Unbounded Case Research Articles

Optimal Primal-Dual Methods for a Class of Saddle Point Problems

We present a novel accelerated primal-dual (APD) method for solving a class of deterministic and stochastic saddle point problems (SPP). The basic idea of this algorithm is to incorporate a multi-step acceleration scheme into the primal-dual method without smoothing the objective function. For deterministic SPP, the APD method achieves the same optimal rate of convergence as Nesterov's smoothing technique. Our stochastic APD method exhibits an optimal rate of convergence for stochastic SPP not only in terms of its dependence on the number of the iteration, but also on a variety of problem parameters. To the best of our knowledge, this is the first time that such an optimal algorithm has been developed for stochastic SPP in the literature. Furthermore, for both deterministic and stochastic SPP, the developed APD algorithms can deal with the situation when the feasible region is unbounded, as long as a saddle point exists. In the unbounded case, we incorporate the modified termination criterion introduced by Monteiro and Svaiter in solving SPP problem posed as monotone inclusion, and demonstrate that the rate of convergence of the APD method depends on the distance from the initial point to the set of optimal solutions.

Core Spreading Vortex Method for Simulating 3D Flow around Bluff Bodies

This paper presents the development of core spreading vortex element method, which is a mesh-free method, for simulating 3D viscous flow over bluff bodies. The developed method simulates external flow around complex geometry by tracking local velocities and vorticities of particles introduced within the fluid domain. The viscous effect is modeled using core spreading method coupled with the splitting spatial adaption scheme, and a smoothing interpolation scheme for overlapping issue and population control, respectively. The particle’s velocity is calculated using Biot-Savart formulation. To accelerate computation, Fast Multipole Method (FMM) is employed. The solver is validated, for both unbounded and bounded flows at low Reynolds numbers, using a number of benchmark problems. For unbounded case, simulation of the collision of two vortex rings was performed. To test the performance of the method in simulating bounded flow problem, simulation of flow around a sphere was carried out. The results are found to be in good agreement with those reported in literatures and also simulations using other diffusion model.

Computing with infinitely many processes

We explore four classic problems in concurrent computing (election, mutual exclusion, consensus, and naming) when the number of processes which may participate is unbounded. Partial information about the number of processes actually participating and the concurrency level is shown to affect the computability and complexity of solving these problems when using only atomic registers. We survey and generalize work carried out in models with known bounds on the number of processes, and prove several new results. These include improved bounds for election when participation is required and a new adaptive starvation-free mutual exclusion algorithm for unbounded concurrency. We also survey results in models with shared objects stronger than atomic registers, such as test&set bits, semaphores or read–modify–write registers, and update them for the unbounded case.

Large deviation principle of stochastic differential equations with non-Lipschitzian coefficients

We study the large deviation principle of stochastic differential equations with non-Lipschitzian and non-homogeneous coefficients. We consider at first the large deviation principle when the coefficients σ and b are bounded, then we generalize the conclusion to unbounded case by using bounded approximation program. Our results are generalization of S. Fang-T. Zhang’s results.

Analytical formulation of 2-D aeroelastic model in weak ground effect

This paper deals with the aeroelastic modeling and analysis of a 2-D oscillating airfoil in ground effect, elastically constrained by linear and torsional springs and immersed in an incompressible potential flow (typical section) at a finite distance from the ground. This work aims to extend Theodorsen theory, valid in an unbounded flow domain, to the case of weak ground effect, i.e., for clearances above half the airfoil chord. The key point is the determination of the aerodynamic loads, first in the frequency domain and then in the time domain, accounting for their dependence on the ground distance. The method of images is exploited in order to comply with the impermeability condition on the ground. The new integral equation in the unknown vortex distribution along the chord and the wake is solved using asymptotic expansions in the perturbation parameter defined as the inverse of the non-dimensional ground clearance of the airfoil. The mathematical model describing the aeroelastic system is transformed from the frequency domain into the time domain and then in a pure differential form using a finite-state aerodynamic approximation (augmented states). The typical section, which the developed theory is applied to, is obtained as a reduced model of a wing box finite element representation, thus allowing comparison with the corresponding aeroelastic analysis carried out by a commercial solver based on a 3-D lifting surface aerodynamic model. Stability (flutter margins) and response of the airfoil both in frequency and time domains are then investigated. In particular, within the developed theory, the solution of the Wagner problem can be directly achieved confirming an asymptotic trend of the aerodynamic coefficients toward the steady-state conditions different from that relative to the unbounded domain case. The dependence of flutter speed and the frequency response functions on ground clearance is highlighted, showing the usefulness of this approach in efficiently and robustly accounting for the presence of the ground when unsteady analysis of elastic lifting surfaces in weak ground effect is required.

Stabilization of positive Takagi–Sugeno fuzzy discrete-time systems with multiple delays and bounded controls

This paper deals with the problem of stabilization by state feedback control of Takagi–Sugeno (T–S) fuzzy discrete-time systems with multiple fixed delays while imposing positivity in closed-loop. The obtained results are presented under linear programming (LP) form. In particular, the synthesis of state feedback controllers is first solved in terms of Linear programming for the unbounded controls case. This result is then extended to the stabilization problem by nonnegative controls, and stabilization by bounded controls. The stabilization conditions are derived using the single Lyapunov–Krasovskii functional (LKF). An example of a real plant is studied to show the advantages of the design procedure. A comparison between linear programming and LMI approaches is presented.

The Bernstein–Orlicz norm and deviation inequalities

We introduce two new concepts designed for the study of empirical processes. First, we introduce a new Orlicz norm which we call the Bernstein–Orlicz norm. This new norm interpolates sub-Gaussian and sub-exponential tail behavior. In particular, we show how this norm can be used to simplify the derivation of deviation inequalities for suprema of collections of random variables. Secondly, we introduce chaining and generic chaining along a tree. These simplify the well-known concepts of chaining and generic chaining. The supremum of the empirical process is then studied as a special case. We show that chaining along a tree can be done using entropy with bracketing. Finally, we establish a deviation inequality for the empirical process for the unbounded case.

Convergence of a misanthrope process to the entropy solution of 1D problems

We prove the convergence, in some strong sense, of a Markov process called “a misanthrope process” to the entropy weak solution of a one-dimensional scalar nonlinear hyperbolic equation. Such a process may be used for the simulation of traffic flows. The convergence proof relies on the uniqueness of entropy Young measure solutions to the nonlinear hyperbolic equation, which holds for both the bounded and the unbounded cases. In the unbounded case, we also prove an error estimate. Finally, numerical results show how this convergence result may be understood in practical cases.

Koplienko Trace Formula

Koplienko (Sib Math J 25(5): 735–743, 1984) gave a trace formula for perturbations of self-adjoint operators by operators of Hilbert–Schmidt class \({\mathcal{B}_2(\mathcal{H})}\). Recently Gesztesy et al. (Basics Z Mat Fiz Anal Geom 4(1):63–107, 2008) gave an alternative proof of the trace formula when the operators involved are bounded. In this article, we give a still another proof and extend the formula for unbounded case by reducing the problem to a finite dimensional one as in the proof of Krein trace formula by Voiculescu (On a Trace Formula of M. G. Krein. Operator Theory: Advances and Applications, vol. 24, pp. 329–332. Birkhauser, Basel, 1987), Sinha and Mohapatra (Proc Indian Acad Sci (Math Sci) 104(4):819–853, 1994).

The -spectrum and the -functional calculus

In some recent papers (called $\mathcal{S}$-functional calculus) for n-tuples of both bounded and unbounded not-necessarily commuting operators. The $\mathcal{S}$-functional calculus is based on the notion of $\mathcal{S}$-spectrum, which naturally arises from the definition of the $\mathcal{S}$-resolvent operator for n-tuples of operators. The $\mathcal{S}$-resolvent operator plays the same role as the usual resolvent operator for the Riesz–Dunford functional calculus, which is associated to a complex operator acting on a Banach space. When one considers commuting operators (bounded or unbounded) there is the possibility of simplifying the computation of the $\mathcal{S}$-spectrum. In fact, in this case we can use the F-spectrum, which is easier to compute than the $\mathcal{S}$-spectrum. In the case of commuting operators, our functional calculus is based on the $\mathcal{F}$-spectrum and will be called $\mathcal{SC}$-functional calculus. We point out that for a correct definition of the $\mathcal{S}$-resolvent operator and of the $\mathcal{SC}$-resolvent operator in the unbounded case we have to face different extension problems. Another reason for a more detailed study of the $\mathcal{F}$-spectrum is that it is related to the $\mathcal{F}$-functional calculus which is based on the integral version of the Fueter mapping theorem. This functional calculus is associated to monogenic functions constructed by starting from slice monogenic functions.

Generalized Modal Expansion of Electromagnetic Field in 2-D Bounded and Unbounded Media

A generalized modal expansion theory is presented to investigate and illustrate the physics of wave-matter interaction within arbitrary two-dimensional (2-D) bounded and unbounded electromagnetic problems. We start with the bounded case where the field excited by any sources is expanded with a complete set of biorthogonal eigenmodes. In regard to non-Hermitian or nonreciprocal problems, an auxiliary system is constructed to seek for the modal-expansion solution. We arrive at the unbounded case when the boundary tends to infinity or is replaced by the perfectly matched layer (PML). Modes are approximately categorized into two types: trapped modes and radiation modes, which respond differently to environment variations. When coupled with the source, these modes contribute to the modal-expansion solution with different weights, which leads to a reduced modal representation of the excited field in some geometries.

The spectral methods for parabolic Volterra integro-differential equations

In this paper we study the numerical solutions to parabolic Volterra integro-differential equations in one-dimensional bounded and unbounded spatial domains. In a bounded domain, the given parabolic Volterra integro-differential equation is converted to two equivalent equations. Then, a Legendre-collocation method is used to solve them and finally a linear algebraic system is obtained. For an unbounded case, we use the algebraic mapping to transfer the problem on a bounded domain and then apply the same presented approach for the bounded domain. In both cases, some numerical examples are presented to illustrate the efficiency and accuracy of the proposed method.

On operators Cauchy dual to 2-hyperexpansive operators: the unbounded case

On operators Cauchy dual to 2-hyperexpansive operators: the unbounded case

Optimal dividend strategies in the diffusion model with stochastic return on investments

This paper studies the optimal dividend problem in the diffusion model with stochastic return on investments. The insurance company invests its surplus in a financial market. More specially, the authors consider the case of investment in a Black-Scholes market with risky asset such as stock. The classical problem is to find the optimal dividend payment strategy that maximizes the expectation of discounted dividend payment until ruin. Motivated by the idea of Thonhauser and Albrecher (2007), we take the lifetime of the controlled risk process into account, that is, the value function considers both the expectation of discounted dividend payment and the time value of ruin. The authors conclude that the optimal dividend strategy is a barrier strategy for the unbounded dividend payment case and is of threshold type for the bounded dividend payment case.

A note on density for the core of unbounded Bergman operators

In this paper, we identify a large collection of open subsets of the complex plane for which the core of corresponding unbounded Bergman operators is densely defined. This result gives the necessary background to investigate the concept of invariant subspaces, index, and cyclicity in the unbounded case.

Paths of bounded length and their cuts: Parameterized complexity and algorithms

We study the parameterized complexity of two families of problems: the bounded length disjoint paths problem and the bounded length cut problem. From Menger’s theorem both problems are equivalent (and computationally easy) in the unbounded case for single source, single target paths. However, in the bounded case, they are combinatorially distinct and are both NP-hard, even to approximate. Our results indicate that a more refined landscape appears when we study these problems with respect to their parameterized complexity. For this, we consider several parameterizations (with respect to the maximum length l of paths, the number k of paths or the size of a cut, and the treewidth of the input graph) of all variants of both problems (edge/vertex-disjoint paths or cuts, directed/undirected). We provide FPT-algorithms (for all variants) when parameterized by both k and l and hardness results when the parameter is only one of k and l . Our results indicate that the bounded length disjoint-path variants are structurally harder than their bounded length cut counterparts. Also, it appears that the edge variants are harder than their vertex-disjoint counterparts when parameterized by the treewidth of the input graph.

The parameterized complexity of [formula omitted]-flip local search for SAT and MAX SAT

Sat and Max Sat are among the most prominent problems for which local search algorithms have been successfully applied. A fundamental task for such an algorithm is to increase the number of clauses satisfied by a given truth assignment by flipping the truth values of at most k variables ( k -flip local search). For a total number of n variables the size of the search space is of order n k and grows quickly in k ; hence most practical algorithms use 1-flip local search only. In this paper we investigate the worst-case complexity of k -flip local search, considering k as a parameter: is it possible to search significantly faster than the trivial n k bound? In addition to the unbounded case we consider instances with a bounded number of literals per clause and instances where each variable occurs in a bounded number of clauses. We also consider the related problem that asks whether we can satisfy all clauses by flipping the truth values of at most k variables.

Commutativity up to a Factor: More Results and the Unbounded Case

We give more results on the question of commutativity up to a factor for bounded operators and which has been recently of interest to a number of mathematicians. We also give some generalizations to unbounded operators.

Outer approximation schemes for generalized semi-infinite variational inequality problems

We introduce and analyse outer approximation schemes for solving variational inequality problems in which the constraint set is as in generalized semi-infinite programming. We call these problems generalized semi-infinite variational inequality problems. First, we establish convergence results of our method under standard boundedness assumptions. Second, we use suitable Tikhonov-like regularizations for establishing convergence in the unbounded case.

Bounded sequence testing from deterministic finite state machines

The W - and W p - methods are the basis for conformance testing from a deterministic finite state machine ( DFSM) when the conformance relation considered is equivalence. However, many DFSM applications use only input sequences of limited length. In such cases, the test data only need to establish that the implementation under test produces the specified responses for sequences of length less than or equal to the upper bound l . This paper extends the W - and W p -methods to the case in which only bounded sequences are allowed. The methods for bounded sequences are stronger than the originals since test suites for the unbounded case can be obtained as a particular case (in which the upper bound l is sufficiently large) from the new formulae. Furthermore, the generalization is not straightforward as it is not sufficient to extract the sequences of length at most l from the test suites produced in the unbounded case, or even all prefixes of length at most l of the original test sequences. The practicality of the methods is also improved in comparison to the unbounded case: the size of the test suites may be considerably reduced while the complexity of the test generation algorithms remains basically unchanged.