Fixed Point Operator Research Articles

Modal characterisation of simulation relations in probabilistic concurrent games

Probabilistic game structures combine both nondeterminism and stochasticity, where players repeatedly take actions simultaneously to move to the next state of the concurrent game. Probabilistic alternating simulation and bisimulation are important tools to compare the behaviour of different probabilistic game structures. In this paper, we present a sound and complete modal characterisation of these two relations by proposing a new modal logic based on probability distributions. This logic enables a player to enforce a property in the next state or distribution. We further extend the logic with fixpoint operators, which also characterises the simulation relations. This logic can express a lot of interesting properties in practical applications. • Modal characterization for Probabilistic Alternating Simulation (and Bisimulation). • A new modal logic for probability distributions in concurrent game structures. • Extends the modal logic with fixpoint operators.

Nonlinear acceleration of momentum and primal-dual algorithms

We describe convergence acceleration schemes for multistep optimization algorithms where the underlying fixed-point operator is not symmetric. In particular, our analysis handles algorithms with momentum terms such as Nesterov’s accelerated method or primal-dual methods. The acceleration technique combines previous iterates through a weighted sum, whose coefficients are computed via a simple linear system. We analyze performance in both online and offline modes, and we study in particular a variant of Nesterov’s method that uses nonlinear acceleration at each iteration. We use Crouzeix’s conjecture to show that acceleration performance is controlled by the solution of a Chebyshev problem on the numerical range of a non-symmetric operator modeling the behavior of iterates near the optimum. Numerical experiments are detailed on logistic regression problems.

Research of ecological indicators of two-way vehicle in stationary conditions

The article concerns the research on the emission of pollutants of a rail-road tractor in two stationary research tests. The purpose of the tests was to carry out control tests of pollutant emissions and their analysis. The object used during the works was approved in accordance with the Stage V standard, which requires measurements of emissivity both in stationary, dynamic and real conditions. Despite the requirement to test engines installed on a vehicle during their normal duty cycle with PEMS, the emission limits measured in this test have not yet been defined. Therefore, the work below focuses on the stationary test cycle. The measurements were carried out in accordance with the combustion engine operating points described in the approval test, and then compared with the modernized NRSC test. It contains modified measuring points and rotational speeds of the crankshaft, adopted on the basis of the most common operating parameters of agricultural tractor combustion engines in real operating conditions. The measurements were performed with the use of a mobile dynamometer and devices for measuring emissions of harmful exhaust gas compounds and recording on-board data. In the test, the vehicle drive system worked at fixed operating points, with defined values of crankshaft rotational speed and load. Based on the recorded data on the concentrations of pollutants in the exhaust gases, the specific emission of the object was determined. In the final stage of the work, these data were used to perform a comparative analysis with the emission limits contained in the standard.

Температурное затухание генерации квантово-каскадных лазеров с частотами 2.3, 3.2, 4.1 ТГц

In this work, we conduct research of spectral and power characteristics of quantum cascade lasers (QCLs) based on a GaAs/Al0.15Ga0.85As active region emitting at 2.3 (A), 3.2 (B) and 4.1 (C) THz. The QCL devices had a double-metal Au waveguide and operated in pulsed mode with 1.5–9 us pulses at 20 Hz repetition rate. Using the integral output power curves measured with different pulse durations, we consider the potential mechanisms of QCL temperature degradation using Arrhenius plots. Moreover, we present the spectra of the lasers measured at fixed operating points for devices A, С and with current scanning for device B in a wide temperature range from 5 to 120 K. We hope that our results will prove useful for research concerning QCL maximum operating temperatures.

Temperature degradation of 2.3, 3.2 and 4.1 THz quantum cascade lasers

In this work, we conduct research of spectral and power characteristics of quantum cascade lasers (QCLs) based on a GaAs/Al0.15Ga0.85As active region emitting at 2.3 (A), 3.2 (B) and 4.1 (C) THz. The QCL devices had a double-metal Au waveguide and operated in pulsed mode with 1.5-9 μs pulses at 20 Hz repetition rate. Using the integral output power curves measured with different pulse durations, we consider the potential mechanisms of QCL temperature degradation using Arrhenius plots. Moreover, we present the spectra of the lasers measured at fixed operating points for devices A, C and with current scanning for device B in a wide temperature range from 5 to 120 K. We hope that our results will prove useful for research concerning QCL maximum operating temperatures. Keywords: quantum cascade laser, terahertz range, quantum well, molecular beam epitaxy, activation energy, temperature degradation.

Separate Fractional (p,q)-Integrodifference Equations via Nonlocal Fractional (p,q)-Integral Boundary Conditions

In this paper, we study a boundary value problem involving (p,q)-integrodifference equations, supplemented with nonlocal fractional (p,q)-integral boundary conditions with respect to asymmetric operators. First, we convert the given nonlinear problem into a fixed-point problem, by considering a linear variant of the problem at hand. Once the fixed-point operator is available, existence and uniqueness results are established using the classical Banach’s and Schaefer’s fixed-point theorems. The application of the main results is demonstrated by presenting numerical examples. Moreover, we study some properties of (p,q)-integral that are used in our study.

Relaxed inertial fixed point method for infinite family of averaged quasi-nonexpansive mapping with applications to sparse signal recovery

It is known that several optimization problems can be converted to a fixed point problem for which the underline fixed point operator is an averaged quasi-nonexpansive mapping and thus the corresponding fixed point method utilizes to solve the considered optimization problem. In this paper, we consider a fixed point method involving inertial extrapolation step with relaxation parameter to obtain a common fixed point of a countable family of averaged quasi-nonexpansive mappings in real Hilbert spaces. Our results bring a unification of several versions of fixed point methods for averaged quasi-nonexpansive mappings considered in the literature and give several implications of our results. We also give some applications to monotone inclusion problem with three-operator splitting method and composite convex and non-convex relaxed inertial proximal methods to solve both convex and nonconvex reweighted \(l_Q\) regularization for recovering a sparse signal. Finally, some numerical experiments are drawn from sparse signal recovery to illustrate our theoretical results.

Sequential Riemann–Liouville and Hadamard–Caputo Fractional Differential Equation with Iterated Fractional Integrals Conditions

In the present research, we initiate the study of boundary value problems for sequential Riemann–Liouville and Hadamard–Caputo fractional derivatives, supplemented with iterated fractional integral boundary conditions. Firstly, we convert the given nonlinear problem into a fixed point problem by considering a linear variant of the given problem. Once the fixed point operator is available, we use a variety of fixed point theorems to establish results regarding existence and uniqueness. Some properties of iteration that will be used in our study are also discussed. Examples illustrating our main results are also constructed. At the end, a brief conclusion is given. Our results are new in the given configuration and enrich the literature on boundary value problems for fractional differential equations.

Acceleration of nonlinear solvers for natural convection problems

Abstract This paper develops an efficient and robust solution technique for the steady Boussinesq model of non-isothermal flow using Anderson acceleration applied to a Picard iteration. After analyzing the fixed point operator associated with the nonlinear iteration to prove that certain stability and regularity properties hold, we apply the authors’ recently constructed theory for Anderson acceleration, which yields a convergence result for the Anderson accelerated Picard iteration for the Boussinesq system. The result shows that the leading term in the residual is improved by the gain in the optimization problem, but at the cost of additional higher order terms that can be significant when the residual is large. We perform numerical tests that illustrate the theory, and show that a 2-stage choice of Anderson depth can be advantageous. We also consider Anderson acceleration applied to the Newton iteration for the Boussinesq equations, and observe that the acceleration allows the Newton iteration to converge for significantly higher Rayleigh numbers that it could without acceleration, even with a standard line search.

Enabling convergence of the iterated penalty Picard iteration with [formula omitted] penalty parameter for incompressible Navier–Stokes via Anderson acceleration

This paper considers an enhancement of the classical iterated penalty Picard (IPP) method for the incompressible Navier–Stokes equations, where we restrict our attention to O(1) penalty parameter, and Anderson acceleration (AA) is used to significantly improve its convergence properties. After showing the fixed point operator associated with the IPP iteration is Lipschitz continuous and Lipschitz continuously (Frechet) differentiable, we apply a recently developed general theory for AA to conclude that IPP enhanced with AA improves its linear convergence rate by the gain factor associated with the underlying AA optimization problem. Results for several challenging numerical tests are given and show that IPP with penalty parameter 1 and enhanced with AA is a very effective solver.

FPGA Implementation of an Efficient FFT Processor for FMCW Radar Signal Processing

This paper presents the design and implementation results of an efficient fast Fourier transform (FFT) processor for frequency-modulated continuous wave (FMCW) radar signal processing. The proposed FFT processor is designed with a memory-based FFT architecture and supports variable lengths from 64 to 4096. Moreover, it is designed with a floating-point operator to prevent the performance degradation of fixed-point operators. FMCW radar signal processing requires windowing operations to increase the target detection rate by reducing clutter side lobes, magnitude calculation operations based on the FFT results to detect the target, and accumulation operations to improve the detection performance of the target. In addition, in some applications such as the measurement of vital signs, the phase of the FFT result has to be calculated. In general, only the FFT is implemented in the hardware, and the other FMCW radar signal processing is performed in the software. The proposed FFT processor implements not only the FFT, but also windowing, accumulation, and magnitude/phase calculations in the hardware. Therefore, compared with a processor implementing only the FFT, the proposed FFT processor uses 1.69 times the hardware resources but achieves an execution time 7.32 times shorter.

Anderson acceleration based on the [formula omitted] Sobolev norm for contractive and noncontractive fixed-point operators

Anderson acceleration (AA) is a technique for accelerating the convergence of fixed-point iterations. In this paper, we apply AA to a sequence of functions and modify the norm in its internal optimization problem to the H−s norm, for some positive integer s, to bias it towards low-frequency spectral content in the residual. We analyze the convergence of AA by quantifying its improvement over Picard iteration. We find that AA based on the H−2 norm is well-suited to solve fixed-point operators derived from second-order elliptic differential operators, including the Helmholtz equation.

Filtration and canonical completeness for continuous modal mu-calculi

The continuous modal mu-calculus is a fragment of the modal mu-calculus, where the application of fixpoint operators is restricted to formulas whose functional interpretation is Scott-continuous, rather than merely monotone. By game-theoretic means, we show that this relatively expressive fragment still allows two important techniques of basic modal logic, which notoriously fail for the full modal mu-calculus: filtration and canonical models. In particular, we show that the Filtration Theorem holds for formulas in the language of the continuous modal mu-calculus. As a consequence we obtain the finite model property over a wide range of model classes. Moreover, we show that if a basic modal logic L is canonical and the class of L-frames admits filtration, then the logic obtained by adding continuous fixpoint operators to L is sound and complete with respect to the class of L-frames. This generalises recent results on a strictly weaker fragment of the modal mu-calculus, viz. PDL.

A Fixed-point Theorem for Horn Formula Equations

We consider constrained Horn clause solving from the more general point of view of solving formula equations. Constrained Horn clauses correspond to the subclass of Horn formula equations. We state and prove a fixed-point theorem for Horn formula equations which is based on expressing the fixed-point computation of a minimal model of a set of Horn clauses on the object level as a formula in first-order logic with a least fixed point operator. We describe several corollaries of this fixed-point theorem, in particular concerning the logical foundations of program verification, and sketch how to generalise it to incorporate abstract interpretations.

An Lpspaces-based formulation yielding a new fully mixed finite element method for the coupled Darcy and heat equations

AbstractIn this work we present and analyse a new fully mixed finite element method for the nonlinear problem given by the coupling of the Darcy and heat equations. Besides the velocity, pressure and temperature variables of the fluid, our approach is based on the introduction of the pseudoheat flux as a further unknown. As a consequence of it, and due to the convective term involving the velocity and the temperature, we arrive at saddle point-type schemes in Banach spaces for both equations. In particular, and as suggested by the solvability of a related Neumann problem to be employed in the analysis, we need to make convenient choices of the Lebesgue and ${\textrm {H}}(div)$-type spaces to which the unknowns and test functions belong. The resulting coupled formulation is then written equivalently as a fixed-point operator, so that the classical Banach theorem, combined with the corresponding Babuška–Brezzi theory, the Banach–Nečas–Babuška theorem, suitable operators mapping Lebesgue spaces into themselves, regularity assumptions and the aforementioned Neumann problem, are employed to establish the unique solvability of the continuous formulation. Under standard hypotheses satisfied by generic finite element subspaces, the associated Galerkin scheme is analysed similarly and the Brouwer theorem yields existence of a solution. The respective a priori error analysis is also derived. Then, Raviart–Thomas elements of order $k\ge 0$ for the pseudoheat and the velocity and discontinuous piecewise polynomials of degree $\le k$ for the pressure and the temperature are shown to satisfy those hypotheses in the two-dimensional case. Several numerical examples illustrating the performance and convergence of the method are reported, including an application into the equivalent problem of miscible displacement in porous media.

Alternating Fixpoint Operator for Hybrid MKNF Knowledge Bases as an Approximator of AFT

AbstractApproximation fixpoint theory (AFT) provides an algebraic framework for the study of fixpoints of operators on bilattices and has found its applications in characterizing semantics for various classes of logic programs and nonmonotonic languages. In this paper, we show one more application of this kind: the alternating fixpoint operator by Knorr et al. for the study of the well-founded semantics for hybrid minimal knowledge and negation as failure (MKNF) knowledge bases is in fact an approximator of AFT in disguise, which, thanks to the abstraction power of AFT, characterizes not only the well-founded semantics but also two-valued as well as three-valued semantics for hybrid MKNF knowledge bases. Furthermore, we show an improved approximator for these knowledge bases, of which the least stable fixpoint is information richer than the one formulated from Knorr et al.’s construction. This leads to an improved computation for the well-founded semantics. This work is built on an extension of AFT that supports consistent as well as inconsistent pairs in the induced product bilattice, to deal with inconsistencies that arise in the context of hybrid MKNF knowledge bases. This part of the work can be considered generalizing the original AFT from symmetric approximators to arbitrary approximators.

On continuation-passing transformations and expected cost analysis

We define a continuation-passing style (CPS) translation for a typed λ-calculus with probabilistic choice, unbounded recursion, and a tick operator — for modeling cost. The target language is a (non-probabilistic) λ-calculus, enriched with a type of extended positive reals and a fixpoint operator. We then show that applying the CPS transform of an expression M to the continuation λ v . 0 yields the expected cost of M . We also introduce a formal system for higher-order logic, called EHOL, prove it sound, and show it can derive tight upper bounds on the expected cost of classic examples, including Coupon Collector and Random Walk. Moreover, we relate our translation to Kaminski et al.’s ert-calculus, showing that the latter can be recovered by applying our CPS translation to (a generalization of) the classic embedding of imperative programs into λ-calculus. Finally, we prove that the CPS transform of an expression can also be used to compute pre-expectations and to reason about almost sure termination.

On equivalence of PFP-operator and PFP-quantifier

In this paper we consider two different logic languages. Both of them are extensions of first order logic. The first semantics is obtained by adding a partial fixed point operator. The second semantics is based on considering a partial fixed point as a non-standard quantifier. For this two semantics we demonstrate that they have an equal expressive power. For this purpose we show how to express an arbitrary formula of one logic with a formula of another.

Sound quality DNA construction according to the scenario and operating condition of diesel engine

The sound quality (SQ) of a diesel engine varies with scenario and operating condition, therefore, the traditional evaluation method based on fixed operating points may not be able to reflect the SQ flexibly. Besides, it is difficult to effectively quantify the SQ performance due to the differences of application scenarios, which is not conducive to conduct targeted acoustic improvement. To address this issue, a quick SQ evaluation approach was proposed based on the DNA construction of scenario and operating condition. In this approach, the effectiveness and diversity of sampling were considered, and fifteen operating conditions of each engine (four in total) including frequently used ones and randomly selected ones were picked for jury tests. The revised grouped pair comparison (RGPC) was developed to improve the evaluation accuracy and efficiency for large scale samples. The fuzzy analytic hierarchy process (FAHP) was introduced to calculate the weights of operating conditions for different application scenarios, and the weighted preference was employed as a comprehensive SQ index. As a result, the DNA of scenario-operating condition were extracted and examined, which can quickly and precisely sequence the SQ of diesel engines in different application scenarios.

A system of nonlinear fractional BVPs with ϕ-Laplacian operators and nonlocal conditions

This work investigates the existence of multiple positive solutions for a system of two nonlinear higher-order fractional differential equations with ϕ-Laplacian operators and nonlocal conditions. A degenerate nonlinearity which obeys some general growth conditions is considered. The singularities are dealt with by approximating the fixed point operator. New existence results are presented by using the fixed point index theory. Examples of applications illustrate the theoretical results.