Multivalued Part Research Articles

On the SCD semismooth* Newton method for generalized equations with application to a class of static contact problems with Coulomb friction

In the paper, a variant of the semismooth^{*} Newton method is developed for the numerical solution of generalized equations, in which the multi-valued part is a so-called SCD (subspace containing derivative) mapping. Under a rather mild regularity requirement, the method exhibits (locally) superlinear convergence behavior. From the main conceptual algorithm, two implementable variants are derived whose efficiency is tested via a generalized equation modeling a discretized static contact problem with Coulomb friction.

Nonlocal semilinear evolution inclusions with time lag

In this paper, nonautonomous semilinear evolution inclusions with time lag and nonlocal initial conditions are studied. We define the so-called limit and weak solutions. We prove the existence of solutions when the multivalued part of the right-hand side and the initial functions are Lipschitz under natural conditions on the Lipschitz constants. Afterwards, under the same conditions, a variant of the important in optimal control relaxation theorem is proved.

On a Semismooth* Newton Method for Solving Generalized Equations

In the paper, a Newton-type method for the solution of generalized equations (GEs) is derived, where the linearization concerns both the single-valued and the multivalued part of the considered GE. The method is based on the new notion of semismoothness${}^*$, which, together with a suitable regularity condition, ensures the local superlinear convergence. An implementable version of the new method is derived for a class of GEs, frequently arising in optimization and equilibrium models.

Nonlocal problem for evolution inclusions with one-sided Perron nonlinearities

We study a class of nonlocal initial problems of evolution inclusions in the form of compact valued perturbations of m-dissipative evolution inclusions in Banach spaces with uniformly convex duals. The multivalued part is assumed to be one-sided Perron. Our approach enables us to use a compactness method technique under non compact types assumptions.

On a Rational Function in a Linear Relation

The behavior of the domain, the range, the kernel and the multi-valued part of a rational function in a linear relation is analyzed, respectively. We give some basic properties of such linear relations, and we prove that the rational form of the spectral mapping theorem holds in terms of ascent, essential ascent, descent and essential descent.

Transition time of nonlinear Landau–Zener model in adiabatic limit

The impact of nonlinear interaction on the loop structure of lower energy level and on the time evolution curve of canonical momentum which corresponds to the lower eigenstate are analyzed respectively. We find that the curve changes from single-valued to multi-valued as nonlinear interaction grows. The fascinating part is that the time range delimited by turning points in the loop of energy level and the period between two inflexion points on the multi-valued part of the evolution curve of canonical momentum are the same. Therefore, we propose a characteristic time in the transition process of nonlinear Landau–Zener model in adiabatic limit. Last, the physical meaning of the transition time as a measure of how much time the system experiences a structural change which directly results in the breakdown of adiabaticity is discussed.

Well-posedness, stability and invariance results for a class of multivalued Lur’e dynamical systems

This paper analyzes the existence and uniqueness issues in a class of multivalued Lur’e systems, where the multivalued part is represented as the subdifferential of some convex, proper, lower semicontinuous function. Through suitable transformations the system is recast into the framework of dynamic variational inequalities and the well-posedness (existence and uniqueness of solutions) is proved. Stability and invariance results are also studied, together with the property of continuous dependence on the initial conditions. The problem is motivated by practical applications in electrical circuits containing electronic devices with nonsmooth multivalued voltage/current characteristics, and by state observer design for multivalued systems.

A perturbation result for q-open quotient morphisms in normed spaces and applications to linear relations

Using some techniques of perturbation theory for quotient morphisms between Banach spaces we obtain necessary and sufficient conditions for the stability of the topological index of a q-open quotient morphism under small (with respect to the gap topology) perturbations with quotient morphisms. As an application we obtain necessary and sufficient conditions for the stability of the topological index of an open linear relation with closed multivalued part between normed spaces under small perturbations with linear relations.

Componentwise and Cartesian decompositions of linear relations

Let $A$ be a, not necessarily closed, linear relation in a Hilbert space $\got H$ with a multivalued part $\mathop{\rm mul} A$. An operator $B$ in $\got H$ with $\mathop{\rm ran} B\perp\mathop{\rm mul} A^{**}$ is said to be an operator part of $A$ when $A

Turbidite reservoir characterization : object-based stochastic simulation meandering channels

Abstract Stochastic imaging has become an important tool for risk assessment and has successfully been applied to oil field management. This procedure aims at generating several possible and equiprobable 3D models of subsurface structures that enhance the available data set. Among these stochastic simulation techniques, object-based approaches consist of defining and distributing objects reproducing underground geobodies. A technical challenge still remains in object-based simulation. Due to advances in deep water drilling technology, new hydrocarbon exploration has been opened along the Atlantic margins. In these turbidite oil fields, segments of meandering channels can be observed on high-resolution seismic horizons. However, no present object-based simulation technique can reproduce exactly such known segments of channel. An improved object-based approach is proposed to simulate meandering turbidite channels conditioned on well observations and such seismic data. The only approaches dealing with meandering channels are process-based as opposed to structure-imitating. They are based on the reproduction of continental river evolution through time. Unfortunately, such process-based approaches cannot be used for stochastic imaging as they are based on equations reflecting meandering river processes and not turbiditic phenomena. Moreover, they incoporate neither shape constraints (such as channel dimensions and sinuosity) nor location constraints, such as well data. Last, these methods generally require hydraulic parameters that are not available from oil field study. The proposed approach aims at stochastically generating meandering channels with specified geometry that can be constrained to pass through well-observations. The method relies on the definition of geometrical parameters that characterize the shape of the expected channels such as dimensions, directions and sinuosity. The meandering channel object is modelled via a flexible parametric shape. The object is defined by a polygonal center-line (called backbone) that supports several sections. Channel sinuosity and local channel profiles are controlled by the backbone and, respectively the sections. Channel generation is performed within a 2D domain, D representing the channel-belt area. The proposed approach proceeds in two main steps. The first step consists in generating a channel center-line (C) defined by an equation v=Z(u) within the domain D. The geometry of this line is simulated using a geostatistical simulation technique that allows the generation of controlled but irregular center-lines conditioned on data points. During the second step, a vector field enabling the curve (C) to be transformed into a meandering curve (C’) is estimated. This vector field acts as a transform that specifies the third degree of channel sinuosity, in other words, the meandering parts of the loops. This field is parameterized by geometrical parameters such as curvature and tangent vectors along the curve (C) and the a priori maximum amplitude of the meander loops of the curve (C’). To make channel objects pass through conditioning points, adjustment vectors are computed at these locations and are interpolated along the curves. Synthetic datasets have been built to check if a priori parameters such as tortuosity are reproduced, and if the simulations are equiprobable. From this dataset, hundred simulations have been generated and enable one to verify that these two conditions are satisfied. Equiprobability is however not always satisfied from data points that are very close and located in a multivalued part of a meander : preferential orientation of the loops may indeed be observed. Solving this issue will be the focus of future works. Nevertheless, the results presented in this paper show that the approach provides satisfying simulations in any other configurations. This approach is moreover well-suited for petroleum reservoir characterization because it only needs specification of geometrical parameters such as dimension and sinuosity that can be inferred from the channel parts seen on seismic horizons or analogues.

Transient eddy current field of current forced three-dimensional conductors

The author presents an efficient finite element technique for modeling the transient eddy current field of three-dimensional current forced conductors. The electromagnetic field is described by the magnetic vector potential inside the conductors and by the magnetic scalar potential in the nonconducting region. This magnetic scalar potential is split into a multivalued and a monovalued potential. The multivalued part can be determined in advance from the geometrical data and in every time step this precalculated multivalued potential multiplied by the instantaneous value of the prescribed current generates the eddy current field.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Eigenfunction expansions for nondensely defined differential operators

The operator L, is closed, densely defined and symmetric with domain ID(&). Let 5, be a p-dimensional (p is finite) subspace of !+j and let S,, be the restriction of L, to a(&,) = a(L,) n !&l. Then S, is a closed operator but not densely defined on 53. By identifying graph and operator we may speak of the adjoint subspace S = S,,* of S, in a2 = fi x $j and then we have S, C S. We consider self-adjoint subspace extensions H of S,,: S,, C H = H* C S. Such a self-adjoint subspace H gives rise to a self-adjoint operator H, in the smaller Hilbert space @ 0 H(O), where H(0) is the multivalued part of H. The above theory is given by Coddington [3]. The same author characterizes the self-adjoint subspace extensions of nondensely defined operators in terms of abstract boundary conditions, cf. [4], [Sj. Now, in the boundary conditions determining H appear integral terms and in the expression for H, appear both boundary and integral terms. For such operators Coddington [6], [7] has shown the existence of a matrix valued spectral distribution function p in terms of which he gave eigenfunction expansions converging in the Hilbertspace norm. The principal aim of this paper is to derive an expansion theorem in which point-wise convergence is allowed. Coddington’s norm convergence theorem is a straightforward consequence of our theorem. We will prove our results by adaptation of a method used by Niessen [lo], cf. Rellich [ll]. This adaptation is required to take care of the boundary and integral terms