Pointwise Constraints Research Articles

Locally Lipschitz Stability of Solutions to a Parametric Parabolic Optimal Control Problem with Mixed Pointwise Constraints

Locally Lipschitz Stability of Solutions to a Parametric Parabolic Optimal Control Problem with Mixed Pointwise Constraints

Semi-supervised pivotal-aware nonnegative matrix factorization with label and pairwise constraint propagation for data clustering

Semi-supervised nonnegative matrix factorization (NMF) methods have found extensive utility in data clustering applications. However, these existing methods encounter challenges in effectively leveraging the limited supervisory information to enhance the performance of clustering. In particular, the majority of these methods tend to solely utilize either the label information or the pairwise constraint information, neglecting the potential benefits of their joint utilization. To address this issue, a novel algorithm named semi-supervised pivotal-aware nonnegative matrix factorization (SPNMF) is proposed in this paper, which utilizes the label information and the pairwise constraint information simultaneously to enhance the performance in data clustering tasks. The proposed method has two main innovations: (1) adopting the dual constraint propagation (DCP) algorithm (i.e., label propagation and pairwise constraint propagation) to effectively utilize a meager amount of label information for learning a compact data representation; (2) incorporating the pivotal-aware technique to mitigate the negative impact of outliers. Notably, the DCP algorithm not only propagates limited label information to a multitude of unlabeled samples, but also disseminates the pairwise constraint supervisory information obtained from the labels to unconstrained samples, allowing for the acquisition of richer supervisory information in the form of pointwise and pairwise constraints. Furthermore, a comprehensive analysis of SPNMF is conducted. Extensive experimental results are executed on eight real-world image datasets. The results underscore the superior performance of SPNMF in comparison to several state-of-the-art methods.

Mass, momentum and energy preserving FEEC and broken-FEEC schemes for the incompressible Navier–Stokes equations

Abstract In this article we propose two finite-element schemes for the Navier–Stokes equations, based on a reformulation that involves differential operators from the de Rham sequence and an advection operator with explicit skew-symmetry in weak form. Our first scheme is obtained by discretizing this formulation with conforming FEEC (Finite Element Exterior Calculus) spaces: it preserves the point-wise divergence free constraint of the velocity, its total momentum and its energy, in addition to being pressure robust. Following the broken-FEEC approach, our second scheme uses fully discontinuous spaces and local conforming projections to define the discrete differential operators. It preserves the same invariants up to a dissipation of energy to stabilize numerical discontinuities. For both schemes we use a middle point time discretization that preserve these invariants at the fully discrete level and we analyze its well-posedness in terms of a CFL condition. While our theoretical results hold for general finite elements preserving the de Rham structure, we consider one application to tensor-product spline spaces. Specifically, we conduct several numerical test cases to verify the high order accuracy of the resulting numerical methods, as well as their ability to handle general boundary conditions.

Optimal control of variably distributed‐order time‐fractional diffusion equation: Analysis and computation

AbstractFractional diffusion equations exhibit competitive capabilities in modeling many challenging phenomena such as the anomalously diffusive transport and memory effects. We prove the well‐posedness and regularity of an optimal control of a variably distributed‐order fractional diffusion equation with pointwise constraints, where the distributed‐order operator accounts for, for example, the effect of uncertainties. We accordingly develop and analyze a fully‐discretized finite element approximation to the optimal control without any artificial regularity assumption of the true solution. Numerical experiments are also performed to substantiate the theoretical findings.

A bi-variant variational model for diffeomorphic image registration with relaxed Jacobian determinant constraints

Diffeomorphic registration is a widely used technique for finding a smooth and invertible transformation between two coordinate systems, which are measured using template and reference images. The point-wise volume-preserving constraint det⁡(∇φ(x))=1 is effective in some cases, but may be too restrictive in others, especially when local deformations are relatively large. This can result in poor matching when enforcing large local deformations. In this paper, we propose a new bi-variant diffeomorphic image registration model that introduces a soft constraint on the Jacobian equation det⁡(∇φ(x))=f(x)>0. This allows local deformations to shrink and grow within a flexible range 0<κm<det⁡(∇φ(x))<κM. The Jacobian determinant of transformation is explicitly controlled by optimizing the relaxation function f(x). To prevent deformation folding and improve the smoothness of the transformation, a positive constraint is imposed on the optimization of the relaxation function f(x), and a regularizer is used to ensure the smoothness of f(x). Furthermore, the positivity constraint ensures that f(x) is as close to one as possible, which helps to achieve a volume-preserving transformation on average. We also analyze the existence of the minimizer for the variational model and propose a penalty-splitting algorithm with a multilevel strategy to solve this model. Numerical experiments demonstrate the convergence of the proposed algorithm and show that the positivity constraint can effectively control the range of relative volume without compromising the accuracy of the registration. Moreover, the proposed model generates diffeomorphic maps for large local deformations and outperforms several existing registration models in terms of performance.

Integral constraint regularization method for fractional optimal control problem with pointwise state constraint

This paper investigates integral constraint regularization approximation of fractional optimal control problem with pointwise state constraint. Firstly the continuous first-order optimality condition is derived. In order to deal with the irregular multiplier, a regularized optimal control problem with finitely many integral state constraints is proposed based on integral constraint regularization method. The error between the original solution and the regularized solution is deduced. Continuous piecewise linear finite element combined with variational discretization approach is used to discretize the regularized optimal control problem. Finite element discretization error combined with the regularization error leads to the final error estimate. Finally, numerical examples are presented to verify the theoretical findings.

On the Use of Ellipsoidal Estimation Techniques in the RRT* Suboptimal Pathfinding Algorithm

The article is devoted to the development of an algorithm for the approximate solution of the time-optimal control problem for a system of ordinary differential equations, under the condition of avoiding collisions with stationary obstacles and subject to the specified pointwise constraints on the possible values of the control parameters. The main idea is to use a modification of the algorithm for finding suboptimal paths using rapidly growing random trees (RRT*). The most difficult part of this algorithm is to find the optimal trajectories for the problems of transferring the system from one fixed position to another, close to it, without taking into account state constraints. This subproblem is proposed to be solved using the methods of ellipsoidal calculus. This approach makes it possible to efficiently search suboptimal trajectories both for linear systems with large state space dimension and for systems with nonlinear dynamics. Algorithms for the linear and non-linear cases are sequentially analyzed in the paper, and the corresponding examples of calculations are presented.

C0 interior penalty methods for an elliptic distributed optimal control problem with general tracking and pointwise state constraints

We consider C0 interior penalty methods for a linear-quadratic elliptic distributed optimal control problem with pointwise state constraints in two spatial dimensions, where the cost function tracks the state at points, curves and regions of the domain. Error estimates and numerical results that illustrate the performance of the methods are presented.

Control of the von Neumann Entropy for an Open Two-Qubit System Using Coherent and Incoherent Drives.

This article is devoted to developing an approach for manipulating the von Neumann entropy S(ρ(t)) of an open two-qubit system with coherent control and incoherent control inducing time-dependent decoherence rates. The following goals are considered: (a) minimizing or maximizing the final entropy S(ρ(T)); (b) steering S(ρ(T)) to a given target value; (c) steering S(ρ(T)) to a target value and satisfying the pointwise state constraint S(ρ(t))≤S¯ for a given S¯; (d) keeping S(ρ(t)) constant at a given time interval. Under the Markovian dynamics determined by a Gorini-Kossakowski-Sudarshan-Lindblad type master equation, which contains coherent and incoherent controls, one- and two-step gradient projection methods and genetic algorithm have been adapted, taking into account the specifics of the objective functionals. The corresponding numerical results are provided and discussed.

On Solvability Conditions for the Cauchy Problem for Non-Volterra Functional Differential Equations with Pointwise and Integral Restrictions on Functional Operators

Cauchy problems are considered for families of, generally speaking, non-Volterra functional differential equations of the second order. For each family considered, in terms of the parameters of this family, necessary and sufficient conditions for the unique solvability of the Cauchy problem for all equations of the family are obtained. Such necessary and sufficient conditions are obtained for the following four kinds of families: integral restrictions are imposed on positive and negative functional operators, namely, operator norms are specified; pointwise restrictions are imposed on positive and negative functional operators in the form of values of operators’ actions on the unit function; an integral constraint is imposed on a positive functional operator, a pointwise constraint is imposed on a negative functional operator; a pointwise constraint is imposed on a positive functional operator, an integral constraint is imposed on a negative functional operator. In all cases, effective conditions for the solvability of the Cauchy problem for all equations of the family are obtained, expressed through some inequalities regarding the parameters of the families. The set of parameters of families of equations for which Cauchy problems are uniquely solvable can be easily calculated approximately with any accuracy. The resulting solvability conditions improve the solvability conditions following from the Banach contraction principle. An example of the Cauchy problem for an equation with a coefficient changing sign is given. Taking into account various restrictions for the positive and negative parts of functional operators allows us to significantly improve the known solvability conditions.

On a Problem of Calculating the Solvability Set for a Linear System with Uncertainty

We consider a linear-convex control system defined by a set of differential equations with continuous matrix coefficients. The system may have control parameters, as well as uncertainties (interference) the possible values of which are subject to strict pointwise constraints. For this system, over a finite period of time, taking into account the constraints, we study the problem of guaranteed hitting the target set from a given initial position despite the effect of uncertainty. The main stage of solving the problem is the construction of an alternating integral and a solvability set. To construct the latter, the greatest computational complexity is the calculation of the geometric difference between the target set and the set determined by the uncertainty. A two-dimensional example of this problem is considered for which a method is proposed for finding the solvability set without the need to calculate the convex hull of the difference of the support functions of the sets.

A Comparative Analysis of the Response-Tracking Techniques in Aerospace Dynamic Systems Using Modal Participation Factors

Mechanical structural systems are subject to multiple dynamic disturbances during service. While several possible scenarios can be examined to determine their design loading conditions, only a relatively small set of such scenarios is considered critical. Therefore, only such particular deterministic set of critical load cases is commonly employed for the structural design and optimization. Nevertheless, during the design and optimization stages, the mass and stiffness distributions of such assemblies vary, and, in consequence, their dynamic response also varies. Thus, it is important to consider the variations in the dynamic loading conditions during the design-and-optimization cycles. This paper studies the modal participation factors at length and proposes an alternative to the current point-wise treatment of the dynamic equations of motion of flexible bodies during design optimization. First, the most relevant-to-structural-dynamics definitions available in the literature are reviewed in depth. Second, the analysis of those definitions that have the potential to be adopted as point-wise constraint equations during structural optimization is extended. Finally, a proof of concept is presented to demonstrate the usability of each definition, followed by a case study in which the potential advantages of the proposed extended analysis are shown.

Discontinuous Galerkin methods for an elliptic optimal control problem with a general state equation and pointwise state constraints

We investigate discontinuous Galerkin methods for an elliptic optimal control problem with a general state equation and pointwise state constraints on general polygonal domains. We show that discontinuous Galerkin methods for general second-order elliptic boundary value problems can be used to solve the elliptic optimal control problems with pointwise state constraints. We establish concrete error estimates and numerical experiments are shown to support the theoretical results.

Numerical study of the coalescence-induced droplet jumping with macrotexture based on single-phase model

The low energy conversion efficiency in coalescence-induced droplet jumping limits its potential for various applications, such as self-cleaning, anti-icing, and energy harvesting. Fortunately, it has been proven that this efficiency can be significantly increased through a sophisticated macrotexture design. In this study, we propose a single-phase model with a moving mesh to simulate the self-jumping process under a ridge. The effect of the ridge is realized by adopting a pointwise constraint on several surface nodes. This effective model is validated by experimental results of droplet velocity. In comparison with volume-of-fluid, a single-phase flow method enhances computational efficiency by at least 33.3%. The kinematics and dynamics of the self-jumping process have been investigated with respect to the influences of ridge height and Ohnesorge number. With the help of the radial distributions of velocity and internal pressure, the self-propelled process can be divided into coalescence-induced and lobe-induced stages. The high ridge brings more symmetry-breaking, accelerating the droplet in the coalescence-induced stage. In the lobe-induced stage, the slingshot effect is weakened under high Ohnesorge number due to the prolate shape caused by viscous dissipation. Moreover, the study's findings demonstrate promising application prospects for other ridge shapes, thereby expanding the potential practical applications of this research.

Locally Hölder Continuity of the Solution Map to a Boundary Control Problem with Finite Mixed Control-State Constraints

The local stability of the solution map to a parametric boundary control problem governed by semilinear elliptic equations with finite mixed pointwise constraints is considered in this paper. We prove that the solution map is locally Hölder continuous in -norm of control variable when the strictly nonnegative second-order optimality conditions are satisfied for the unperturbed problem.

Integral Identity and Estimate of the Deviation of Approximate Solutions of a Biharmonic Obstacle Problem

We show that the integral identity obtained by D.E. Apushkinskaya and S.I. Repin (2020) for approximate solutions of the biharmonic obstacle problem that satisfy a pointwise constraint on the second divergence is valid for arbitrary approximate solutions. Using this result, we obtain a new estimate for the deviation of approximate solutions from exact ones in the case when the approximate solutions do not satisfy the pointwise constraint on the second divergence.

A relaxation-based probabilistic approach for PDE-constrained optimization under uncertainty with pointwise state constraints

A relaxation-based probabilistic approach for PDE-constrained optimization under uncertainty with pointwise state constraints

A Symmetric Interior Penalty Method for an Elliptic Distributed Optimal Control Problem with Pointwise State Constraints

Abstract We construct a symmetric interior penalty method for an elliptic distributed optimal control problem with pointwise state constraints on general polygonal domains. The resulting discrete problems are quadratic programs with simple box constraints that can be solved efficiently by a primal-dual active set algorithm. Both theoretical analysis and corroborating numerical results are presented.

Multigrid methods for an elliptic optimal control problem with pointwise state constraints

We design multigrid methods for an elliptic distributed optimal control problem with pointwise state constraints. They are based on the P1 finite element method from Brenner et al. (2021), where the optimal control problem was reformulated as a variational inequality for the state variable. The performance of the multigrid methods is illustrated by numerical experiments.

ON THE BANG-BANG PRINCIPLE FOR PARABOLIC OPTIMAL CONTROL PROBLEMS

Optimal control problems for the linear heat equation with final observation and pointwise constraints on the control are considered, where the control depends only on the time. It is shown that to each finite number of given switching points, there is a final target such that the optimal objective value is positive, the optimal control is bang bang, and has the desired switching structure. The theory is completed by numerical examples.