Closed-form Representation Research Articles

Controlled Invariant Sets: Implicit Closed-Form Representations and Applications

Controlled Invariant Sets: Implicit Closed-Form Representations and Applications

An extreme worst-case risk measure by expectile

Abstract Expectiles have received increasing attention as a risk measure in risk management because of their coherency and elicitability at the level $\alpha\geq1/2$ . With a view to practical risk assessments, this paper delves into the worst-case expectile, where only partial information on the underlying distribution is available and there is no closed-form representation. We explore the asymptotic behavior of the worst-case expectile on two specified ambiguity sets: one is through the Wasserstein distance from a reference distribution and transforms this problem into a convex optimization problem via the well-known Kusuoka representation, and the other is induced by higher moment constraints. We obtain precise results in some special cases; nevertheless, there are no unified closed-form solutions. We aim to fully characterize the extreme behaviors; that is, we pursue an approximate solution as the level $\alpha $ tends to 1, which is aesthetically pleasing. As an application of our technique, we investigate the ambiguity set induced by higher moment conditions. Finally, we compare our worst-case expectile approach with a more conservative method based on stochastic order, which is referred to as ‘model aggregation’.

ANALYTICALLY PRICING EUROPEAN OPTIONS UNDER A TWO-FACTOR STOCHASTIC INTEREST RATE MODEL WITH A STOCHASTIC LONG-RUN EQUILIBRIUM LEVEL

AbstractWe construct a new stochastic interest rate model with two stochastic factors, by introducing a stochastic long-run equilibrium level into the Vasicek interest rate model which follows another Ornstein–Uhlenbeck process. With the interest rate under the Black–Scholes model being assumed to follow the newly proposed model, a closed-form representation of European option prices is successfully presented, when the analytical characteristic function of the underlying log-price under a forward measure is derived. To assess the model performance, a preliminary empirical study is conducted using S&P 500 index and its options, with the Vasicek model and an alternative two-factor Vasicek model taken as benchmarks.

Solutions for Poissonian stopping problems of linear diffusions via extremal processes

We develop a general yet simple technique for solving Poissonian timing problems of linear diffusions by relying on the close connection of the extremal processes and the first passage times of the underlying diffusion. We provide a closed-form representation of the expected value gained by employing an ordinary first passage time-based stopping strategy. This approach simplifies the determination of the optimal policy, transforming it into an analysis of ordinary first-order optimality conditions. We relate our findings to various existing approaches for solving stopping problems of linear diffusions and express the optimality conditions in a single boundary setting in a form familiar from optimal stopping of Lévy-processes.

Gradient-Based Optimization for Intent Conflict Resolution

The evolving landscape of network systems necessitates automated tools for streamlined management and configuration. Intent-driven networking (IDN) has emerged as a promising solution for autonomous network management by prioritizing declaratively defined desired outcomes over traditional manual configurations without specifying the implementation details. This paradigm shift towards flexibility, agility, and simplification in network management is particularly crucial in addressing inefficiencies and high costs linked to manual management, notably in the radio access part. This paper explores the concurrent operation of multiple intents, acknowledging the potential for conflicts, and proposes an innovative reformulation of these conflicts to enhance network administration effectiveness. Following the initial detection of conflicts among intents using a gradient-based approach, our work employs the Multiple Gradient Descent Algorithm (MGDA) to minimize all loss functions assigned to each intent simultaneously. In response to the challenge posed by the absence of a closed-form representation for each key performance indicator in a dynamic environment for computing gradient descent, the Stochastic Perturbation Stochastic Approximation (SPSA) is integrated into the MGDA algorithm. The proposed method undergoes initial testing using a commonly employed toy example in the literature before being simulated for conflict scenarios within a mobile network using the ns3 network simulator.

Polarized representation for depolarization-dominant materials.

The light-matter interactions which occur in common indoor environments are strongly depolarizing, but the relatively small polarization attributes can be informative. This information is used in applications such as physics-based rendering and shape-from-polarization. Look-up table polarized bidirectional reflectance distribution functions (pBRDFs) for indoor materials are available, but closed-form representations are advantageous for their ease of use in both forward and inverse problems. First-surface Fresnel reflection, diffuse partial polarization, and ideal depolarization are popular terms used in closed-form pBRDF representations. The relative contributions of these terms are highly dependent on material, albedo/wavelength, and scattering geometry. Complicating matters further, current pBRDF representations incoherently combine Mueller matrices (MM) for Fresnel and polarized diffuse terms which couples into depolarization. In this work, a pBRDF representation is introduced where first-surface Fresnel reflection and diffuse polarization are coherently combined using Jones calculus to avoid affecting depolarization. The first-surface and diffuse reflection terms are combined using an analytic function which takes as input the scattering geometry as well as geometry-independent material parameters. Agreement with wide-field-of-view polarimetric measurements is demonstrated using the new pBRDF which has only six physically meaningful parameters: the scalar-valued depolarization parameter and average reflectance which are geometry-dependent and four geometry-independent material constants. In general, depolarization is described by nine parameters but a triply-degenerate (TD) model simplifies depolarization to a single parameter. To test this pBRDF representation, the material constants for a red 3D printed sphere are assumed and the geometry-dependent depolarization parameter is estimated from linear Stokes images. The geometry-averaged error of the depolarization parameter is 4.2% at 662 nm (high albedo) and 11.7% at 451 nm (low albedo). The error is inversely proportional to albedo and depolarization, so the TD-MM model is considered appropriate for depolarization-dominant materials. The robustness of the pBRDF representation is also demonstrated by comparing measured and extrapolated Mueller images of a Stanford bunny of the same red 3D printing material. The comparison is performed by using Mueller calculus to simulate polarimetric measurements based on the measured and extrapolated data.

Bayesian analysis on a natural conjugate prior for the nonhomogeneous Poisson process with a power-law intensity under time-truncated sampling

The core content in this paper discusses the Bayesian approach, which essentially estimates and predicts the reliability of a repairable system during the actual testing process. As specified in previous literature, the nonhomogeneous Poisson process (NHPP) with a power-law intensity function has often been employed as a model for describing the failure process of repairable systems since it provides a reasonable way of presenting the rate of change for the reliability of a system as a function of time. This research proposed the same time-truncated sampling assumption to mainly ensure that the joint posterior distribution of the power-law failure process has a considerably closed-form representation. This proposed approach facilitates the analysis of the actual performance, with which analysts can provide the prior information about the authentic means and variations of the shape and scale parameters of the power-law intensity. The corresponding posterior means and variations can then be significantly and easily obtained without any complex integral arithmetic. Using Monte Carlo simulation for comparison, the Bayesian estimation with the proposed prior outperforms the maximum likelihood estimation, particularly when the prior aging of the repairable is underestimated.

Optimal asset allocation for DC pension subject to allocation and terminal wealth constraints under a remuneration scheme

. We investigate the optimal investment problem faced by a defined contribution (DC) pension fund manager under simultaneous allocation and expected shortfall (ES) constraints. Under a non concave utility, a Value-at-Risk (VaR) constraint does not lead to the full prevention of moral hazard. As a widely employed risk management tool, whether an ES constraint can provide a more effective protection than a VaR constraint has been a focus point of research. We apply a dual control approach and a concavification technique to solve the ES-constrained optimization problem for a DC pension plan under an incentive scheme and derive the closed-form representations of the optimal wealth and portfolio processes. Furthermore, we compare the effect of an ES constraint on the optimal investment behavior with that under a VaR constraint in the presence of an option-like scheme for the DC pension members. Theoretical and numerical results show that for a relatively low protection level, a joint VaR and an ES constraints induce the same structure of the optimal solution, which implies that for a non concave optimization problem, the ES-based risk management has lost its advantage over the VaR-based risk management. Therefore, it needs to design a more efficient risk measure to improve the risk management for a DC pension plan.

Creep of a closed cylindrical tank at hydrostatic pressure

Objective. When constructing the resolving relations of the theory of shells, the validity of the basic assumptions about the material of the structure under consideration is assumed, which is considered homogeneous, isotropic and viscoelastic, i.e. obeying the Maxwell-Gurevich law. The subject to study is a polymer cylindrical shell, rigidly clamped at the base and subject to internal hydrostatic pressure. Method. The problem is reduced to an inhomogeneous differential equation of the fourth degree with respect to the displacement of the middle surface w along the z axis. Since the closed form representation of the solution to this equation is extremely difficult, the search for it is presented numerically, in particular, using the grid method. The creep strain components ε*x, ε*θ, γ*xθ were determined as a linear approximation of the velocity by the Runge-Kutta method. Result. In the process of calculating the shell using moment theory, it was found that as a result of shell creep, tangential stresses increased by more than 12 percent. Conclusion. The proposed technique makes it possible to simulate changes in the mechanical properties of the shell (for example, indirect heterogeneity) caused by the influence of physical fields.

Experimental Assessment of Chance-Constrained Motion Planning for Small Uncrewed Aircraft

This work extends the experimental evaluation of chance-constrained motion planning algorithms to fielded fixed-wing small uncrewed aircraft systems (sUAS). Despite advances in planning algorithms, certain challenges remain to producing trajectories for nonholonomic mobile robotic systems such as sUAS. These challenges include nonlinear dynamics which create a complex mapping from inputs to outputs, initialization uncertainty in online motion planning due to compute time of planners and latency in data transfer, environmental uncertainty that has non-Gaussian impact on the robot’s trajectory, and system uncertainty that arises from incomplete models of complex systems. Small UAS often have proprietary components such as commercial, off-the-shelf autopilots which prevent motion planning models from accurately capturing system behavior in all regions of the state space. These challenges can be addressed by leveraging probabilistic motion planners that accurately represent and reason over dynamics and uncertainty. Chance-constrained motion planning offers a method of reasoning over uncertainty as feasibility constraints, and Monte Carlo sampling within a motion planning algorithm offers a method of representing complex uncertainty that may not have a closed form representation. This work extends the chance-constrained motion planning problem to reason over constraint on the trajectory and constraints on the state which ensure that the system model remains valid. Dynamical and systems models are formulated to extend to a broad class of fixed-wing UAS through the experimental derivation of input distributions and system parameters. Uncertainty is quantified using a data-driven approach to modeling input distributions, and Monte Carlo uncertainty sampling deployed within a Rapidly-exploring Random Tree motion planning algorithm is used to plan a trajectory containing a representation of uncertainty. The motion planning algorithm’s ability to accurately reason over layered chance constraints is evaluated experimentally in 61 fielded missions. The results show that when the flight conditions fall within the domain of the uncertainty distributions, the motion planning system is able to accurately reason over the chance constraints.

Solutions and local stability of the Jacobsthal system of difference equations

<abstract><p>We presented a comprehensive theory for deriving closed-form expressions and representations of the general solutions for a specific case of systems involving Riccati difference equations of order $ m+1 $, as discussed in the literature. However, our focus was on coefficients dependent on the Jacobsthal sequence. Importantly, this system of difference equations represents a natural extension of the corresponding one-dimensional difference equation, uniquely characterized by its theoretical solvability in a closed form. Our primary objective was to demonstrate a direct linkage between the solutions of this system and Jacobsthal and Lucas-Jacobsthal numbers. The system's capacity for theoretical solvability in a closed form enhances its distinctiveness and potential applications. To accomplish this, we detailed offer theoretical explanations and proofs, establishing the relationship between the solutions and the Jacobsthal sequence. Subsequently, our exploration addressed key aspects of the Jacobsthal system, placing particular emphasis on the local stability of positive solutions. Additionally, we employed mathematical software to validate the theoretical results of this novel system in our research.</p></abstract>

The thermodynamic properties of diatomic gases. A rigorous analytical approach

The paper reports a closed-form analytical representation of the thermodynamic functions of diatomic molecules in the ground electronic state. For this purpose, analytical formulas for vibrational and rotational partition functions are derived. A rigorous method is proposed for factorizing the vibrational and rotational contributions to the ro-vibrational partition function using an average vibrational quantum number. The behavior of thermodynamic functions as a function of temperature is discussed for carbon monoxide as a test system. The reliability of the analytical method is verified by comparison with experimental data and with the direct summation method, which uses theoretical values of energy levels.

A matrix unified framework for deriving various impulse responses in Markov switching VAR: Evidence from oil and gas markets

We propose a new method to compute various impulse response functions (IRF) for a Markov switching VAR model in terms of neat matrix expressions in closed form. The key is to derive a suitable closed form representation for Markov switching VAR models using a state-space representation. By this representation, the IRF analysis can be processed with respect to either an asymmetric discrete or a symmetric continuous shocks. A simulation study demonstrates the actual advantages of the proposed matrix methodology. To illustrate the feasibility and the usefulness of our approach, we present empirical applications to oil and natural gas markets showing the relevance of accommodating asymmetries in the relationship between their price shocks and economic activities.

A simple spectral representation of a second-order symmetric tensor and its variation

The spectral decomposition of a second-order, symmetric tensor is widely adopted in many fields of Computational Mechanics. As an example, in elasto-plasticity under large strain and rotations, given the Cauchy deformation tensor, it is a fundamental step to compute the logarithmic strain tensor.Recently, this approach has also been adopted in small-strain isotropic plasticity to reconstruct the stress tensor as a function of its eigenvalues, allowing the formulation of predictor–corrector return algorithms in the invariants space. These algorithms not only reduce the number of unknowns at the constitutive level, but also allow the correct handling of stress states in which the plastic normals are undefined, thus ensuring a better convergence with respect to the standard approach.While the eigenvalues of a symmetric, second-order tensor can be simply computed as a function of the tensor invariants, the computation of its eigenbasis can be more difficult, especially when two or more eigenvalues are coincident. Moreover, when a Newton–Raphson algorithm is adopted to solve nonlinear problems in Computational Mechanics, also the tensorial derivatives of the eigenbasis, whose computation is still more complicated, are required to assemble the tangent matrix.A simple and comprehensive method is presented, which can be adopted to compute a closed form representation of a second-order tensor, as well as their derivatives with respect to the tensor itself, allowing a simpler and numerically accurate implementation of spectral decomposition of a tensor in Computational Mechanics applications.

Theoretical and Non-Dimensional Investigations into Vibration Control Using Viscoelastic and Endochronic Elements

Theoretical and non-dimensional investigations have been performed to study the vibration control potential of approaches that are not only based on viscoelastic but also on endochronic elements. The latter are known from the endochronic theory of plasticity and provide the possibility of establishing rate-independent schemes for vibration control. The main question that has to be answered is: Can rate-independent damping be efficiently used to reduce mechanical vibrations? To answer this question, non-dimensional models for dynamical systems are derived and analyzed numerically in the time domain as well as in the frequency domain. The results are used to compare the performance of an optimally tuned endochronic absorber to the performance of an optimally tuned dynamic absorber with viscoelastic damping. Based on a novel closed-form representation for non-linear systems with endochronic elements, it has been possible to prove that the rate-independent control of vibration results in an overall control profit that is close to the control profit obtained by the application of well-established approaches. It has also been found that the new concept is advantageous if anti-resonances have to be considered in broadband vibration control. Based on these novel findings, a practical realization in the context of active vibration control is proposed in which the rate-independent control law is implemented with an appropriate signal processing hardware.

Necessary and Sufficient Conditions for Stabilizing an Uncertain Stochastic Multidelay System

We focus on the problem of asymptotically mean-square stabilization in discrete-time stochastic systems that exhibit plant uncertainty, multiple input delays, and multiplicative noises. Our innovative contributions are described as follows. First, we employ a reduction method to transform the original model into a delay-free auxiliary system, and establish an equivalent proposition for stabilization based on this reformulation. On the basis of the reformulated model, we propose two stabilization criteria for the uncertainty-free case, including both Lyapunov-type and Riccati-type criteria. More generally, we extend the stabilization result to the uncertain model, and propose a necessary and sufficient stabilization criterion utilizing matrix homogeneous polynomials. Finally, we explore the existence and uniqueness of a delay margin under certain structural restrictions, and provide a closed-form representation of this margin.

Micromechanical model for effective thermoelectric properties of thermoelectric composite containing ellipsoidal inclusions with interfacial resistances

The effective design of thermoelectric composites is of great significance in improving performance of thermoelectric composites. Herein, we develop a homogenization theory to predict the effective thermoelectric properties of thermoelectric composite containing ellipsoidal inclusions whereby interfacial thermal and electrical resistances are considered. Firstly, the closed-form representations of the effective thermoelectric properties are derived under the assumption of a small temperature difference. Then the effective properties of the thermoelectric composite subjecting to a large temperature difference are obtained by cutting the composite to a serial connection of pieces with a small temperature difference. The relationship between the interfacial resistances and the effective properties of thermoelectric composites is discussed systematically under the different conditions of the aspect ratio, volume fraction, size of inclusion and temperature difference in the composite. Our theoretical predictions well match the effective properties obtained from finite element analysis.

Planning with Hidden Parameter Polynomial MDPs

For many applications of Markov Decision Processes (MDPs), the transition function cannot be specified exactly. Bayes-Adaptive MDPs (BAMDPs) extend MDPs to consider transition probabilities governed by latent parameters. To act optimally in BAMDPs, one must maintain a belief distribution over the latent parameters. Typically, this distribution is described by a set of sample (particle) MDPs, and associated weights which represent the likelihood of a sample MDP being the true underlying MDP. However, as the number of dimensions of the latent parameter space increases, the number of sample MDPs required to sufficiently represent the belief distribution grows exponentially. Thus, maintaining an accurate belief in the form of a set of sample MDPs over complex latent spaces is computationally intensive, which in turn affects the performance of planning for these models. In this paper, we propose an alternative approach for maintaining the belief over the latent parameters. We consider a class of BAMDPs where the transition probabilities can be expressed in closed form as a polynomial of the latent parameters, and outline a method to maintain a closed-form belief distribution for the latent parameters which results in an accurate belief representation. Furthermore, the closed-form representation does away with the need to tune the number of sample MDPs required to represent the belief. We evaluate two domains and empirically show that the polynomial, closed-form, belief representation results in better plans than a sampling-based belief representation.

Data-driven breakage mechanics: Predicting the evolution of particle-size distribution in granular media

This paper presents a model-free data-driven framework for breakage mechanics. In contrast with continuum breakage mechanics, the de facto approach for the macroscopic analysis of crushable granular media, the present framework does not require the definition of constitutive models and phenomenological assumptions, relying on material behavior that is known only through empirical data. For this purpose, we revisit the recent developments in model-free data-driven computing for history-dependent materials and extend the main ideas to materials with particle breakage. A systematic construction of the modeling framework is presented, starting from the closed-form representation of continuum breakage mechanics and arriving at alternative model-free representations. The predictive ability of the data-driven framework is highlighted and contrasted with continuum breakage mechanics on different boundary value problems. Moreover, an application to a real experimental test in crushable sand is presented, where the data is furnished by high-fidelity grain-scale simulations, indicating that the proposed framework provides an accurate prediction of the mechanics of crushable materials including the state of comminution.

A general class of enriched methods for the simplicial linear finite elements

Low-order elements are widely used and preferred for finite element analysis, specifically the three-node triangular and four-node tetrahedral elements, both based on linear polynomials in barycentric coordinates. They are known, however, to under-perform when nearly incompressible materials are involved. The problem may be circumvented by the use of higher degree polynomial elements, but their application become both more complex an computationally expensive. For this reason, non-polynomial enriched finite element methods have been proposed for solving engineering problems. In line with previous researches, the main contribution of this paper is to present a general strategy for enriching the standard simplicial linear finite element by non-polynomial functions. A key role is played by a characterization result, given in terms of the non-vanishing of a certain determinant, which provides necessary and sufficient conditions, on the enrichment functions and functionals, that guarantee the existence of families of such enriched elements. We show that the enriched basis functions admit a closed form representation in terms of enrichment functions and functionals. Finally, we provide concrete examples of admissible enrichment functions and perform some numerical tests.