A Fast Implicit Numerical Method for Time Dependent Viscous Flows

The iterative method used by Esch (1964) and Pearson (1964, 1965a, b), for the solution of an implicit finite difference approximation to the Navier Stokes equation, is analysed. A more general iteration method is suggested that may require many iteration parameters, and it is shown how these parameters can be computed. It was found that when the non-dimensional number vΔt/2L² is small, a single optimum iteration parameter exists (v being the kinematic viscosity, Δt the time step and L a characteristic length). An approximate expression for the “best” parameter is developed, and a procedure is described for improving that estimate. With the improved estimate and extrapolation in time, convergence is achieved in one or two iterations per time step on the average. In some cases the time step used was 200 times bigger than the time step required for stability of explicit schemes.     

求线性规划对偶问题最优解的一种方法

线性规划对偶问题的最优解有重要的经济意义,中给出了一种较为简捷的求对偶问题最优解的方法。     

The Solution of Time Dependent Problems by Galerkin Methods

Evolutionary problems involving parabolic and hyperbolic equationsin any number of space dimensions are formulated in such a waythat Galerkin methods, using a local basis, can be used to findapproximate solutions. These methods are particularly suitedto problems involving general regions and derivative boundaryconditions.     

Relaxation of Optimal Control Problems Involving Time Dependent Subdifferential Operators

This article provides in the setting of infinite dimensional Hilbert space, a result concerning the existence and uniqueness of solutions for Lipschitz single-valued perturbations of evolution problems associated with time-dependent subdifferential operators. The result is used to extend to optimal control problems associated with such equations the relaxation theorems with Young measures established in Casting et al. [7 C. Castaing , P. Raynaud de Fitte , and M. Valadier , M. ( 2004 ). Young Measures on Topological Spaces with Applications in Control Theory and Probability Theory . Kluwer Academic Publishers , Dordrecht . [Google Scholar], 11 J. F. Edmond and L. Thibault ( 2005 ). Relaxation of an optimal control problem involving a perturbed sweeping process . Math. Program Ser. B 104 : 347 – 373 .[Crossref], [Web of Science ®] , [Google Scholar]] and Edmond and Thibault [11 J. F. Edmond and L. Thibault ( 2005 ). Relaxation of an optimal control problem involving a perturbed sweeping process . Math. Program Ser. B 104 : 347 – 373 .[Crossref], [Web of Science ®] , [Google Scholar]].     

Solution of Time Dependent Electrochemical Machining Problems by a Co-ordinate Transformation

A numerical procedure for calculating the solution to two-dimensionaltime dependent electrochemical machining problems is derived.This method can be used in problems where the cathode containsinsulated portions. The electrostatic potential problem is solvedby transforming the variational integral to a domain where theboundaries become straight lines, and discretizing the resultantintegral. An implicit finite difference technique is used tointegrate the free surface equation. A variable spatial meshis incorporated into the scheme. The results are compared toa Kantorovich solution of the same problem.     

Singular Perturbation Solution of Time Dependent Mass-Transfer with Non-linear Chemical Reaction

In this work the author presents two analyses of time dependentprocesses in which diffusion and non-linear reaction take placesimultaneously. The first is an analysis of the distributionof a single reactant in a stagnant medium when the reactionrate is proportional to the concentration squared. In the secondanalysis the time dependent distributions of two reactants aretraced, when the reaction rate is porportional to the productof the concentrations. The processes under consideration areassumed to take place in unbounded containers and attentionis focused on the cases in which the initial concentrationshave non-zero space average. The analyses are carried out assumingthat the space variations of the initial distributions and thediffusivities are large while the reaction constant(s) is small.Therefore, initially the changes in the distribution due todiffusion are much faster than the rate of absorption in thereaction. However, eventually the processes become reactiondominated. The two distinct behaviours are represented by correspondingasymptotic expansions which are matched in the usual manner.On the other hand, if the initial average concentrations vanish,then, in as much as the processes are diffusion dominated initially,they are diffusion dominated thoughout. The distributions canthen be solved-for by regular perturbations. The treatment of the single component case shows that when theinitial average concentration does not vanish, the residualamount of reactant is inversely proportional to the time thatelapsed since the beginning of the process multiplied by thereaction constant. This residual amount is altogether independentof the bulk quantity supplied initially. The decay of the binaryprocess is similar provided the bulk quantities supplied initiallyare well proportioned. If there is a disproportion, the decayis exponential rather than algebraic. In all events, late inthe process the residual amount(s) is independent of the diffusivity.     

Optimal Pricing for Optimal Transport

Suppose that c(x, y) is the cost of transporting a unit of mass from x ∈ X to y ∈ Y and suppose that a mass distribution μ on X is transported optimally (so that the total cost of transportation is minimal) to the mass distribution ν on Y. Then, roughly speaking, the Kantorovich duality theorem asserts that there is a price f(x) for a unit of mass sold (say by the producer to the distributor) at x and a price g(y) for a unit of mass sold (say by the distributor to the end consumer) at y such that for any x ∈ X and y ∈ Y, the price difference g(y) ? f(x) is not greater than the cost of transportation c(x, y) and such that there is equality g(y) ? f(x) = c(x, y) if indeed a nonzero mass was transported (via the optimal transportation plan) from x to y. We consider the following optimal pricing problem: suppose that a new pricing policy is to be determined while keeping a part of the optimal transportation plan fixed and, in addition, some prices at the sources of this part are also kept fixed. From the producers’ side, what would then be the highest compatible pricing policy possible? From the consumers’ side, what would then be the lowest compatible pricing policy possible? We have recently introduced and studied settings in c-convexity theory which gave rise to families of c-convex c-antiderivatives, and, in particular, we established the existence of optimal c-convex c-antiderivatives and explicit constructions of these optimizers were presented. In applications, it has turned out that this is a unifying language for phenomena in analysis which used to be considered quite apart. In the present paper we employ optimal c-convex c-antiderivatives and conclude that these are natural solutions to the optimal pricing problems mentioned above. This type of problems drew attention in the past and existence results were previously established in the case where X = Y = ? ⁿ under various specifications. We solve the above problem for general spaces X, Y and real-valued, lower semicontinuous cost functions c. Furthermore, an explicit construction of solutions to the general problem is presented.     

Optimal Control Problems Governed by Time Dependent Subdifferential Operators with Delay

This article is devoted to the study, in the infinite dimensional setting, of a Bolza-type problem governed by a class of functional evolution inclusion which involves a time-dependent subdifferential operator with a time-delay perturbation. We present a relaxation result associated with such equations where the controls are Young measures in order to show the existence of an optimal solution under a suitable convexity assumption.     

A Generalized Conditional Gradient Method for Dynamic Inverse Problems with Optimal Transport Regularization

We develop a dynamic generalized conditional gradient method (DGCG) for dynamic inverse problems with optimal transport regularization. We consider the framework introduced in Bredies and Fanzon (ESAIM: M2AN 54:2351–2382, 2020), where the objective functional is comprised of a fidelity term, penalizing the pointwise in time discrepancy between the observation and the unknown in time-varying Hilbert spaces, and a regularizer keeping track of the dynamics, given by the Benamou–Brenier energy constrained via the homogeneous continuity equation. Employing the characterization of the extremal points of the Benamou–Brenier energy (Bredies et al. in Bull Lond Math Soc 53(5):1436–1452, 2021), we define the atoms of the problem as measures concentrated on absolutely continuous curves in the domain. We propose a dynamic generalization of a conditional gradient method that consists of iteratively adding suitably chosen atoms to the current sparse iterate, and subsequently optimizing the coefficients in the resulting linear combination. We prove that the method converges with a sublinear rate to a minimizer of the objective functional. Additionally, we propose heuristic strategies and acceleration steps that allow to implement the algorithm efficiently. Finally, we provide numerical examples that demonstrate the effectiveness of our algorithm and model in reconstructing heavily undersampled dynamic data, together with the presence of noise.

     

Necessary Conditions for Free End-Time, Measurably Time Dependent Optimal Control Problems with State Constraints

Recently, necessary conditions have been derived for fixed-time optimal control problems with state constraints, formulated in terms of a differential inclusion, under very weak hypotheses on the data. These allow the multifunction describing admissible velocities to be unbounded and possibly nonconvex valued. This paper extends the earlier necessary conditions, to allow for free end-times. A notable feature of the new free end-time necessary conditions is that they cover problems with measurably time dependent data. For such problems, standard analytical techniques for deriving free-time necessary conditions, which depend on a transformation of the time variable, no longer work. Instead, we use variational methods based on the calculus of 'essential values".     

Solution Extensions: Optimal Parameter Method in Optimal Control Problems

Using the extension of solution by the optimal parameter method, we obtain a numerical solution for a certain class of optimal control problems.     

Heuristic Solution Methods for the Multilevel Generalized Assignment Problem

The multilevel generalized assignment problem is a problem of assigning agents to tasks where the agents can perform tasks at more than one efficiency level. A profit is associated with each assignment and the objective of the problem is profit maximization. Two heuristic solution methods are presented for the problem. The heuristics are developed from solution methods for the generalized assignment problem. One method uses a regret minimization approach whilst the other method uses a repair approach on a relaxation of the problem. The heuristics are able to solve moderately large instances of the problem rapidly and effectively. Procedures for deriving an upper bound on the solution of the problem are also described. On larger and harder instances of the problem one heuristic is particularly effective.     

An Explicit-Implicit Predictor-Corrector Domain Decomposition Method for Time Dependent Multi-Dimensional Convection Diffusion Equations

The numerical solution of large scale multi-dimensional convection diffusion equations often requires efficient parallel algorithms. In this work, we consider the extension of a recently proposed non-overlapping domain decomposition method for two dimensional time dependent convection diffusion equations with variable coefficients. By combining predictor-corrector technique, modified upwind differences with explicitimplicit coupling, the method under consideration provides intrinsic parallelism while maintaining good stability and accuracy. Moreover, for multi-dimensional problems, the method can be readily implemented on a multi-processor system and does not have the limitation on the choice of subdomains required by some other similar predictorcorrector or stabilized schemes. These properties of the method are demonstrated in this work through both rigorous mathematical analysis and numerical experiments.     

一类三阶变系数线性系统的解

在文［１］的基础上，本文给出一类三阶变系数线性系统的解的求法及解的表达式．     

Direct Shooting Method for the Numerical Solution of Higher-Index DAE Optimal Control Problems

A method for the numerical solution of state-constrained optimal control problems subject to higher-index differential-algebraic equation (DAE) systems is introduced. For a broad and important class of DAE systems (semiexplicit systems with algebraic variables of different index), a direct multiple shooting method is developed. The multiple shooting method is based on the discretization of the optimal control problem and its transformation into a finite-dimensional nonlinear programming problem (NLP). Special attention is turned to the mandatory calculation of consistent initial values at the multiple shooting nodes within the iterative solution process of (NLP). Two different methods are proposed. The projection method guarantees consistency within each iteration, whereas the relaxation method achieves consistency only at an optimal solution. An illustrative example completes this article.     

A Reduced Complexity Min-Plus Solution Method to the Optimal Control of Closed Quantum Systems

The process of obtaining solutions to optimal control problems via mesh based techniques suffers from the well known curse of dimensionality. This issue is especially severe in quantum systems whose dimensions grow exponentially with the number of interacting elements (qubits) that they contain. In this article we develop a min-plus curse-of-dimensionality-free framework suitable to a new class of problems that arise in the control of certain quantum systems. This method yields a much more manageable complexity growth that is related to the cardinality of the control set. The growth is attenuated through \(\max \) -plus projection at each propagation step. The method’s efficacy is demonstrated by obtaining an approximate solution to a previously intractable problem on a two qubit system.     

A Remark on the Potentials of Optimal Transport Maps

Optimal maps, solutions to the optimal transportation problems, are completely determined by the corresponding c-convex potential functions. In this paper, we give simple sufficient conditions for a smooth function to be c-convex when the cost is given by minimizing a Lagrangian action.     

On a Strong Solution in the Method of Spherical Harmonics for a Nonstationary Transport Equation

We prove existence of a strong generalized solution in the Sobolev space W ₂ ¹ to the nonstationary problem for the system of the method of spherical harmonics (MSH) corresponding to the radiation transport problem.     

Optimal Transport for the System of Isentropic Euler Equations

We introduce a new variational time discretization for the system of isentropic Euler equations. In each timestep the internal energy is reduced as much as possible, subject to a constraint imposed by a new cost functional that measures the deviation of particles from their characteristic paths.