent operations such as L\uders measurements). A bona fide measure of partial coherence is introduced. As an application, we address the monotonicity problem of $K$-coherence (a quantifier for coherence in terms of Wigner-Yanase skew information) [Girolami, Phys. Rev. Lett. 113, 170401 (2014)], which is introduced to realize a measure of coherence as axiomatized by Baumgratz, Cramer, and Plenio [Phys. Rev. Lett. 113, 140401 (2014)]. Since $K$-coherence fails to meet the necessary requirement of monotonicity under incoherent operations, it is desirable to remedy this monotonicity problem. We show that if we modify the original measure by taking skew information with respect to the spectral decomposition of an observable, rather than the observable itself, as a measure of coherence, then the problem disappears, and the resultant coherence measure satisfies the monotonicity. Some concrete examples are discussed and related open issues are indicated.

Closeness matters: monotonicity failure in IRV elections with three candidates

A striking attribute of instant runoff voting (IRV) is that it is subject to monotonicity failure—that is, getting more (first-preference) votes may result in defeat for a candidate who would otherwise have won and getting fewer votes may result in victory for a candidate who otherwise would have lost. Proponents of IRV have argued that monotonicity failure, while a mathematical possibility, is highly unlikely to occur in practice. This paper specifies the precise conditions under which this phenomenon arises in three-candidate elections and applies them to a number of large simulated data sets in order to get a sense of the likelihood of IRV’s monotonicity problem in varying circumstances. The basic finding is that the problem is significant in many circumstances and very substantial when IRV elections are closely contested by three candidates.

Log-majorization and Lie–Trotter formula for the Cartan barycenter on probability measure spaces

We extend Ando–Hiai's log-majorization for the weighted geometric mean of positive definite matrices into that for the Cartan barycenter in the general setting of probability measures on the Riemannian manifold of positive definite matrices equipped with trace metric. The main key is the settlement of the monotonicity problem of the Cartan barycenteric map on the space of probability measures with finite first moment for the stochastic order induced by the cone. We also derive a version of Lie–Trotter formula and related unitarily invariant norm inequalities for the Cartan barycenter as the main application of log-majorization.

RULEM: A novel heuristic rule learning approach for ordinal classification with monotonicity constraints

In many real world applications classification models are required to be in line with domain knowledge and to respect monotone relations between predictor variables and the target class, in order to be acceptable for implementation. This paper presents a novel heuristic approach, called RULEM, to induce monotone ordinal rule based classification models. The proposed approach can be applied in combination with any rule- or tree-based classification technique, since monotonicity is guaranteed in a post-processing step. RULEM checks whether a rule set or decision tree violates the imposed monotonicity constraints and existing violations are resolved by inducing a set of additional rules which enforce monotone classification. The approach is able to handle non-monotonic noise, and can be applied to both partially and totally monotone problems with an ordinal target variable. Two novel justifiability measures are introduced which are based on RULEM and allow to calculate the extent to which a classification model is in line with domain knowledge expressed in the form of monotonicity constraints. An extensive benchmarking experiment and subsequent statistical analysis of the results on 14 public data sets indicates that RULEM preserves the predictive power of a rule induction technique while guaranteeing monotone classification. On the other hand, the post-processed rule sets are found to be significantly larger which is due to the induction of additional rules. E.g., when combined with Ripper a median performance difference was observed in terms of PCC equal to zero and an average difference equal to −0.66%, with on average 5 rules added to the rule sets. The average and minimum justifiability of the original rule sets equal respectively 92.66% and 34.44% in terms of the RULEMF justifiability index, and 91.28% and 40.1% in terms of RULEMS, indicating the effective need for monotonizing the rule sets.

Application of Virtual Reality Technology in Swimming Teaching

Swimming is one of the most popular spots among college students, though it is hard to master. With virtual reality (VR) technology, problems of the low-cross mutual inductance and low efficiency can be solved. A physical model and the software framework of the virtual reality system was proposed. The swimming teaching system can solve the problem that perception-action separation, improve the study ecological validity, and hold great promise for the study of perception- action. A comparison analysis between students' correct and wrong swimming movements is carried out, and the analysis results show that the virtual reality system solves the problem of monotonicity in real swimming teaching.

Flash Memories: ISPP Renewal Theory and Flash Design Tradeoffs

In the write process of multilevel per cell (MLC) flash memories, an iterative approach is used to mitigate the monotonicity problem. The monotonicity in programming is considered to be the major restriction in MLC flash. To solve this issue, an iterative approach called incremental step pulse programming (ISPP) is used to concurrently program lots of cells in small steps. In this paper, we are mostly concerned with deriving a mathematical model for iterative programming using the framework of renewal theory . We obtain a closed-form approximation for the probability distribution of the number of steps required in the ISPP process. We also bound the maximal error between the true distribution and our approximation. Moreover, the results obtained help to accurately analyze the effect of inter-cell interference in this type of memory. Finally, we devise an adaptive step size approach for write process to strike a balance between latency and lifetime under fixed bit error rate constraints or information rate constraints.

Iterative Galerkin discretizations for strongly monotone problems

In this article we investigate the use of fixed point iterations to solve the Galerkin approximation of strictly monotone problems. As opposed to Newton’s method, which requires information from the previous iteration in order to linearize the iteration matrix (and thereby to recompute it) in each step, the alternative method used in this article exploits the monotonicity properties of the problem, and only needs the iteration matrix calculated once for all iterations of the fixed point method. We outline the abstract a priori and a posteriori analyses for the iteratively obtained solutions, and apply this to a finite element approximation of a second-order elliptic quasilinear boundary value problem. We show both theoretically, as well as numerically, how the number of iterations of the fixed point method can be restricted in dependence of the mesh size, or of the polynomial degree, to obtain optimal convergence. Using the a posteriori error analysis we also devise an adaptive algorithm for the generation of a sequence of Galerkin spaces (adaptively refined finite element meshes in the concrete example) to minimize the number of iterations on each space.

Monotonicity of the period of a non linear oscillator

We revisit the problem of monotonicity of the period function for the differential equation u″−u+up=0 and give a simple proof of recent results of Miyamoto and Yagasaki.

The generalized L p -mixed affine surface area

In this article, we put forward the concept of the (i, j)-type L p -mixed affine surface area, such that the notion of L p -affine surface area which be shown by Lutwak is its special cases. Furthermore, applying this concept, the Minkowski inequality for the (i,−p)-type L p -mixed affine surface area and the extensions of the well-known L p -Petty affine projection inequality are established, respectively. Besides, we give an affirmative answer for the generalized L p -Winterniz monotonicity problem.

Solving Helmholtz problems with the boundary element method using direct radial basis function interpolation

In the present study, a direct interpolation technique that uses radial basis functions is applied to the boundary element method integral term, which refers to inertia, in the Helmholtz equation; consequently, free vibration frequencies and corresponding amplitudes can be determined from an eigenvalue problem solution. The proposed method, which has already been successfully applied to scalar problems governed by the Poisson equation, does not require standard domain integration procedures, which employ cell discretisation, and is more robust than the dual-reciprocity technique. Although similar to the latter in some aspects, because it uses radial basis functions and their primitives for interpolation, the proposed methodology is more general. It allows the immediate use of interpolation functions of any type, and there are no convergence or monotonicity problems as the number of basis points is increased.

Convergence analysis of the proximal point algorithm for pseudo-monotone equilibrium problems

In this paper, we study the weak and strong convergence of the proximal point algorithm for equilibrium problems of pseudo-monotone type in Hilbert spaces. We prove the weak convergence of the generated sequence to a common solution of two equilibrium problems and some strong convergence results with additional assumptions on pseudo-monotone bifunctions. Then we study a regularization of Halpern-type and prove the strong convergence of the generated sequence to an equilibrium point of two pseudo-monotone bifunctions without any additional assumption on bifunctions. Finally, some examples of pseudo-monotone bifunctions from pseudo-monotone operators and Nash-Cournot oligopolistic equilibrium models are also presented. Our results extend some similar results in the literature for monotone and pseudo-monotone equilibrium problems and also the related results for variational inequalities associated with monotone and pseudo-monotone operators.

Optimal Control in Matrix-Valued Coefficients for Nonlinear Monotone Problems: Optimality Conditions II

In this paper we study an optimal control problem for a nonlinear monotone Dirichlet problem where the controls are taken as the matrix-valued coefficients in L^{\infty}(\Omega; \mathbb R^{N \times N}) . Given a suitable cost function, the objective is to provide a substantiation of the first order optimality conditions using the concept of convergence in variable spaces. While in the first part [Z. Anal. Anwend. 34 (2015), 85–108] optimality conditions have been derived and analysed in the general case under some assumptions on the quasi-adjoint states, in this second part, we consider diagonal matrices and analyse the corresponding optimality system without such assumptions.