Second-order Conditions Research Articles

Variational Networks: An Optimal Control Approach to Early Stopping Variational Methods for Image Restoration

We investigate a well-known phenomenon of variational approaches in image processing, where typically the best image quality is achieved when the gradient flow process is stopped before converging to a stationary point. This paradox originates from a tradeoff between optimization and modeling errors of the underlying variational model and holds true even if deep learning methods are used to learn highly expressive regularizers from data. In this paper, we take advantage of this paradox and introduce an optimal stopping time into the gradient flow process, which in turn is learned from data by means of an optimal control approach. After a time discretization, we obtain variational networks, which can be interpreted as a particular type of recurrent neural networks. The learned variational networks achieve competitive results for image denoising and image deblurring on a standard benchmark data set. One of the key theoretical results is the development of first- and second-order conditions to verify optimal stopping time. A nonlinear spectral analysis of the gradient of the learned regularizer gives enlightening insights into the different regularization properties.

$$L^\infty $$-Stability of a Parametric Optimal Control Problem Governed by Semilinear Elliptic Equations

This paper studies local stability of a parametric optimal control problem governed by semilinear elliptic equations with mixed pointwise constraints. We show that if the unperturbed problem satisfies the strictly nonnegative second-order optimality conditions, then the solution map is upper Holder continuous in $$L^\infty $$-norm of control variable.

Pressure Drop and Flow Development in the Entrance Region of Microchannels With Second-Order Velocity Slip Condition and the Requirement for Development Length

AbstractIn this investigation, Lfd* and Δp in the entrance region of circular and parallel plate microchannels have been determined for 10−2≤Re≤104 and 10−4≤Kn≤0.2, employing the second-order velocity slip condition at the wall with C1=1 and 0≤C2≤0.5. Results indicate that although local velocity slip at the wall is always higher than that for the fully developed section, local wall shear stress for higher Kn and C2 could be lower than its fully developed value, which is also more prominent for lower Re. Therefore, depending upon the operating condition, K(x) and Kfd could assume negative values, implying that pressure gradient in the developing region could even be less than that in the fully developed section. It has been further observed that both Lfd* and Kfd are characterized by the low and the high Re asymptotes, using which extremely accurate correlations have been proposed for both geometries.

Normal approximation for weighted sums under a second-order correlation condition

Under correlation-type conditions, we derive an upper bound of order $(\log n)/n$ for the average Kolmogorov distance between the distributions of weighted sums of dependent summands and the normal law. The result is based on improved concentration inequalities on high-dimensional Euclidean spheres. Applications are illustrated on the example of log-concave probability measures.

Cosine-type weighted hybrid absorbing boundary based on the second-order Higdon boundary condition and its GPU implementation

AbstractIn this paper, we analyze the absorption effect of the hybrid absorbing boundary condition (ABC) with various hybrid modes, and propose a cosine-type optimized weighted hybrid mode taking into account the boundary reflection intensity of the inner and outer boundaries of transition area. Additionally, we derive a new finite-difference scheme of the second-order Higdon ABC and the corner equation for Graphic Processing Unit (GPU) acceleration. On this basis, a new type of second-order Higdon hybrid ABC applicable for GPU acceleration is established in the acoustic finite-difference modeling. Numerical experiments demonstrate that the proposed cosine-type weighted hybrid mode can achieve a better absorption effect compared with other weighted hybrid modes; the second-order Higdon ABC based on the proposed new finite-difference scheme can effectively improve the GPU speed-up ratio and is more effective in large-scale wavefield high-precision simulation based on GPU acceleration.

Second-order optimality conditions for set optimization using coradiant sets

The aim of this paper is to study second-order optimality conditions for a set-valued optimization problem with set criterion, where the order relation is induced by a set belong to a class of specific coradiant sets and is not necessarily a preorder. We introduce a notion of generalized second-order radial set, different from the classical second-order radial set, it is defined on a set, not on a point. Using the generalized second-order radial set, we introduce a new type of generalized second-order radial derivatives for set-valued maps and apply them to establish some necessary and sufficient conditions for the lower strict minimal solution of the set optimization problem.

Second-order slip condition considering Langmuir isothermal adsorption for rarefied gas microflows

Effect of the slip boundary condition on rarefied gas flow simulations plays an important role to understand the behaviour of gas microflows in MEMS. Several second-order slip conditions were proposed by the models of the kinetic theory of gases to simulate the rarefied gas microflows, in which the so-called classical second-order slip condition was derived from the Karniadakis et al. model. In this paper, a new second-order slip condition is proposed to employ with the Navier-Stokes-Fourier equations for simulating the rarefied gas flows in microchannels. It is derived by combining the Langmuir isothermal adsorption and the Karniadakis et al. model, with the aim of achieving a more realistic physical model. The pressure-driven back-forward-step, the Couette and pressure-driven Poiseulle rarefied gas flows in microchannels are investigated to validate our new second-order slip condition. Slip velocities using our new second-order slip condition are better than those using the conventional Maxwell and the so-called classical second-order slip conditions, and are in very good agreement with the DSMC data for all cases considered.

First and Second Order Conditions for Optimal Control Problems with an $L^0$ Term in the Cost Functional

In this paper, we investigate optimal control problems subject to a semilinear elliptic partial differential equation. The cost functional contains a term that measures the size of the support of the control, which is the so-called $L^0$-norm. We provide necessary and sufficient optimality conditions of second order. The sufficient second-order condition is obtained by analyzing a partially convexified problem. Interestingly, the structure of the problem yields second-order conditions with different bilinear forms for the necessary and for the sufficient conditions.

Second-Order Optimality Conditions for Multiobjective Optimal Control Problems with Mixed Pointwise Constraints and Free Right End Point

In this paper, we derive first- and second-order necessary optimality conditions of Karush--Kuhn--Tucker type for H-locally weak solutions in the sense of Pareto of multiobjective optimal control problems with free right end point and free end-time and with mixed pointwise constraints. To deal with the problem, we first establish first- and second-order optimality conditions for a reduced problem with fixed end-time. We then obtain optimality conditions for the original problem.

Iodometric determination of cystamine dihydrochloride in tablets using diperoxyadipic acid

The kinetics of cystamine dihydrochloride oxidation by diperoxyadipic acid (DPAA) was studied in aqueous buffer solutions of pH 2.9, 6.9, 8.0, 8.5 and 9.2 under second-order conditions at the temperature 293 K. The second order rate constants, k (L mol−1 min−1) were calculated from kinetic values. A suitable mechanism scheme based on these observations was proposed. The possibility of application of DPAA as reagent for oxidimetric determination of cystamine dihydrochloride in tabtets was investigated. Cystamine dihydrochloride was determined by indirect titration with DPAA. The required amount of Cystamine dihydrochloride was dissolved in water, pH 8.0 buffer solution and DPAA solution was added. After 7 min, the solution was acidified. The excess DPAA was determined via iodometric titration. The advantages of the applied analytical techniques in the determination of cystamine dihydrochloride in tablets «Cystamine 0.2 g» was presented. The recovery of сystamine dihydrochloride was 99.32±1.45%. A paired t-test showed that all results obtained for bulk drug and in tablets «Cystamine 0.2 g», using the proposed procedure and the official procedure respectively, agreed at the 95% confidence level.

Joint Power Allocation for NOMA-Based Diamond Relay Networks With and Without Cooperation

Non-orthogonal multiple access (NOMA)-based diamond relaying (NDR) is an efficient approach for combining NOMA and relaying techniques in such a way as to enhance the achievable rate from the source to the destination in the network. This paper examines the problem of joint power allocation among all the transmission phases during the operation of such networks. Based on the Karush-Kuhn-Tucker (KKT) condition and the second-order sufficient condition (SOSC), at least the local optimal solution is derived and analyzed. In addition, a new protocol for NDR networks based on cooperative communications is introduced and the associated joint power-allocation problem is examined. It is shown that this cooperative NDR (C-NDR) network further improves the achievable rate in some typical placements of the relays. Simulation results verify the correctness of the deviation and confirm the effectiveness of the proposed joint power-allocation method for both NDR and C-NDR networks.

On Optimal Control Problems with Controls Appearing Nonlinearly in an Elliptic State Equation

An optimal control problem for a semilinear elliptic equation is discussed, where the control appears nonlinearly in the state equation but is not included in the objective functional. The existence of optimal controls is proved by a measurable selection technique. First-order necessary optimality conditions are derived and two types of second-order sufficient optimality conditions are established. A first theorem invokes a well-known assumption on the set of zeros of the switching function. A second relies on coercivity of the second derivative of the reduced objective functional. The results are applied to the convergence of optimal state functions for a finite element discretizion of the control problem.

Metric sub-regularity in optimal control of affine problems with free end state

The paper investigates the property of Strong Metric sub-Regularity (SMsR) of the mapping representing the first order optimality system for a Lagrange-type optimal control problem which is affine with respect to the control. The terminal time is fixed, the terminal state is free, and the control values are restricted in a convex compact set U. The SMsR property is associated with a reference solution of the optimality system and ensures that small additive perturbations in the system result in solutions which are at distance to the reference one, at most proportional to the size of the perturbations. A general sufficient condition for SMsR is obtained for appropriate space settings and then specialized in the case of a polyhedral set U and purely bang-bang reference control. Sufficient second-order optimality conditions are obtained as a by-product of the analysis. Finally, the obtained results are utilized for error analysis of the Euler discretization scheme applied to affine problems.

Solving Large-Scale Cubic Regularization by a Generalized Eigenvalue Problem

Cubic regularization methods have several favorable properties. In particular under mild assumptions, they are globally convergent towards critical points with second-order necessary conditions sat...

Second-order slip condition considering Langmuir isothermal adsorption for rarefied gas microflows

Second-order slip condition considering Langmuir isothermal adsorption for rarefied gas microflows

Bounds on time-optimal concatenations of arcs for two-input driftless 3D systems

We study a driftless system on a three-dimensional manifold driven by two scalar controls. We assume that each scalar control has an independent bound on its modulus and we prove that, locally around every point where the controlled vector fields satisfy some suitable nondegeneracy Lie bracket condition, every time-optimal trajectory has at most five bang or singular arcs. The result is obtained using first- and second-order necessary conditions for optimality.

Optimal control theory and advanced optimality conditions for a diffuse interface model of tumor growth

We investigate a distributed optimal control problem for a diffuse interface model for tumor growth. The model consists of a Cahn–Hilliard type equation for the phase field variable, a reaction diffusion equation for the nutrient concentration and a Brinkman type equation for the velocity field. These PDEs are endowed with homogeneous Neumann boundary conditions for the phase field variable, the chemical potential and the nutrient as well as a “no-friction” boundary condition for the velocity. The control represents a medication by cytotoxic drugs and enters the phase field equation. The aim is to minimize a cost functional of standard tracking type that is designed to track the phase field variable during the time evolution and at some fixed final time. We show that our model satisfies the basics for calculus of variations and we present first-order and second-order conditions for local optimality. Moreover, we present a globality condition for critical controls and we show that the optimal control is unique on small time intervals.

Numerical Modelling of Microchannel Gas Flows in the Transition Flow Regime Using the Cascaded Lattice Boltzmann Method.

In this article, a lattice Boltzmann (LB) method for studying microchannel gas flows is developed in the framework of the cascaded collision operator. In the cascaded lattice Boltzmann (CLB) method, the Bosanquet-type effective viscosity is employed to capture the rarefaction effects, and the combined bounce-back/specular-reflection scheme together with the modified second-order slip boundary condition is adopted so as to match the Bosanquet-type effective viscosity. Numerical simulations of microchannel gas flow with periodic and pressure boundary conditions in the transition flow regime are carried out to validate the CLB method. The predicted results agree well with the analytical, numerical, and experimental data reported in the literature.

Analysis of vibration monitoring data of an onshore wind turbine under different operational conditions

A wind turbine structure works under the complicated environment and its operational state is complex. In this study, the operational state of a 1.5 MW wind turbine and the vibration response of the tower have been monitored for a long time. The vibration characteristic of the wind turbine under different operational conditions is discussed in detail, and it was observed that the rated rotation speed condition has the highest vibration level. The Campbell diagrams of wind turbine both in the FA direction and SS direction are drawn based on the natural frequencies results, which are identified by data-driven stochastic subspace identification method. It was observed from the Campbell diagrams that when the rotation speed is close to the grid-connected rotation speed, the blade-passing frequency 3f (0.43 Hz) is equal to the fundamental frequency of wind turbine which will easily lead to the resonance. This phenomenon is confirmed by the time history diagrams of the operational condition and vibration speed of a certain startup process. Lastly, the relationship between first-order, second-order modal parameters and operational conditions of the wind turbine are discussed in detail respectively. The mean value of first-order natural frequency is affected slightly by the different operational conditions while the value of second-order natural frequency increases significantly with the increase of rotation speed. The work in this paper can help us to better understand the response of the wind turbine under different operational conditions, which can also lay the foundation for the structure design, modal validation and damage diagnosis of wind turbine.

A dynamic apparent permeability model for shale microfractures: Coupling poromechanics, fluid dynamics, and sorption-induced strain

The widely distributed microfractures play an important role in shale gas production. However, limited studies focus on gas flow behavior in microfractures, and ignore the complex transport mechanisms, leading to a large error for gas permeability evaluation. In this work, a newly dynamic apparent permeability (AP) model, coupling poromechanics, sorption-induced strain, and gas slippage, has been proposed to effectively reveal the gas flow mechanisms through microfractures of shale. Specifically, a dynamic aperture is innovatively incorporated into the Navier-Stokes (N–S) equation using the second-order slip boundary condition to calculate the gas velocity and volume flux in single microfracture. Based on that, the gas transport model for microfracture networks considering the distributions of aperture and tortuosity is derived using the fractal theory. The newly developed model is verified well with experimental data and network simulation. Results indicate that the gas conductance highly depends on the structure of microfracture networks (i.e., the maximum aperture and fractal dimensions). There are three different AP evolutions under various boundary conditions (i.e., constant confining pressure (Pc), constant pore pressure (Pp), and constant effective stress (σeff)) resulting from the coupling transport mechanisms. The AP presents a similar shape of “V” at reservoir conditions (i.e., constant Pc ), indicating the “negative contribution” of poromechanics at an early stage, and the “positive contribution” for both gas slippage and sorption-induced strain at the late stage should be underlined during gas production. Moreover, the “negative factor” of poromechanics is positively correlated with fracture compressibility coefficient but negatively associated with Biot's coefficient at high pressures (>15 [MPa]). Increasing gas desorption capacity, fracture spacing, and internal swelling coefficient can enhance the “positive factor” of sorption-induced strain at low pressures (<15 [MPa]). This work provides a theoretical guidance to develop shale gas effectively.