Saddle Point Research Articles

An energy-efficient GMRES–multigrid solver for space-time finite element computation of dynamic poroelasticity

AbstractWe present and analyze computationally Geometric MultiGrid (GMG) preconditioning techniques for Generalized Minimal RESidual (GMRES) iterations to space-time finite element methods (STFEMs) for a coupled hyperbolic–parabolic system modeling, for instance, flow in deformable porous media. By using a discontinuous temporal test basis, a time marching scheme is obtained. Higher order approximations that offer the potential to inherit most of the rich structure of solutions to the continuous problem on computationally feasible grids increase the block partitioning dimension of the algebraic systems, comprised of generalized saddle point blocks. Our V-cycle GMG preconditioner uses a local Vanka-type smoother. Its action is defined in an exact mathematical way. Due to nonlocal coupling mechanisms of 348 unknowns, the smoother is applied on patches of elements. This ensures damping of higher order error frequencies. By numerical experiments of increasing complexity, the efficiency of the solver for STFEMs of different polynomial order is illustrated and confirmed. Its parallel scalability is analyzed. Beyond this study of classical performance engineering, the solver’s energy efficiency is investigated as an additional and emerging dimension in the design and tuning of algorithms on the hardware.

Characteristics of the Complex Saddle Point of Polymer Field Theory

Characteristics of the Complex Saddle Point of Polymer Field Theory

An approach to duality via image space analysis and applications to linear vector optimization problems

In Yao et al. [Yao CL, Wang SQ, Tammer C. Construction of vector-valued weak separation functions with applications to conjugate duality in vector optimization; 2024. Submitted.], based on the idea of image space analysis and abstract convexity, we proposed a uniform framework of conjugate duality for the constrained vector optimization problems. This paper aims to apply this framework to linear vector optimizations. With the aid of a special collection of separation functions, a set-valued dual problem, which was also derived by Heyde, Löhne, Tammer in [Heyde F, Löhne A, Tammer C. The attainment of the solution of the dual program in vertices for vectorial linear programs. In: Barichard V, Ehrgott M, Gandibleux X, T'Kindt V, editors. Multiobjective programming and goal programming. 2009. p. 13–24. (Lecture Notes in Economics and Mathematical Systems; 618).; Heyde F, Löhne A, Tammer C. Set-valued duality theory for multiple objective linear programs and application to mathematical finance. Math Meth Oper Res. 2009;69:159–179.], is obtained by this approach. Within the uniform framework, we provide a new way to construct this dual model, and to investigate the duality theory of it. In this method, the strong duality is established by the separation of sets in the image space and the generalized saddle points.

Extended convergence analysis of the Scholtes-type regularization for cardinality-constrained optimization problems

AbstractWe extend the convergence analysis of the Scholtes-type regularization method for cardinality-constrained optimization problems. Its behavior is clarified in the vicinity of saddle points, and not just of minimizers as it has been done in the literature before. This becomes possible by using as an intermediate step the recently introduced regularized continuous reformulation of a cardinality-constrained optimization problem. We show that the Scholtes-type regularization method is well-defined locally around a nondegenerate T-stationary point of this regularized continuous reformulation. Moreover, the nondegenerate Karush–Kuhn–Tucker points of the corresponding Scholtes-type regularization converge to a T-stationary point having the same index, i.e. its topological type persists. As consequence, we conclude that the global structure of the Scholtes-type regularization essentially coincides with that of CCOP.

A class of new extended generalized shift-splitting preconditioners for $$3\times 3$$ block saddle point problems

A class of new extended generalized shift-splitting preconditioners for $$3\times 3$$ block saddle point problems

A hybrid complex spectral conjugate gradient learning algorithm for complex-valued data processing

Complex-valued neural networks (CVNNs) have become a powerful modelling tool for complex-valued data processing. Because most of the critical points of CVNNs are saddle points, the gradient-based learning algorithms for CVNNs enjoy more chances to reach the global minima while suffering from slow convergence. To this end, we propose a hybrid complex spectral conjugate gradient learning algorithm for fast training CVNNs in this paper. The proposed algorithm combines the scaled negative gradient with a Barzilai–Borwein stepsize and an optimized conjugate term to define a new training direction, thus providing an accurate approximation of the second-order curvature of the objective function. The complex Wolfe conditions are employed to adaptively determine the optimal training stepsize. Under mild conditions, the descent property of the training direction and the convergence of the proposed algorithm are theoretically established. Simulation results on a number of benchmark complex-valued data processing problems demonstrate the efficiency of the proposed algorithm.

Hybrid hyperplane gradient learning algorithm for RBF neural network

Gradient-based algorithms are still popularly used for training radial basis function neural network (RBFNN). However, these algorithms may lead to the vanishing gradient (saddle point or local minimum) problem, which is one of the common problems that limit learning performance. To cope with this, a hybrid hyperplane gradient learning algorithm (HHGLA) is proposed to improve the learning performance of the RBFNN in this paper. First, a hyperplane gradient (HPG), based on a hyperspace constructed by a solution population (SP), is introduced. Thus, the search process can cross the surface of the cost function to avoid the appearance of vanishing gradient. Second, an adaptive learning rate is designed to restrict the search process into this hyperspace. Then, during the learning process, the hyperspace is constantly contracted to approximate the global optimal solution. Third, the convergence property of HHGLA-based RBFNN is thoroughly analyzed to ensure its application success. Finally, empirical comparisons of the proposed HHGLA-RBFNN are given to illustrate its superiority. The simulation results show the feasibility of HHGLA-RBFNN of accuracy.

Stochastic path power and the Laplace transform

Abstract Transition probabilities for stochastic systems can be expressed in terms of a functional integral over paths taken by the system. Approximately evaluating this integral by the saddle point method in the weak-noise limit leads to a remarkable mapping between dominant stochastic paths through the potential V and conservative, Hamiltonian mechanics with an effective potential −│▽V│2. The conserved ``energy'' in this effective system has dimensions of power. We show that this power, H, can be identified with the Laplace parameter of the time-transformed dynamics. As H→0, corresponding to the long-time limit, the equilibrium Boltzmann density is recovered. However, keeping H finite leads to insights into the non-equilibrium behaviour of the system. Moreover, it facilitates the explicit summation over families of trajectories, which is far harder in the time domain, and turns out to be essential for making contact with the long-time limit in some cases. We illustrate the validity of these results using simple examples that can be explicitly solved by other means.

First-principles study of behaviors of point defects in Mo-14 Re alloy

Molybdenum alloys with rhenium additions of 14 % (Mo-14 Re) are regarded as structural materials for advanced nuclear reactors. The performance of Mo-14 Re alloy under irradiation is crucial for its application. Irradiation performance is related to the behaviors of point defects produced by irradiation. The behaviors of point defects in Mo-14 Re alloy were investigated using first-principles methods in this work. Compared with the results in pure Mo alloy, the formation energy and the migration energy of vacancy in Mo-14 Re were both reduced, and the migration energy of interstitial dumbbell increased while the formation energy decreased in Mo-14 Re. In Mo-14 Re, vacancy preferred to form and diffuse faster in Re-enriched environment. Knowing the number of Re atoms in the 1NN (first nearest neighbor) shell of the vacancy and saddle point, the vacancy formation energy and migration energy can be estimated. Since the interstitial atom has a certain orientation, the distribution of the atoms around it is anisotropic. To estimate the interstitial formation energy, it was necessary to count the number of Re atoms at the tensile and compressive sites at the 1NN shell around the interstitial. The interstitial motion was three-dimensional (3D) in Mo-14 Re. At 1200 K, the diffusion coefficient of Re atom was a little higher than that of Mo atom, which could lead to the segregation of Re atoms at defect sinks and the formation of Re-enriched phase. It is possible to investigate the longer-term evolution of microstructure using cluster dynamics (CD) or kinetic monte carlo (KMC) techniques based on these results.

Parameter space of leptogenesis in polynomial inflation

Polynomial inflation is a very simple and well motivated scenario. A potential with a concave “almost” saddle point at field value ϕ = ϕ 0 fits well the cosmic microwave background (CMB) data and makes testable predictions for the running of the spectral index and the tensor to scalar ratio. In this work we analyze leptogenesis in the polynomial inflation framework. We delineate the allowed parameter space giving rise to the correct baryon asymmetry as well as being consistent with data on neutrino oscillations. To that end we consider two different reheating scenarios. (i) If the inflaton decays into two bosons, the reheating temperature can be as high as  T rh ∼ 1014 GeV without spoiling the flatness of the potential, allowing vanilla N 1 thermal leptogenesis to work if  T rh > M 1 where N 1 is the lightest right-handed neutrino and  M 1 its mass. Moreover, if the dominant decay of the inflaton is into Higgs bosons of the Standard Model, we find that rare three-body inflaton decays into a Higgs boson plus one light and one heavy neutrino allow leptogenesis even for T rh < M 1 if the inflaton mass is of order 1012 GeV or higher; in the polynomial inflation scenario this requires ϕ 0 ≳ 2.5 MP . This novel mechanism of non-thermal leptogenesis is quite generic, since the coupling leading to the three-body final state is required in the type I see-saw mechanism. (ii) If the inflaton decays into two fermions, the flatness of the potential implies a lower reheating temperature. In this case inflaton decay to two N 1  still allows successful non-thermal leptogenesis if  ϕ 0 ≳ 0.1 MP and T rh ≳ 106 GeV.

Convergence of a Ramshaw-Mesina iteration

In 1991 Ramshaw and Mesina introduced a clever synthesis of penalty methods and artificial compression methods. Its form makes it an interesting option to replace the pressure update in the Uzawa iteration. The result, for the Stokes problem, is (1)Step1:−△un+1+∇pn=f(x),inΩ,un+1|∂Ω=0,Step2:pn+1−pn+β∇⋅(un+1−un)+α2∇⋅un+1=0.For saddle point problems, including Stokes, this iteration converges under a condition similar to the one required for Uzawa iteration.

Dynamic Regret of Distributed Online Saddle Point Problem

Dynamic Regret of Distributed Online Saddle Point Problem

Study on the application of 3D imaging based on the γ-photon silk thread method for visualisation and detection of flow fields

As a flow display technique, the silk thread method can clearly show the state of the flow field. However, displaying the state of the flow field in industrial closed metal cavities is impossible, and results of the traditional silk thread method fail to present depth information. To address these limitations, we propose a display method based on γ-photon silk thread 3D imaging. Firstly, a flow field visualization experimental platform applicable for industrial closed metal cavity was designed. Then, aiming at the shortcomings of insufficient γ-photon scanning data and limited imaging quality in flow field visualization, an adaptive sinogram interpolation-iterative filtering reconstruction algorithm was proposed to expand the data via the adaptive interpolation processing of the sinogram in image reconstruction. Beltrami filtering was also embedded in the iterative algorithm to effectively reduce noise and artifacts in the imaging image. Finally, the internal flow field inspection of the NACA0018 airfoil and the industrial confinement turbine were used as examples to visualise the wing upper surface flow and turbine blade passage secondary flow, respectively, and the results were consistent with those of the numerical simulation analysis. As the angle of attack of the wing increased, laminar flow separation occurred, and the silk thread structure state changed to the point where the aircraft stalled. After the imaging map analysis, the stall angle ranged between (15°,19°), which is consistent with the simulation results. In the turbine blade passage secondary flow detection, the imaging map formed a horseshoe vortex with airflow separation at the central saddle point and silk threads clearly presenting the structural state of the flow field along different directions.

Структура вихревых полей в волноводе пекериса с учетом дислокаций

Vortex fields are the most complex and least studied objects in the acoustics of layered media. They significantly complicate the structure of the sound field due to the appearance of nodal and saddle points in it, in the vicinity of which the concept of the phase of the sound wave becomes uncertain. Therefore, such special points of the sound field are usually called phase front dislocations. In turn, the dislocation structure of the vortex field is most complex in waveguides, which play an extremely important role in theoretical and applied hydroacoustics. In this work, the vortex structure of the sound field and the corresponding dislocation structure of the vortex field are analyzed using the example of a Pekeris waveguide, the sound field in which is constructed within the framework of a non-self-adjoint model formulation of the corresponding boundary value problem.

Multi-dimensional phase portraits of stochastic fractional derivatives for nonlinear dynamical systems with solitary wave formation

This manuscript delves into the examination of the stochastic fractional derivative of Drinfel’d-Sokolov-Wilson equation, a mathematical model applicable in the fields of electromagnetism and fluid mechanics. In our study, the proposed equation is through examined through various viewpoints, encompassing soliton dynamics, bifurcation analysis, chaotic behaviors, and sensitivity analysis. A few dark and bright shaped soliton solutions, including the unperturbed term, are also examined, and the various 2D and 3D solitonic structures are computed using the Tanh-method. It is found that a saddle point bifurcation causes the transition from periodic behavior to quasi-periodic behavior in a sensitive area. Further analysis reveals favorable conditions for the multidimensional bifurcation of dynamic behavioral solutions. Different types of wave solutions are identified in certain solutions by entering numerous values for the parameters, demonstrating the effectiveness and precision of Tanh-methods. A planar dynamical system is then created using the Galilean transformation, with the actual model serving as a starting point. It is observed that a few physical criteria in the discussed equation exhibit more multi-stable properties, as many multi-stability structures are employed by some individuals. Moreover, sensitivity behavior is employed to examine perturbed dynamical systems across diverse initial conditions. The techniques and findings presented in this paper can be extended to investigate a broader spectrum of nonlinear wave phenomena.

Augmented Lagrangian Acceleration of Global-in-Time Pressure Schur Complement Solvers for Incompressible Oseen Equations

This work is focused on an accelerated global-in-time solution strategy for the Oseen equations, which highly exploits the augmented Lagrangian methodology to improve the convergence behavior of the Schur complement iteration. The main idea of the solution strategy is to block the individual linear systems of equations at each time step into a single all-at-once saddle point problem. By elimination of all velocity unknowns, the resulting implicitly defined equation can then be solved using a global-in-time pressure Schur complement (PSC) iteration. To accelerate the convergence behavior of this iterative scheme, the augmented Lagrangian approach is exploited by modifying the momentum equation for all time steps in a strongly consistent manner. While the introduced discrete grad-div stabilization does not modify the solution of the discretized Oseen equations, the quality of customized PSC preconditioners drastically improves and, hence, guarantees a rapid convergence. This strategy comes at the cost that the involved auxiliary problem for the velocity field becomes ill conditioned so that standard iterative solution strategies are no longer efficient. Therefore, a highly specialized multigrid solver based on modified intergrid transfer operators and an additive block preconditioner is extended to solution of the all-at-once problem. The potential of the proposed overall solution strategy is discussed in several numerical studies as they occur in commonly used linearization techniques for the incompressible Navier–Stokes equations.

Research on the efficient EM modeling method from multilayered anisotropic medium-metal composite targets

The paper presents a high-frequency modeling approach tailored for the electromagnetic (EM) scattering characteristics from electrically large radar targets coated with multi-layered anisotropic mediums (MLAMs). The approach begins by deriving the plane-wave spectrum expression for the incident EM field within MLAMs. It then employs spectral domain full-wave analysis method (SDFWAM) to obtain an analytical representation of the scattering field, further leveraging saddle point evaluation (SPE) to derive asymptotic solutions in the spatial domain. By integrating principles of physical optics (PO) and the tangent plane approximation, the far-field scattering characteristics of target enveloped in the specified medium are efficiently delineated. Validations against standard structure and the Misty satellite model reveal the method's pronounced alignment with the method of moments – finite element method (MoM-FEM) hybrid numerical algorithm, underscoring its notable computational efficiency. Furthermore, in conjunction with scattering sources decomposition technique, the approach is applied to optimize the radar cross-section (RCS) of the Su-57 aircraft, achieving precise and intelligent RCS control at minimal material cost. In conclusion, this research offers pivotal technological and theoretical foundations for EM scattering prediction, stealth design, and performance assessment in radar target domains.

Topological phases and non-Hermitian topology in tunable nonreciprocal cyclic three-mode optical systems

We propose a method for simulating a 1D non-Hermitian Su-Schrieffer-Heeger model with modulated nonreciprocal hopping using a cyclic three-mode optical system. The current system exhibits different localization of topologically nontrivial phases, which can be characterized by the winding number. We find that the eigenenergies of such a system undergo a real-complex transition as the nonreciprocal hopping changes, accompanied by a non-Bloch parity-time symmetry breaking. We explain this phase transition by considering the evolution of saddle points on the complex energy plan and the ratio of complex eigenenergies. Additionally, we demonstrate that the skin states resulting from the non-Hermitian skin effect possess higher-order exceptional points under the critical point of the non-Bloch parity-time phase transition. Furthermore, we investigate the non-Hermitian skin phase transition by the directional mean inverse participation ratio and the generalized Brillouin zone. This work provides an alternative way to investigate the novel topological and non-Hermitian effects in nonreciprocal optical systems.

A matching Schur complement preconditioning technique for inverse source problems

Numerical discretization of a regularized inverse source problem leads to a non-symmetric saddle point linear system. Interestingly, the Schur complement of the non-symmetric saddle point system is Hermitian positive definite (HPD). Then, we propose a preconditioner matching the Schur complement (MSC). Theoretically, we show that the preconditioned conjugate gradient (PCG) method for a linear system with the preconditioned Schur complement as coefficient has a linear convergence rate independent of the matrix size and value of the regularization parameter involved in the inverse problem. Fast implementations are proposed for the matrix-vector multiplication of the preconditioned Schur complement so that the PCG solver requires only quasi-linear operations. To the best of our knowledge, this is the first solver with guarantee of linear convergence for the inversion of Schur complement arising from the discrete inverse problem. Combining the PCG solver for inversion of the Schur complement and the fast solvers for the forward problem in the literature, the discrete inverse problem (the saddle point system) is solved within a quasi-linear complexity. Numerical results are reported to show the performance of the proposed matching Schur complement (MSC) preconditioning technique.

Dynamic Modeling of Environmental Subsidies

In this research, the intertemporal optimal management of subsidies offered by the environmental regulator and the dynamic conflict between two groups of economic agents involved in environmental quality are discussed. First the environmental model is examined in its optimal control management form with two state variables. The analysis of the improved model, inspecting the social planner’s decision (policy) variable, the variable which influences not only environmental quality but the national budget stock, reveals that is dependent on the growth of the national budget stock. A negative growth rate leads to the long run saddle point equilibrium, while an incremental growth rate, but less than the model’s discount rate, leads to an interesting complicated limit cycle equilibrium, for which under certain parameter values the orbit’s phase portrait can be drawn. For the dynamic game model between the social planner and natural resource exploiters, the equilibrium conditions are examined. It is rather a richer than the point equilibrium for which the cyclical strategies have great importance. Therefore, the conditions for that rich equilibrium are found. As a continuation, the paper concludes that the equilibrium condition is that the players’ discount rates are both greater than the national budget interest rate. Finally, the certain values of the equilibrium strategies and national budget stock are provided.