Saddle Point Approach Research Articles

Hypersoft Sets in a Game Theory-Based Decision Making Model

In this study, we offer a hypersoft set theory-based game model for handling uncertainties. The term ”hypersoft game” refers to this newly suggested game. Four techniques of game solution are identified: hypersoft saddle points, hypersoft upper and lower values, hypersoft dominated strategy and hypersoft nash equilibrium. We build a two-person hypersoft game first. Additionally, we present real-world problems that the hypersoft saddle point approaches and hypersoft dominating strategy are used to tackle. In conclusion, we expand the hypersoft games from two players to n players.

Saddle-Point Approach to Large-Time Volatility Smile

Saddle-Point Approach to Large-Time Volatility Smile

A full discretization for the saddle-point approach of a degenerate parabolic problem involving a moving body

This paper aims to study a degenerate parabolic problem for a solenoidal vector field in which the time derivative acts on a moving body. We propose a fully-discrete finite element scheme combined with backward Euler’s method for the saddle-point variational formulation. The convergence of this numerical scheme is proved and error estimates for some stable finite element pairs are also established.

Asymptotics and statistics on Fishburn matrices and their generalizations

A direct saddle-point analysis (without relying on any modular forms or functional equations) is developed to establish the asymptotics of Fishburn matrices and a large number of other variants with a similar sum-of-finite-product form for their (formal) generating functions. In addition to solving some conjectures, the application of our saddle-point approach to the distributional aspects of statistics on Fishburn matrices is also examined with many new limit theorems characterized, representing the first of their kind for such structures.

A saddle point approach to an optimal boundary control problem for steady Navier-Stokes equations

In this paper we propose a saddle point approach to solve boundary control problems for the steady Navier-Stokes equations with mixed Dirichlet-Neumann boundary conditions, both in two and three dimensions. We provide a comprehensive theoretical framework to address (i) the well posedness analysis for the optimal control problem related to this system and (ii) the derivation of a system of first-order optimality conditions. We take advantage of a suitable treatment of boundary Dirichlet controls (and data) realized by means of Lagrange multipliers. In spite of the fact that this approach is rather common, a detailed analysis is still missing for mixed boundary conditions. We consider the minimization of quadratic cost (e.g., tracking-type or vorticity) functionals of the velocity. A descent method is then applied for numerical optimization, exploiting the Galerkin finite element method for the discretization of the state equations, the adjoint (Oseen) equations and the optimality equation. Numerical results are shown for simplified two-dimensional fluid flows in a tract of blood vessel where a bypass is inserted; to avoid to simulate the whole bypass configuration, we represent its action by a boundary velocity control.

On a weakly C-$\epsilon$-vector saddle point approach in weak vector problems

On a weakly C-$\epsilon$-vector saddle point approach in weak vector problems

Feedback Synthesis for a Terminal Control Problem

A terminal control problem with linear controlled dynamics on a fixed time interval is considered. A boundary value problem in the form of a linear programming problem is stated in a finite-dimensional terminal space at the right endpoint of the interval. The solution of this problem implicitly determines a terminal condition for the controlled dynamics. A saddle-point approach to solving the problem is proposed, which is reduced to the computation a saddle point of the Lagrangian. The approach is based on saddle-point inequalities in terms of primal and dual variables. These inequalities are sufficient optimality conditions. A method for computing a saddle point of the Lagrangian is described. Its monotone convergence with respect to some of the variables on their direct product is proved. Additionally, weak convergence with respect to controls and strong convergence with respect to phase and adjoint trajectories and with respect to terminal variables of the boundary value problem are proved. The saddle-point approach is used to synthesize a feedback control in the case of control constraints in the form of a convex closed set. This result is new, since, in the classical case of the theory of linear regulators, a similar assertion is proved without constraints imposed on the controls. The theory of linear regulators relies on matrix Riccati equations, while the result obtained is based on the concept of a support function (mapping) for the control set.

Measuring risks in the tail: The extreme VaR and its confidence interval

Contrary to the current regulatory trend regarding risks, the purpose of this paper is to emphasize the necessity of considering the Value-at-Risk (VaR) with confidence levels like 99.9%, as an alternative way of measuring risks in the extreme tail. Although the mathematical definition of the VaR is trivial, its computation is challenging in practice, because the uncertainty of the VaR may not be negligible for a finite amount of data. We begin to build confidence intervals around the unknown VaR. We build them using two different approaches, the first one uses the asymptotic Gaussian result and the second saddlepoint approach, the latter proves to be more robust when we use finite samples. We compare our approach with other methodologies which are based on bootstrapping techniques, focusing on the estimation of the quantiles of a distribution. Finally, we apply these confidence intervals to perform a stress testing exercice with historical stock returns during the financial crisis, in order to identify potential violations of the VaR during periods of turmoil on financial markets.

Densities and distribution functions of the ratio of two linear functions and the product of Gamma variates: A saddlepoint approach

ABSTRACTSaddlepoint approximations for the densities and the distribution functions of the ratio of two linear functions of gamma random variables and the product of gamma random variables are derived. Ratios of linear functions with positive and negative weights and non identical gamma variables are considered. The saddlepoint approximations are very accurate in the tails as in the center of the distribution. Extensive simulation studies are used to evaluate the accuracy of the proposed methods.

Four-loop static contribution to the gravitational interaction potential of two point masses

We compute a subset of three, velocity-independent four-loop (and fourth post-Newtonian) contributions to the harmonic-coordinates effective action of a gravitationally interacting system of two point-masses. We find that, after summing the three terms, the coefficient of the total contribution is rational, due to a remarkable cancellation between the various occurrences of $\pi^2$. This result, obtained by a classical field-theory calculation, corrects the recent effective-field-theory-based calculation by Foffa et al. [arXiv:1612.00482]. Besides showing the usefulness of the saddle-point approach to the evaluation of the effective action, and of x-space computations, our result brings a further confirmation of the current knowledge of the fourth post-Newtonian effective action. We also show how the use of the generalized Riesz formula [Phys. Rev. D 57, 7274 (1998)] allows one to analytically compute a certain four-loop scalar master integral (represented by a four-spoked wheel diagram) which was, so far, only numerically computed.

Transition radiation at the boundary of a chiral isotropic medium.

This study analyzes the radiation produced by a point charge intersecting the interface between a vacuum and a chiral isotropic medium. We deduce analytical expressions for the Fourier components of an electromagnetic field in both vacuum and medium for arbitrary charge velocity. The main focus is on investigating the far field in a vacuum. The distinguishing feature of the interface with a chiral isotropic medium is that the field in the vacuum area contains both copolarization (coinciding with the polarization of the self-field of a charge) and cross-polarization (orthogonal to the polarization of the self-field). Using a saddle-point approach, we obtain asymptotic representations for the field components in the far-field zone for typical frequency ranges of the Condon model of the chiral medium. We note that a so-called lateral wave is generated in a vacuum for certain parameters. The main contribution to the radiation at large distances is presented by two (co- and cross-) spherical waves of transition radiation. These waves are coherent and result in a total spherical wave with elliptical polarization, with the polarization coefficient being determined by the chirality of the medium. We present typical radiation patterns and ellipses of polarization.

Monte Carlo simulation of the crossover from Bose glass to Bragg glass phase in layered BSCCO with columnar defects

Monte Carlo simulations of layered BSCCO samples are used to investigate the behavior of vortex matter at low fields, particularly in connection with the possible occurrence of a Bragg glass (BrG) phase at low density of columnar defects, a phenomenon characterized by the prevalence of short-range over long-range order. In this dislocation-free topological phase the translational order correlation function displays a power law decay. For magnetic induction B = 0.1 kG the analysis of the data for the first Bragg peak of the planar structure factor, the hexatic order parameter, and the Delaunay triangulation shows that, as the density of columnar defects is lowered, a crossover (or transition) from Bose glass to BrG phase takes place in this highly anisotropic high-T superconductor. Most importantly, an analysis of the low-temperature 3D vortex–vortex correlation function in terms of the structure factor, calculated via a saddle point approach and the use of the numerical data as input, provides clear-cut evidence of the power law decay of the divergent Bragg peaks in the BrG phase, a fundamental feature that was inequivocally verified only in isotropic compounds.

Inf-Sup Stable Finite Element Methods for the Landau--Lifshitz--Gilbert and Harmonic Map Heat Flow Equations

In this paper we propose and analyze a finite element method for both the harmonic map heat and Landau--Lifshitz--Gilbert equations, the time variable remaining continuous. Our starting point is to set out a unified saddle point approach for both problems in order to impose the unit sphere constraint at the nodes. A proper inf-sup condition is proved for the Lagrange multiplier leading to the well-posedness of the unified formulation. A priori energy estimates are shown for the proposed method. When time integrations are combined with the saddle point finite element approximation some extra elaborations are required in order to ensure both a priori energy estimates for the director or magnetization vector depending on the model and an inf-sup condition for the Lagrange multiplier. These extra elaborations are needed due to the fact that any crude time integration either does not keep the unit length at the nodes or does not satisfy an energy law. We will carry out a linear Euler--like time-stepping method...

Bivariate symmetry tests for complete and competing risks data: a saddlepoint approach

A class of bivariate symmetry tests for complete data and competing risks data is considered. Saddlepoint approximation for the exact p-values of the underlying permutation distribution of these tests is derived. Several simulation studies are conducted to evaluate the performance of the saddlepoint approximation and the asymptotic approximation. The saddlepoint approximation was found to be highly accurate and superior to the asymptotic approximations in replicating the exact permutation significance.

Exact results for Wilson loops in orbifold ABJM theory**Supported by NSFC (11222549, 11575202), K. C. Wong Education Foundation and Youth Innovation Promotion Association of CAS (2011016)

We investigate the exact results for circular 1/4 and 1/2 BPS Wilson loops in the d = 3 mathcal = 4 super Chern-Simons-matter theory that could be obtained by orbifolding Aharony-Bergman-Jafferis-Maldacena (ABJM) theory. The partition function of the mathcal = 4 orbifold ABJM theory has been computed previously in the literature. In this paper, we re-derive it using a slightly different method. We calculate the vacuum expectation values of the circular 1/4 BPS Wilson loops in fundamental representation and of circular 1/2 BPS Wilson loops in arbitrary representations. We use both the saddle point approach and Fermi gas approach. The results for Wilson loops are in accord with the available gravity results.

A saddlepoint approach to estimating joint extreme value distributions for vector non-stationary Gaussian processes

The focus of this study is on estimating the multivariate extreme value distributions associated with a vector of mutually correlated non-stationary Gaussian processes. This involves computing the joint crossing statistics of the vector processes by assuming the crossings to be Poisson counting processes. A mathematical artifice is adopted to take into account the dependencies that exist between the crossings of the processes. The crux in the formulation lies in the evaluation of a high-dimensional integral, which can be computationally expensive. This difficulty is bypassed by using saddlepoint approximation. The developments are illustrated through two numerical examples and are validated using Monte Carlo simulations. In the second example, reliability analysis is carried out for a multiply supported pipeline on an offshore jacket structure subjected to wave loading using the proposed formulation.

Saddle point approach to solving problem of optimal control with fixed ends

In a Hilbert space, the problem of terminal control with linear dynamics and fixed ends of the trajectory is considered. The integral objective functional has a quadratic form. In contrast to the traditional approach, the problem of terminal control is interpreted not as an optimization problem, but as a saddle-point problem. The solution to this problem is a saddle point of the Lagrange function with components in the form of controls, phase and conjugate trajectories. A saddle-point method is proposed, the convergence of the method in all components of the solution is proved.

Spatiotemporal instability of an electrified falling film.

The linear spatiotemporal response to an infinitesimal disturbance is analyzed for an electrified falling film based on the Orr-Sommerfeld boundary value problem. Four spatial branches are identified, which lose their spatial symmetry in the presence of an electric field. In addition, the absolute and convective instabilities are studied within the frameworks of the saddle point approach and the collision criterion. The collision criterion offers two unstable branches that characterize the exact range of the unstable wave packet that is diminished in the presence of an electric field. In contrast, the saddle point approach offers only one unstable branch. The associated unstable range of the wave packet is similar to that predicted by the collision criterion only when the electric field is enhanced. It is shown that the electrified falling film is convectively unstable.

Photodetachment of F− by mid-infrared circularly polarized few-cycle pulses

SynopsisAbove threshold detachment (ATD) of the valence p-state electrons of F− by strong circularly polarized few-cycle laser pulses is analysed within the framework of the adiabatic saddle point approach. We consider the emission spectra of the ejected photoelectrons and examine the qualitative effects arising from variation of the carrier-envelope phase (CEP), number of optical cycles and the orientation of the orbital angular momentum of the ground state.

Dynamic method of multipliers in terminal control

A method for solving the terminal control problem with a fixed time interval and fixed initial conditions is proposed. The solution to the boundary value problem posed at the right end of the time interval determines the terminal conditions. This boundary value problem is a finite-dimensional convex programming problem. The dynamics of the terminal control problem is described by a linear controllable system of differential equations. This system is interpreted as a conventional system of linear equality constraints. Then the terminal control problem can be regarded as a dynamic convex programming problem posed in an infinite-dimensional functional Hilbert space. In this paper, the functional problem is treated as a saddle-point problem rather than optimization problem. Accordingly, a saddle-point approach to solving the problem is proposed. This approach is based on maximizing the dual function generated by the modified Lagrangian function of the convex programming problem posed in the functional space. The convergence of the proposed methods is also proved in the functional space. This convergence has the additional property of being monotone in norm with respect to controls, phase trajectories, adjoint functions, as well as finite-dimensional terminal variables.