Mixed Finite Element Approximation of Periodic Hamilton--Jacobi--Bellman Problems With Application to Numerical Homogenization

We theoretically study the dynamics and resonance shift of the Rabi model with frequency modulation, i.e., the Rabi model driven additionally by a slow longitudinal field, by using the counterrotating-hybridized rotating-wave (CHRW) method, aiming to illustrate the effects of the counterrotating (CR) terms of the transverse field. The CHRW method is based on a unitary transformation and reduces the aperiodic Hamiltonian to an effective periodic Hamiltonian that can be efficiently treated by Floquet theory. The validity of the effective Hamiltonian and widely used rotating-wave approximation (RWA) Hamiltonian is carefully examined compared to the numerically exact results over a wide parameter range. It is found that the effective Hamiltonian gives a correct description, while the RWA breaks down in the strong driving regime. Interestingly, we show that under certain conditions the longitudinal field can be used to modify resonance widths such that resonance widths can be comparable to the magnitude of the Bloch-Siegert (BS) shift, which in turn makes the CR-induced BS shift significant and leads to the complete breakdown of the RWA even in a moderately strong driving regime (in which the RWA holds for the Rabi model without frequency modulation). In addition, by using the effective Hamiltonian, we can efficiently access resonance positions for the bichromatically driven qubit and study how the resonance shifts due to the combined effects of the CR terms and frequency modulation. For a weak longitudinal field, we show that resonance positions can be analytically calculated from the effective Rabi frequency for the effective Hamiltonian, which are in excellent agreement with the numerically exact results.

We obtain error bounds for monotone approximation schemes of Hamilton--Jacobi--Bellman equations. These bounds improve previous results of Krylov and the authors. The key step in the proof of these new estimates is the introduction of a switching system which allows the construction of approximate, (almost) smooth supersolutions for the Hamilton--Jacobi--Bellman equation.

Effective Hamiltonians governing the time evolution in a subspace of unstable states can be found using more or less accurate approximations. A convenient tool for deriving them is the evolution equation for a subspace of state space sometime called the Królikowski–Rzewuski (KR) equation. KR equation results from the Schrödinger equation for the total system under considerations. We will discuss properties of approximate effective Hamiltonians derived using KR equation for n-particle, two-particle and for one-particle subspaces. In a general case these effective Hamiltonians depend on time t. We show that at times much longer than times at which the exponential decay take place the real part of the exact effective Hamiltonian for the one-particle subsystem (that is the instantaneous energy) tends to the minimal energy of the total system when t → ∞ whereas the imaginary part of this effective Hamiltonian tends to zero as t → ∞.

This paper presents a progressive type of variational principle for topological shape optimization. The principle makes it possible to derive a finite element method (FEM) for searching structural shapes with minimal weight and high strength in the optimization operation. The principle is based on that of minimum potential energy. In the formulation of this principle, we first assume both a search domain and an optimum one. The search domain includes the optimum one. Then, the search domain is locally eliminated according to the information of minimum strain energy in the domain. When a maximum equivalent stress in the search domain reaches the yield stress as a constraint condition, it is evaluated that the search domain agrees with the optimum one. Therefore, we obtain the variational principle for the topological optimization. Next, we derive the FEM based on this principle. Finally, we obtain the optimum topological shapes of two-dimensional finite element models using the FEM. Hence, the validity of the variational principle is confirmed from the numerical solutions of the models.

Efficient solver for mixed finite element method and control-volume mixed finite element method in 3-D on Hexahedral grids

Numerical analysis of a free-boundary singular control problem in financial economics

A procedure for the numerical approximation of high-dimensional Hamilton-Jacobi-Bellman (HJB) equations associated to optimal feedback control problems for semilinear parabolic equations is proposed. Its main ingredients are a pseudospectral collocation approximation of the PDE dynamics, and an iterative method for the nonlinear HJB equation associated to the feedback synthesis. The latter is known as the Successive Galerkin Approximation. It can also be interpreted as Newton iteration for the HJB equation. At every step, the associated linear Generalized HJB equation is approximated via a separable polynomial approximation ansatz. Stabilizing feedback controls are obtained from solutions to the HJB equations for systems of dimension up to fourteen.

The phonon-assisted process of energy transfer aiming at exploring the newly emerging frontier between biology and physics is an issue of central interest. This article shows the important role of the intramolecular vibrational modes for excitation energy transfer in the photosynthetic systems. Based on a dimer system consisting of a donor and an acceptor modeled by two two-level systems, in which one of them is coupled to a high-energy vibrational mode, we derive an effective Hamiltonian describing the vibration-assisted coherent energy transfer process in the polaron frame. The effective Hamiltonian reveals in the case that the vibrational mode dynamically matches the energy detuning between the donor and the acceptor, the original detuned energy transfer becomes resonant energy transfer. In addition, the population dynamics and coherence dynamics of the dimer system with and without vibration-assistance are investigated numerically. It is found that, the energy transfer efficiency and the transfer time depend heavily on the interaction strength of the donor and the high-energy vibrational mode, as well as the vibrational frequency. The numerical results also indicate that the initial state and dissipation rate of the vibrational mode have little influence on the dynamics of the dimer system. Results obtained in this article are not only helpful to understand the natural photosynthesis, but also offer an optimal design principle for artificial photosynthesis.

In this study, an optimal adaptive control approach is established to solve the robust output tracking problem of a class of continuous time uncertain linear systems based on the policy iteration (PI) in actor-critic algorithm. First, by augmenting the integral variables of the tracking error into state variables, the robust tracking problem is transformed into a robust control problem of an augmented uncertain linear system. It is proven that the robust control law of the augmented system enables the output of the considered system to track a polynomial time signal asymptotically. Second, an optimal control method in the corresponding auxiliary nominal system is established, and based on the Bellman optimality principle, PI algorithms are proposed to solve online tracking controllers for the matched and the mismatched uncertain systems. Finally, for testing the availability of the proposed approach and theoretical results, two numerical experiments are provided.

The energy convergence of stress singular problems using the mixed finite element method is established. A mixed finite element method for determination of elastic crack tip stress intensity factor by energy release rate is also presented. It is shown through numerical examples that the mixed method has excellent potential for approximate analysis of stress singular problems.

Geometrically complex geological features, such as faults, cross stratified beds and large scale inclined strata, can have very significant effects on the flow of reservoir fluids. Standard finite difference methodology lacks the geometric flexibility required for modeling flow through or around such structures. Triangle based discretization methods, by contrast, offer an attractive means to discretize and simulate flow through geometrically complex features accurately. In this paper, a triangle based mixed finite element – finite volume technique for compositional reservoir simulation is developed. The formulation is of IMPES type and is applicable to three phase, multicomponent systems in two dimensions. Most previous mixed finite element formulations were restricted to two phase or fully miscible systems. The compositional approach presented here extends these earlier techniques though it retains the general structure of the more limited formulations. Specifically, the governing equations are cast in terms of a pressure equation and a series of component mass balance equations expressed in terms of the ‘total’ velocity. A triangle based mixed finite element method is applied to the solution of the pressure equation while a finite volume technique is employed for solution of the component mass balance equations. The overall methodology is particularly well suited for the modeling of flow through complex geological strata characterized by heterogeneous, generally anisotropic permeability fields. This is demonstrated explicitly through application of a preliminary implementation of the method to the simulation of flow through a faulted cross section with a complex permeability description. Extension of the formulation to the three dimensional case is also discussed.

Consider a model of a flexible rnanufacturing system with failure-prone machines. The control variables are the rates of maintenance, repair, and production. The objective is to choose these control variables over time that minimize the total cost of over/under stock, repair, and maintenance with either discounted cost or average cost per unit time. The focus of this work is on the modeling and numerical methods for a wear process formulation. A Markov chain approach is used to deal with the approximation problem. The dynamic programming equation is used only for suggesting a good numerical approximation scheme. The convergence result is obtained by deriving the weak convergence of the associated Markov chain. A one machine one part type model is given as an illustration for the numerical experiment

Certain finite difference methods on rectangular grids for second-order elliptic equations are known to yield superconvergent flux approximations. A class of related finite difference methods recently have been defined for triangular meshes by applying special quadrature rules to an extended version of a mixed finite element method by Arbogast, Dawson, and Keenan [Mixed Finite Element Methods as Finite Difference Methods for Solving Elliptic Equations on Triangular Elements, Tech. report 93-53, Dept. of Computational and Applied Mathematics, Rice University, Houston, TX, 1993; in Computational Methods in Water Resources, Kluwer Academic Publishers, Norwell, MA, 1994]; the usual hybrid mixed method can also be applied to meshes of triangular and tetrahedral elements. Unfortunately, the flux vectors from these methods are only first-order accurate. Empirical evidence indicates that a local postprocessing technique described by Keenan in [An Efficient Postprocessor for Velocities from Mixed Methods on Triangular Elements, Dept. of Computational and Applied Mathematics, Tech. report 94-22, Rice University, Houston, TX, 1994] recovers second-order accurate velocities at special points. In this paper, a class of local postprocessing techniques generalizing the one in [An Efficient Postprocessor for Velocities from Mixed Methods on Triangular Elements, Tech. report 94-22, Rice University, Houston, TX, 1994] are presented and analyzed. These postprocessors are shown to recover second-order accurate velocity fields on three lines meshes. Numerical experiments illustrate these results and investigate more general situations, including meshes of tetrahedral elements.

This article presents a novel dynamic programming approach to determine the robust controllability of Boolean control networks (BCNs) subject to stochastic disturbances. By applying Bellman's optimality principle, we derive the recurrence relation for computing the optimal time matrix, a crucial concept characterizing robust reachability between two arbitrary states. We develop a finite-termination dynamic programming algorithm to calculate the optimal time matrix exactly and efficiently, with a rigorously certified iteration count. Sufficient and necessary conditions for robust controllability are then established based on the optimal time matrix. Furthermore, for any pair of reachable states, we construct time-optimal state feedback control laws to steer the system from the initial state to the target state, regardless of disturbances. Finally, extensive numerical experiments with biological networks validate the effectiveness of the proposed approach, showing significant improvements in computational efficiency. Additionally, we introduce a Q-learning-based algorithm and compare its performance, highlighting the advantages of our dynamic programming approach in terms of both efficiency and solution quality.

Transportation network design for maximizing flow-based accessibility