Monotone Domain Research Articles

Convex Monotone Semigroups on Lattices of Continuous Functions

We consider convex monotone C_0 -semigroups on a Banach lattice, which is assumed to be a Riesz subspace of a \sigma -Dedekind complete Banach lattice. Typical examples include the space of all bounded uniformly continuous functions and the space of all continuous functions vanishing at infinity. We show that the domain of the classical generator of a convex semigroup is typically not invariant. Therefore, we propose alternative versions for the domain, such as the monotone domain and the Lipschitz set, for which we prove invariance under the semigroup. As a main result, we obtain the uniqueness of the semigroup in terms of an extended version of the generator. The results are illustrated with several examples related to Hamilton–Jacobi–Bellman equations, including nonlinear versions of the shift semigroup and the heat equation. In particular, we determine their symmetric Lipschitz sets, which are invariant and allow us to define the generators in a weak sense.

Incorporating monotonic domain knowledge in support vector learning for data mining regression problems

A common problem of data-driven data mining methods is that they might lack considering domain knowledge, despite possibly having high accuracy with respect to the data. As such, prior knowledge plays an important role in many data mining applications. Incorporating prior knowledge into data mining techniques is not trivial and remains a partially open issue drawing much attention. In this paper, we propose a new support vector regression (SVR) model that takes into account the prior knowledge of domain experts in the form of inequalities, which reflect the monotonic relationship between the output and some of the attributes of the input. A dual quadratic programming problem corresponding to the SVR model is derived, along with algorithms for solving it and creating constraints, respectively. The experiment results, which were conducted on two artificial and two practical datasets, show that the proposed model, which considers the monotonicity defined by domain experts, performs better than the original SVR. Moreover, the proposed method is also suitable for prior domain knowledge of piecewise monotonicity.

Monotone domain decomposition iterative technique for boundary value problems of a nonlinear fractional differential equation

In this paper, we present a monotone domain decomposition iterative technique for boundary value problems of a nonlinear fractional differential equation. We construct two monotone domain decomposition sequences of upper and lower solutions which converge uniformly to the actual solution of the problem. The accuracy and efficiency of our new approach are demonstrated through two examples.

Monotonicity and revenue equivalence domains by monotonic transformations in differences

In a mechanism design setting with quasilinear preferences, a domain D of admissible valuations of an agent is called a monotonicity domain if every 2-cycle monotone allocation rule is truthfully implementable (in dominant strategies). D is called a revenue equivalence domain if every implementable allocation rule satisfies revenue equivalence. Carbajal and Müller (2015) introduced the notions of monotonic transformations in differences and showed that if D admits these transformations then it is a revenue equivalence and monotonicity domain. Here, we show that various economic domains, with countable or uncountable allocation sets, admit monotonic transformations in differences. Our applications include public and private supply of divisible public goods, multi-unit auction-like environments with increasing valuations, and allocation problems with externalities. Single-peaked domains admit only a modified version of monotonic transformations in differences. We show that this property implies too that single-peaked domains are revenue and monotonicity domains.

Prototype selection to improve monotonic nearest neighbor

Student surveys occupy a central place in the evaluation of courses at teaching institutions. At the end of each course, students are requested to evaluate various aspects such as activities, methodology, coordination or resources used. In addition, a final qualification is given to summarize the quality of the course. The prediction of this final qualification can be accomplished by using monotonic classification techniques. The outcome offered by these surveys is particularly significant for faculty and teaching staff associated with the course.The monotonic nearest neighbor classifier is one of the most relevant algorithms in monotonic classification. However, it does suffer from two drawbacks, (a) inefficient execution time in classification and (b) sensitivity to no monotonic examples. Prototype selection is a data reduction process for classification based on nearest neighbor that can be used to alleviate these problems. This paper proposes a prototype selection algorithm called Monotonic Iterative Prototype Selection (MONIPS) algorithm. Our objective is two-fold. The first one is to introduce MONIPS as a method for obtaining monotonic solutions. MONIPS has proved to be competitive with classical prototype selection solutions adapted to monotonic domain. Besides, to further demonstrate the good performance of MONIPS in the context of a student survey about taught courses.

Infinite-Horizon Optimal Control for Deploying a Tethered Sub-satellite

An infinite-horizon optimal control problem for the deployment of a tethered subsatellite under complex nonlinear constraints is presented. The nonlinearities in the system model and the constraints of states and control are taken into consideration. Legendre-Gauss-Radau pseudo spectral method is described for solving the infinitehorizon optimal control problem numerically. By using a smooth, monotonic domain transformation technique, an infinite-horizon optimal control problem is approximated with a finite-horizon problem. The resulting problem in the finite interval is transcribed to a nonlinear programming problem. The proposed methods yield a series of approximations to the state on the entire horizon. The optimal control law can be determined via a nonlinear programming, without any iteration. A numerical example is included to demonstrate the high effectiveness and hard real-time of the proposed schemes.

Implementability under monotonic transformations in differences

In a social choice setting with quasilinear preferences and monetary transfers, a domain D of admissible valuations is called a monotonicity domain if every 2-cycle monotone allocation rule is truthfully implementable (in dominant strategies). D is called a revenue equivalence domain if every implementable allocation rule satisfies the revenue equivalence property. We introduce the notions of monotonic transformations in differences, which can be interpreted as extensions of Maskin's monotonic transformations to quasilinear environments, and show that if D admits these transformations then it is a monotonicity and revenue equivalence domain. Our proofs are elementary and do not rely on strenuous additional machinery. We illustrate monotonic transformations in differences for settings with finite and infinite allocation sets.

Monotone iterates for solving coupled systems of nonlinear parabolic equations

This paper deals with numerical solutions of coupled nonlinear parabolic equations. Using the method of upper and lower solutions, monotone sequences are constructed for difference schemes which approximate coupled systems of nonlinear parabolic equations. This monotone convergence leads to existence-uniqueness theorems. An analysis of convergence rates of the monotone iterative method is given. A monotone domain decomposition algorithm which combines the monotone approach and an iterative domain decomposition method based on the Schwarz alternating is proposed. A convergence analysis of the monotone domain decomposition algorithm is presented. An application to a gas–liquid interaction model is given.

Numerical solutions of coupled systems of nonlinear elliptic equations

AbstractThis article deals with numerical solutions of a general class of coupled nonlinear elliptic equations. Using the method of upper and lower solutions, monotone sequences are constructed for difference schemes which approximate coupled systems of nonlinear elliptic equations. This monotone convergence leads to existence‐uniqueness theorems for solutions to problems with reaction functions of quasi‐monotone nondecreasing, quasi‐monotone nonincreasing and mixed quasi‐monotone types. A monotone domain decomposition algorithm which combines the monotone approach and an iterative domain decomposition method based on the Schwarz alternating, is proposed. An application to a reaction‐diffusion model in chemical engineering is given. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 621–640, 2012

Monotonicity and Implementability

Consider an environment with a finite number of alternatives, and agents with private values and quasilinear utility functions. A domain of valuation functions for an agent is a monotonicity domain if every finite-valued monotone randomized allocation rule defined on it is implementable in dominant strategies. We fully characterize the set of all monotonicity domains.

Monotone iterates for solving systems of semilinear elliptic equations and applications

Consider monotone finite difference iterative algorithms for solving coupled systems of semilinear elliptic equations. A monotone domain decomposition algorithm based on a modification of Schwarz alternating method and on decomposition of a computational domain into nonoverlapping subdomains is constructed. Advantages of the algorithm are that the algorithm solves only linear discrete systems at each iterative step, converges monotonically to the exact solution of the nonlinear discrete problem, and is potentially parallelisable. The monotone domain decomposition algorithm is applied to a gas-liquid interaction model. Numerical experiments confirm theoretical results.

Block monotone domain decomposition methods for a nonlinear anisotropic convection-diffusion equation

This article deals with discrete monotone iterative methods for solving a quasi-linear, singularly perturbed, convection-diffusion problem. A block monotone domain decomposition method based on a Schwarz alternating method and on block (line) successive under-relaxation iterative method is constructed. The advantages of this monotone method are that the method solves only linear discrete systems at each iterative step of the iterative process and converges monotonically to the exact solution of the quasi-linear problem. Numerical experiments are presented. References I. Boglaev, A monotone Schwarz algorithm for a semilinear convection-diffusion problem. J. Numer. Math., 12:169--191, 2004. doi:10.1016/j.cam.2004.03.011 I. P. Boglaev, V. V. Sirotkin, A. Ya. Vilenkin, Ch. V. Kopetskii, A. V. Serebryakov, On the numerical simulation of heat transfer in the process of casting amorphous metal ribbons. Preprint, Institute of Solid State Physics, USSR Academy of Sciences, Chernogolovka, 1983. I. Boglaev and S. Pack, On block monotone domain decomposition algorithms for solving a nonlinear singularly perturbed convection-diffusion problem. Reports in Mathematics 17, IFS, Massey University, 2007. T. Chan and T. Mathew, Domain decomposition algorithms. Acta Numerica, 4:61--143, 1994. B. Smith, P. Bjorstad, and W. Gropp, Domain decomposition. Cambridge University Press, Cambridge, 1996. http://www.cambridge.org/0521602866 R. S. Varga, Matrix Iterative Analysis. Second Edition. Springer--Verlag, Berlin Heidelberg, 2000.

A monotone domain decomposition algorithm for nonlinear parabolic difference schemes

A monotone domain decomposition algorithm for a nonlinear algebraic system, which is a finite difference approximation of a nonlinear reaction-diffusion problem of parabolic type, is presented and is shown to converge monotonically either from above or from below to a solution of the system. The algorithm is based on a modification of the Schwarz alternating method and the method of upper and lower solutions. Advantages of the algorithm are that the algorithm solves only linear discrete systems at each iterative step, converges monotonically to the exact solution of the system, and is potentially parallelisable. Numerical experiments for a model problem from chemical engineering are presented.

The solution of a semilinear evolutionary convection–diffusion problem by a monotone domain decomposition algorithm

This paper deals with a monotone domain decomposition algorithm for solving a semilinear singularly perturbed convection–diffusion problem of parabolic type. On each time level, the monotone method (known as the method of lower and upper solutions) is applied to computing a nonlinear upwind difference scheme obtained after discretization of the continuous problem. A monotone domain decomposition algorithm based on a modification of the Schwarz alternating method is constructed. The rate of convergence of the monotone algorithm is estimated. Uniform convergence properties of the monotone domain decomposition algorithm are studied. Numerical experiments are presented.

Monotone algorithms for solving nonlinear monotone difference schemes of parabolic type in the canonical form

This paper deals with monotone iterative algorithms for solving nonlinear monotone difference schemes of parabolic type. Firstly, the monotone method (known as the method of lower and upper solutions) is applied to computing the nonlinear monotone difference schemes in the canonical form. Secondly, a monotone domain decomposition algorithm based on a modification of the Schwarz alternating method is constructed. This monotone algorithm solves only linear discrete systems at each iterative step and converges monotonically to the exact solution of the nonlinear monotone difference schemes. Numerical experiments are presented.

Monotone Iterates for Solving Nonlinear Monotone Difference Schemes

This paper is concerned with monotone iterative algorithms for solving nonlinear monotone difference schemes of elliptic type. Firstly, the monotone method (known as the method of lower and upper solutions) is applied to computing the nonlinear monotone difference schemes in the canonical form. Secondly, a monotone domain decomposition algorithm based on a modification of the Schwarz alternating method is constructed. This monotone algorithm solves only linear discrete systems at each iterative step and converges monotonically to the exact solution of the nonlinear monotone difference schemes. Numerical experiments are presented.

Monotone finite difference domain decomposition algorithms and applications to nonlinear singularly perturbed reaction-diffusion problems

This paper deals with monotone finite difference iterative algorithms for solving nonlinear singularly perturbed reaction-diffusion problems of elliptic and parabolic types. Monotone domain decomposition algorithms based on a Schwarz alternating method and on box-domain decomposition are constructed. These monotone algorithms solve only linear discrete systems at each iterative step and converge monotonically to the exact solution of the nonlinear discrete problems. The rate of convergence of the monotone domain decomposition algorithms are estimated. Numerical experiments are presented.

Monotone Schwarz iterates for a semilinear parabolic convection–diffusion problem

This paper deals with a discrete monotone iterative algorithm for solving a nonlinear singularly perturbed convection–diffusion problem of parabolic type. On each time level, the monotone method (known as the method of lower and upper solutions) is applied to computing a nonlinear upwind difference scheme obtained after discretisation of the continuous problem. A monotone domain decomposition algorithm based on a modification of the Schwarz alternating method is constructed. The rate of convergence of the monotone Schwarz method is estimated. Uniform convergence properties of the monotone domain decomposition algorithm are studied. Numerical experiments are presented.

Monotone iterative algorithms for a nonlinear singularly perturbed parabolic problem

This paper deals with discrete monotone iterative algorithms for solving a nonlinear singularly perturbed parabolic reaction–diffusion problem. Firstly, the monotone method (known as the method of lower and upper solutions) is applied to computing a nonlinear difference scheme obtained after discretisation of the continuous problem. Secondly, a monotone domain decomposition algorithm based on a modification of the Schwarz alternating method is constructed. This monotone algorithm solves only linear discrete systems at each iterative step of the iterative process. The rate of convergence of the monotone domain decomposition algorithm is estimated. Numerical experiments are presented.

A monotone Schwarz algorithm for a semilinear convection–diffusion problem

This paper deals with discrete monotone iterative algorithms for solving a nonlinear singularly perturbed convection–diffusion problem. Firstly, the monotone method (known as the method of lower and upper solutions) is applied to computing a nonlinear difference scheme obtained after discretisation of the continuous problem. Secondly, a monotone domain decomposition algorithm based on a modification of the Schwarz alternating method is constructed. This monotone algorithm solves only linear discrete systems at each iterative step of the iterative process. The rate of convergence of the monotone Schwarz method is estimated. Numerical experiments are presented.