Approximate Variational Principle Research Articles

A variational principle for a fractal nano/microelectromechanical (N/MEMS) system

PurposeThe variational principle views a complex problem in an energy way, it gives good physical understanding of an iteration method, and the variational-based numerical methods always have a conservation scheme with a fast convergent rate. The purpose of this paper is to establish a variational principle for a fractal nano/microelectromechanical (N/MEMS) system.Design/methodology/approachThis paper begins with an approximate variational principle in literature for the studied problem, and a genuine variational principle is obtained by the semi-inverse method.FindingsThe semi-inverse method is a good mathematical tool to the search for a genuine fractal variational formulation for the N/MEMS system.Research limitations/implicationsThe established variational principle can be used for both analytical and numerical analyses of the N/MEMS systems, and it can be extended to some more complex cases.Practical implicationsThe variational principle can be used for variational-based finite element methods and energy-based analytical methods.Originality/valueThe new and genuine variational principle is obtained. This paper discovers the missing piece of the puzzle for the establishment of a variational principle from governing equations for a complex problem by the semi-inverse method. The new variational theory opens a new direction in fractal MEMS systems.

A short remark on Chien’s variational principle of maximum power losses for viscous fluids

Purpose – The purpose of this paper is to point out a paradox in variational theory for viscous flows. Chien (1984) claimed that a variational principle of maximum power loses for viscous fluids was established, however, it violated the well-known Helmholtz’s principle. Design/methodology/approach – Restricted variables are introduced in the derivation, the first order and the second order of variation of the restricted variables are zero. Findings – An approximate variational principle of minimum power loses is established, which agrees with the Helmholtz’s principle, and the paradox is solved. Research limitations/implications – This paper focusses on incompressible viscose flows, and the theory can be extended to compressible one and other viscose flows. It is still difficult to obtain a variational formulation for Navier-Stokes equations. Practical implications – The variational principle of minimum power loses can be directly used for numerical methods and analytical analysis. Originality/value – It is proved that Chien’s variational principle is a minimum principle.

Geometric reconstruction in bioluminescence tomography

In bioluminescence tomography the location as well as the radiation intensity of a photon source (marked cell clusters) inside an organism have to be determined given the outside photon count. This inverse source problem is ill-posed: it suffers not only from strong instability but also from non-uniqueness. To cope with these difficulties the source is modeled as a linear combination of indicator functions of measurable domains leading to a nonlinear operator equation. The solution process is stabilized by a Tikhonov like functional which penalizes the perimeter of the domains. For the resulting minimization problem existence of a minimizer, stability, and regularization property are shown. Moreover, an approximate variational principle is developed based on the calculated domain derivatives which states that there exist smooth almost stationary points of the Tikhonov like functional near to any of its minimizers. This is a crucial property from a numerical point of view as it allows to approximate the searched-for domain by smooth domains. Based on the theoretical findings numerical schemes are proposed and tested for star-shaped sources in 2D: computational experiments illustrate performance and limitations of the considered approach.

NEW ESTIMATES FOR EVANS' VARIATIONAL APPROACH TO WEAK KAM THEORY

We consider a recent approximate variational principle for weak KAM theory proposed by Evans. As in the case of classical integrability, for one-dimensional mechanical Hamiltonian systems all the computations can be carried out explicitly. In this setting, we illustrate the geometric content of the theory and prove new lower bounds for the estimates related to its dynamic interpretation. These estimates also extend to the case of n degrees of freedom.

Some new PDE methods for weak KAM theory

We discuss a new approximate variational principle for weak KAM theory. The advantage of this approach is that we build both a minimizing measure and a solution of the generalized eikonal equation at the same time. Furthermore the approximations are smooth, and so we can derive some interesting formulas upon differentiating the Euler-Lagrange equation. Our method is inspired by the ”calculus of variations in the sup-norm” ideas of Aronsson, Jensen, Barron and others.

Extended and strongly extended well-posedness of set-valued optimization problems

In this paper, we deal with the extended well-posedness and strongly extended well-posedness of set-valued optimization problems. These two concepts are generalizations of the extended well-posedness of real-valued optimization probems defined by Zolezzi. We obtain some criteria and characterizations of these two types of extended well-posedness, further generalizing most results obtained by Zolezzi for the extended well-posedness of scalar optimization problems. In the mean time, many results obtained by us for the extended well-posedness of vector optimization problems have been generalized to set-valued optimization. Finally, we present an approximate variational principle for set-valued maps, derive a necessary approximate optimality condition for set-valued optimization, based on which we introduce a condition, which is somewhat analogous to the Palais-Smale condition (C), and provide sufficient conditions for the extended and strongly extended well-posedness of set-valued optimization problems.

Equivalents of a general approximate variational principle for set-valued maps and application to efficiency

In this paper, three equivalents of a corollary of the general approximate variational principle for set-valued maps which we obtained in [5] are derived. Applying a corollary of the general approximate variational principle, we obtain existence theorems for efficient solutions to vector optimization problems.

Ekeland's ε-variational principle for set-valued mappings

In this paper, we introduce the concept of approximate solutions for set-valued mappings and provide a sufficient condition for the existence of approximate solutions of set-valued mappings. We obtain an approximate variational principle for set-valued mappings.

An approximate variational method for improved thermodynamics of molecular fluids

For a certain class of thermodynamic perturbation theories, a generalization of the Gibbs–Bogoliubov inequality holds through second order of perturbation theory and for a subset of terms the inequality is true to infinite order. Using this approximate variational principle, a perturbation theory is chosen for which the Helmholtz free energy of the reference system is minimized under the constraint that the first order term is identically zero. We apply these ideas to the determination of effective spherical potentials that accurately reproduce the thermodynamics of nonspherical molecular potentials. For a diatomic-Lennard-Jones (DLJ) potential with l/σ=0.793, the resulting spherical reference potential is identical to the median average over angles for the repulsive part of the potential, but differs in the attractive well. The variational effective spherical potential leads to more accurate thermodynamics than the median, however, particularly in the triple point region.

Existence and regularity of extrema

Consider the minimization of a possibly noncoercive Gâteaux differentiable functional F:X→ R . A modified notion of coercivity is introduced which may be usable to show existence of a minimum. Alternatively, F ̂ :D→ R has a minimum at yε D ( F ̂ not differentiable but the restriction F of F ̂ to X ⊂ D differentiable), one may be able to show y ̄ is actually in X . The latter case is related to justification of formally calculated “necessary conditions” for optimal controls. The arguments are applications of Ekeland's “approximate variational principle” ( J. Math. Anal. Appl. 47 (1974) , 324–353).

A characterization of tyhonov well-posedness for minimum problems, with applications to variational inequalities(∗)

We give a characterization of Tyhonov well-posedness for the problem of minimising a convex lower-semieontinuous function f on a closed convex set K. To get this result, we use Ekeland's theorem on the“approximate variational principle”[21; when f is differentiable, the condition is and diam, where . We prove also some results relating the diameter of level sets of a sub- homogeneous function g with a function which characterizes the minimum increase of g above the minimum value. This result allows us to relate our characterization with a former one due to Zolezzi [10] Then we use the condition on diam(T∊) to give a definition of“well-posedness”for variational inequalities and prove some related results.

A variational approach to perturbed soliton equations

A general perturbation method is presented for soliton equations modified by small but arbitrary perturbation terms. Equations for the adiabatic time evolution of the soliton parameters are derived from an approximate variational principle. The method is applied to the Korteweg-de Vries and nonlinear Schrödinger equations modified by various dissipative perturbation terms.

Approximate Variational Principle in Quantum Statistics

Approximate Variational Principle in Quantum Statistics