Function Minimization Problem Research Articles

Continuation problem of the electromagnetic field in the frequency domain for horizontally layred media

The paper presents analytical expressions for solving the problem of continuation of the electromagnetic field in the frequency domain for a horizontally layered medium. The numerical solution algorithm uses layerwise recalculation of the required quantities, for which analytical expressions are presented in a form that allows calculations and layerwise recalculation without the accumulation of rounding errors. The solution to the continuation problem is obtained as a solution to the cost functional minimization problem. The strong convexity of the functional is proven, which implies the existence of a unique solution to the problem posed.

A second-order projection neurodynamic approach with exponential convergence for sparse signal reconstruction

Recently, a class of second-order neurodynamic approaches with convergence rates of O(1t) or O(1t2) has been developed to address the sparse signal reconstruction problem. In this paper, we propose a second-order projection neurodynamic approach (SOPNA) with exponential convergence to reconstruct a sparse signal by solving a modified inverted Gaussian function (MIGF) minimization problem. The existence, uniqueness, and feasibility of the solution to SOPNA are detailedly investigated, and the exponential convergence rate of O(exp(−μt)),μ>0 is proved. This implies that our proposed SOPNA can achieve a significantly superior convergence performance than several existing second-order neurodynamic approaches. Numerical experiments also confirm its effectiveness and superiority. Finally, the applications of the proposed SOPNA in real signal and real image reconstructions validate its practical feasibility.

Second-order Sobolev gradient flows for computing ground state of ultracold Fermi gases

Based on density functional theory, the ground state for ultracold Fermi gases with dipole–dipole interaction is a functional minimization problem with orthonormality constraints. Many methods such as direct minimization, self-consistent iteration method and first-order gradient flow had been proposed to solve such problem. In this paper, we introduce the second-order Sobolev gradient flows, solve them to the steady state and get the ground state of ultracold Fermi gases numerically. Two contributions have been made: firstly we extend the usual first-order gradient flows to the second-order gradient flows; secondly, we extend the usual L2 gradient to the Sobolev gradient and get the Sobolev gradient flows. We prove that the second-order Sobolev gradient flows have the properties of orthonormality preserving and ‘modified’ energy diminishing, which are desirable for the computation of the ground state solution. Several numerical techniques have been used to discretize the time-dependent second-order Sobolev gradient flows. These include explicit leap-frog method and stabilized semi-implicit method for temporal discretization and Fourier pseudo-spectral method for spatial variables. Extensive numerical examples for the ground state are reported to demonstrate the power of the numerical methods.

Modified Memoryless Spectral-Scaling Broyden Family on Riemannian Manifolds

This paper presents modified memoryless quasi-Newton methods based on the spectral-scaling Broyden family on Riemannian manifolds. The method involves adding one parameter to the search direction of the memoryless self-scaling Broyden family on the manifold. Moreover, it uses a general map instead of vector transport. This idea has already been proposed within a general framework of Riemannian conjugate gradient methods where one can use vector transport, scaled vector transport, or an inverse retraction. We show that the search direction satisfies the sufficient descent condition under some assumptions on the parameters. In addition, we show global convergence of the proposed method under the Wolfe conditions. We numerically compare it with existing methods, including Riemannian conjugate gradient methods and the memoryless spectral-scaling Broyden family. The numerical results indicate that the proposed method with the BFGS formula is suitable for solving an off-diagonal cost function minimization problem on an oblique manifold.

A minimum objective function trim procedure for VTOLs noise reduction

The present paper proposes a novel approach for multi-rotor acoustic nuisance mitigation based on the identification of optimal control settings. This strategy is possible thanks to the peculiarity of multi-rotor systems (like, for instance, those involved in urban air mobility applications) to have redundant controls. The idea is to take full advantage of the control redundancy by defining the control settings that guarantee the desired steady-state flight conditions while, at the same time, optimizing selected target functions such as performance or noise emissions. To this aim, the trim problem is recast into a constrained cost function minimization problem. Since the objective is to reduce as much as possible the noise disturbance of trimmed VTOL, an aeroacoustic tool based on the chordwise-compact form of the Farassat 1A formulation for the evaluation of noise radiation is suitably coupled with a trim solver. The numerical investigations consider quadcopter and hexacopter configurations in level flight at several advancing speeds. Comparisons of the proposed minimum-noise trim strategy with two other trim strategies based on the minimization of rotor torque and control effort prove its capability to provide an overall reduction of noise throughout the entire velocity range examined.

An extrapolation approach within the Wiener path integral technique for efficient stochastic response determination of nonlinear systems

An extrapolation approach within the Wiener path integral (WPI) technique is developed for determining the stochastic response of diverse nonlinear dynamical systems. Specifically, the WPI technique treats the system response joint transition probability density function (PDF) as a functional integral over the space of all possible paths connecting the initial and the final states of the response vector. Further, the functional integral is evaluated, ordinarily, by considering the contribution only of the most probable path. This corresponds to an extremum of the functional integrand, and is determined by solving a functional minimization problem that takes the form of a deterministic boundary value problem (BVP). This BVP corresponds to a specific grid point of the response PDF domain. Remarkably, the BVPs corresponding to two neighboring grid points not only share the same equations, but also the boundary conditions differ only slightly. This unique aspect of the technique is exploited herein. Specifically, it is shown that solution of a BVP and determination of the response PDF value at a specific grid point can be used for extrapolating and estimating efficiently the PDF values at neighboring points without the need for considering additional BVPs. Notably, the herein developed approach enhances significantly the computational efficiency of the WPI technique without, practically, affecting the associated degree of accuracy. Two numerical examples relating to a Duffing nonlinear oscillator subjected to combined stochastic and deterministic periodic loading, and to an oscillator with asymmetric nonlinearities and fractional derivative elements are considered to demonstrate the reliability of the extrapolation approach. Juxtapositions with pertinent Monte Carlo simulation data are included as well.

On the existence, uniqueness and calculation of Wiener path integral most probable paths for determining the stochastic response of nonlinear dynamical systems

Various techniques are developed for addressing the existence, uniqueness and numerical calculation of Wiener path integral (WPI) most probable path solutions. Specifically, the WPI technique for determining the stochastic response of diverse nonlinear dynamical systems treats the system response joint transition probability density function as a functional integral over the space of all possible paths connecting the initial and the final states of the response vector. This functional integral is evaluated, ordinarily, by resorting to an approximate approach that considers the contribution only of the most probable path. The most probable path corresponds to an extremum of the functional integrand and is determined by solving a functional minimization problem that takes the form of a deterministic boundary value problem (BVP).In this paper, first, it is shown that for the commonly considered case of the system nonlinearity being of polynomial form, there exist globally optimal solutions corresponding to the most probable path BVP. Further, relying on algebraic geometry concepts and tools, a condition is derived for determining if the BVP for the most probable path exhibits a unique solution over a specific region. Furthermore, a novel solution approach is developed for the BVP by relying on Sylvester’s dialytic method of elimination. Notably, the method reduces the complexity of the BVP system of coupled multivariate polynomial equations by eliminating one or more variables. Various numerical examples pertaining to diverse nonlinear oscillators are included for demonstrating the capabilities of the developed techniques.

Based on Gradient Algorithm for the Inverse Source Problem of a Class of Time-space Fractional Diffusion Equations

Since entering the 21st century, the establishment of fractional-order diffusion equations in various fields has been of great value and has garnered widespread attention. This study focuses on inverse source term problem for time-space fractional diffusion equation (TSFDE) using given boundary data. First, the identification source problem is transformed into a functional minimization problem utilize the Tikhonov-type regularization method. Then, the sensitivity and the adjoint problem are derived, and the gradient of functional is obtained. The conjugate gradient algorithm is used to solve the minimization problem. Finally, three xamplel with different types of source terms are used to stated the effectiveness and stability, the impact of various parameters on the numerical results is analyzed.

Q-Learning-Aided Offloading Strategy in Edge-Assisted Federated Learning over Industrial IoT

Federated learning (FL) is a key solution to realizing a cost-efficient and intelligent Industrial Internet of Things (IIoT). To improve training efficiency and mitigate the straggler effect of FL, this paper investigates an edge-assisted FL framework over an IIoT system by combining it with a mobile edge computing (MEC) technique. In the proposed edge-assisted FL framework, each IIoT device with weak computation capacity can offload partial local data to an edge server with strong computing power for edge training. In order to obtain the optimal offloading strategy, we formulate an FL loss function minimization problem under the latency constraint in the proposed edge-assisted FL framework by optimizing the offloading data size of each device. An optimal offloading strategy is first derived in a perfect channel state information (CSI) scenario. Then, we extend the strategy into an imperfect CSI scenario and accordingly propose a Q-learning-aided offloading strategy. Finally, our simulation results show that our proposed Q-learning-based offloading strategy can improve FL test accuracy by about 4.7% compared to the conventional FL scheme. Furthermore, the proposed Q-learning-based offloading strategy can achieve similar performance to the optimal offloading strategy and always outperforms the conventional FL scheme in different system parameters, which validates the effectiveness of the proposed edge-assisted framework and Q-learning-based offloading strategy.

Inverse problems of identifying the unknown transverse shear force in the Euler–Bernoulli beam with Kelvin–Voigt damping

Abstract In this paper, we study the inverse problems of determining the unknown transverse shear force g ⁢ ( t ) {g(t)} in a system governed by the damped Euler–Bernoulli equation ρ ⁢ ( x ) ⁢ u t ⁢ t + μ ⁢ ( x ) ⁢ u t + ( r ⁢ ( x ) ⁢ u x ⁢ x ) x ⁢ x + ( κ ⁢ ( x ) ⁢ u x ⁢ x ⁢ t ) x ⁢ x = 0 , ( x , t ) ∈ ( 0 , ℓ ) × ( 0 , T ] , \rho(x)u_{tt}+\mu(x)u_{t}+(r(x)u_{xx})_{xx}+(\kappa(x)u_{xxt})_{xx}=0,\quad(x,% t)\in(0,\ell)\times(0,T], subject to the boundary conditions u ⁢ ( 0 , t ) = 0 , u x ⁢ ( 0 , t ) = 0 , [ r ⁢ ( x ) ⁢ u x ⁢ x + κ ⁢ ( x ) ⁢ u x ⁢ x ⁢ t ] x = ℓ = 0 , - [ ( r ⁢ ( x ) ⁢ u x ⁢ x + κ ⁢ ( x ) ⁢ u x ⁢ x ⁢ t ) x ] x = ℓ = g ⁢ ( t ) , u(0,t)=0,\quad u_{x}(0,t)=0,\quad[r(x)u_{xx}+\kappa(x)u_{xxt}]_{x=\ell}=0,% \quad-[(r(x)u_{xx}+\kappa(x)u_{xxt})_{x}]_{x=\ell}=g(t), for t ∈ [ 0 , T ] {t\in[0,T]} , from the measured deflection ν ⁢ ( t ) := u ⁢ ( ℓ , t ) {\nu(t):=u(\ell,t)} , t ∈ [ 0 , T ] {t\in[0,T]} , and from the bending moment ω ⁢ ( t ) := - ( r ⁢ ( 0 ) ⁢ u x ⁢ x ⁢ ( 0 , t ) + κ ⁢ ( 0 ) ⁢ u x ⁢ x ⁢ t ⁢ ( 0 , t ) ) , t ∈ [ 0 , T ] , \omega(t):=-(r(0)u_{xx}(0,t)+\kappa(0)u_{xxt}(0,t)),\quad t\in[0,T], where the terms ( κ ⁢ ( x ) ⁢ u x ⁢ x ⁢ t ) x ⁢ x {(\kappa(x)u_{xxt})_{xx}} and μ ⁢ ( x ) ⁢ u t {\mu(x)u_{t}} account for the Kelvin–Voigt damping and external damping, respectively. The main purpose of this study is to analyze the Kelvin–Voigt damping effect on determining the unknown transverse shear force (boundary input) through the given boundary measurements. The inverse problems are transformed into minimization problems for Tikhonov functionals, and it is shown that the regularized functionals admit unique solutions for the inverse problems. By suitable regularity on the admissible class of shear force g ⁢ ( t ) {g(t)} , we prove that these functionals are Fréchet differentiable, and the derivatives are expressed through the solutions of corresponding adjoint problems posed with measured data as boundary data associated with the direct problem. The solvability of these adjoint problems is obtained under the minimal regularity of the boundary data g ⁢ ( t ) {g(t)} , which turns out to be the regularizing effect of the Kelvin–Voigt damping in the direct problem. Furthermore, using the Fréchet derivative of the more regularized Tikhonov functionals, we obtain remarkable Lipschitz stability estimates for the transverse shear force in terms of the given measurement by a feasible condition only on the Kelvin–Voigt damping coefficient.

Orbital-free approach for large-scale electrostatic simulations of quantum nanoelectronics devices

The route to reliable quantum nanoelectronic devices hinges on precise control of the electrostatic environment. For this reason, accurate methods for electrostatic simulations are essential in the design process. The most widespread methods for this purpose are the Thomas-Fermi (TF) approximation, which provides quick approximate results, and the Schrödinger-Poisson (SP) method, which better takes into account quantum mechanical effects. The mentioned methods suffer from relevant shortcomings: the TF method fails to take into account quantum confinement effects that are crucial in heterostructures, while the SP method suffers severe scalability problems. This paper outlines the application of an orbital-free approach inspired by density functional theory. By introducing gradient terms in the kinetic energy functional, our proposed method incorporates corrections to the electronic density due to quantum confinement while it preserves the scalability of a theory that can be expressed as a functional minimization problem. This method offers a new approach to addressing large-scale electrostatic simulations of quantum nanoelectronic devices.

A mechanics model in design of flexible nozzle with multiple movable hinge boundaries

In this paper, we propose a modified variational approach to predict the morphology of the flexible nozzle used in wind tunnel. Different from previous studies, the movements of the multiple hinges are considered as movable displacement boundary conditions during establishing the potential energy functional. The cubic spline interpolation method is employed to supply the supplementary boundary conditions in calculation of the functional minimization problem. Current analytical model is verified by experiments carried out on a fixed-flexible nozzle structure whose geometries and materials are the same as those from a commissioned supersonic nozzle. The maximum deviation between the predictions from theoretical method and laser displacement testing does not exceed 0.5 mm. This method can also deal with the large deflection beam problem with multiple movable boundaries.

Investigation of the well-posedness of the cauchy problem for the helmholtz equation

In this paper we consider the the initial boundary value problem for the Helmholtz equation. The well-possedness of the Cauchy problem for the Helmholtz equation is investigated and the uniqueness of the solution of the original problem in the class of functions represented as Fourier series is proved. For a direct problem, a proof of the stability theorem of the generalized solution is presented, and an estimate of the stability of the generalized solution is obtained. To solve the problem, the original problem is reduced to the inverse problem and is written in operator form. We reduce this problem to the functional minimization problem, the Landweber iteration method is used in the minimization problem. A numerical study of the stability of the direct and initial problems is carried out. The results of the study are shown in tables and figures.

Minimizing Convex Functions with Rational Minimizers

Given a separation oracle SO for a convex function f defined on ℝ n that has an integral minimizer inside a box with radius R , we show how to find an exact minimizer of f using at most O(n (n log log (n)/ log (n) + log ( R ))) calls to SO and poly ( n , log ( R )) arithmetic operations, or O(n log (nR) calls to SO and exp ( O(n) ) ⋅ poly (log (R) ) arithmetic operations. When the set of minimizers of f has integral extreme points, our algorithm outputs an integral minimizer of f . This improves upon the previously best oracle complexity of O(n 2 ( n + log ( R ))) for polynomial time algorithms and O(n 2 log ( nR ) for exponential time algorithms obtained by [Grötschel, Lovász and Schrijver, Prog. Comb. Opt. 1984, Springer 1988] over thirty years ago. Our improvement on Grötschel, Lovász and Schrijver’s result generalizes to the setting where the set of minimizers of f is a rational polyhedron with bounded vertex complexity. For the Submodular Function Minimization problem, our result immediately implies a strongly polynomial algorithm that makes at most O(n 3 log log ( n )/log ( n )) calls to an evaluation oracle, and an exponential time algorithm that makes at most O(n 2 log ( n )) calls to an evaluation oracle. These improve upon the previously best O(n 3 log 2 ( n )) oracle complexity for strongly polynomial algorithms given in [Lee, Sidford and Wong, FOCS 2015] and [Dadush, Végh and Zambelli, SODA 2018], and an exponential time algorithm with oracle complexity O(n 3 log ( n )) given in the former work. Our result is achieved via a reduction to the Shortest Vector Problem in lattices. We show how an approximately shortest vector of an auxiliary lattice can be used to effectively reduce the dimension of the problem. Our analysis of the oracle complexity is based on a potential function that simultaneously captures the size of the search set and the density of the lattice, which we analyze via tools from convex geometry and lattice theory.

Parallel Algorithm for Parametric Identification of Dynamical Systems with Interval Parameters

The paper presents a parallel algorithm for the parametric identification of dynamical systems with interval parameters. The algorithm is based on the previously developed, substantiated and tested adaptive interpolation algorithm, which makes it possible to explicitly obtain the dependence of the states of a dynamic system on interval parameters. The solution of the problem of parametric identification is reduced to the problem of minimizing a certain objective function in the space of boundaries of interval parameter estimates. Due to the use of the adaptive interpolation algorithm when calculating the gradient of the objective function, there is no need for additional analysis and modeling of the original dynamic system, so it is convenient to use first-order methods for optimization. However, the task of calculating the objective function and the gradient includes a set of conditional minimization problems for explicit functions that can be solved independently of each other. The article discusses the main aspects and features of parallelization and implementation of the parametric identification algorithm and tests it on several representative examples. The acceleration and efficiency of parallelization are analyzed.

Skull Surface Reconstruction with Euler’s Elastica Model from Point Cloud Data

In order to reconstruct accurate smooth implicit surfaces from skull point cloud data with complex topological relations, a variational level set method based on Euler’s Elastica model is proposed. Firstly, Euler’s Elastica is introduced into the variational energy functional as a regularization constraint, and the variational energy functional minimization problem is transformed into a series of subproblems by the augmented Lagrange method. Secondly, each subproblem is solved by generalized soft threshold formula, analytical formula and fast Fourier transform respectively. Finally, the projection algorithm is used during the computation to avoid the re-initialization of the level set function effectively. The experimental results show that the proposed variational level set method based on Euler’s Elastica has better convergence and effectiveness than the existing methods, which can improve the reconstruction accuracy and avoid missing surface edges.

An Adaptive Alternating Direction Method of Multipliers

The alternating direction method of multipliers (ADMM) is a powerful splitting algorithm for linearly constrained convex optimization problems. In view of its popularity and applicability, a growing attention is drawn toward the ADMM in nonconvex settings. Recent studies of minimization problems for nonconvex functions include various combinations of assumptions on the objective function including, in particular, a Lipschitz gradient assumption. We consider the case where the objective is the sum of a strongly convex function and a weakly convex function. To this end, we present and study an adaptive version of the ADMM which incorporates generalized notions of convexity and penalty parameters adapted to the convexity constants of the functions. We prove convergence of the scheme under natural assumptions. To this end, we employ the recent adaptive Douglas–Rachford algorithm by revisiting the well-known duality relation between the classical ADMM and the Douglas–Rachford splitting algorithm, generalizing this connection to our setting. We illustrate our approach by relating and comparing to alternatives, and by numerical experiments on a signal denoising problem.

Minimizing submodular functions on diamonds via generalized fractional matroid matchings

In this paper we show the first polynomial-time algorithm for the problem of minimizing submodular functions on the product of diamonds of finite size. This submodular function minimization problem is reduced to the membership problem for an associated polyhedron, which is equivalent to the optimization problem over the polyhedron, based on the ellipsoid method. The latter optimization problem is a generalization of the weighted fractional matroid matching problem. We give a combinatorial polynomial-time algorithm for this optimization problem by extending the result by Gijswijt and Pap (2013) [9].

On the Structure of a Quasi-Optimal Algorithm for Circuit Designing

The formulation of the process of analog system design has been done on the basis of the control theory application as the problem of a special functional minimization. This approach generalizes the design process and produces the different design trajectories inside the same optimization procedure. Numerical examples show that the potential computer time gain of the optimal design strategy with respect to the traditional design strategy increases when the size and the complexity of the system increase. Some properties of an additional acceleration effect of the design process were analyzed. The special selection of the optimization process start point provides the acceleration effect with a great probability. The positions of the optimal switch points of the control vector were found on the basis of the analysis of the special Lyapunov function of the design process. The combination of the acceleration effect with the start point preliminary selection and the optimal switch points of the control vector serve as the principal ideas to the time-optimal design algorithm construction.

Minimization Problems for Functionals Depending on Generalized Proportional Fractional Derivatives

In this work we study variational problems, where ordinary derivatives are replaced by a generalized proportional fractional derivative. This fractional operator depends on a fixed parameter, acting as a weight over the state function and its first-order derivative. We consider the problem with and without boundary conditions, and with additional restrictions like isoperimetric and holonomic. Herglotz’s variational problem and when in presence of time delays are also considered.