Coisotropic variational problems

In this paper we study constrained variational problems in one independent variable defined on the space of integral curves of a Frenet system in a homogeneous space G/H. We prove that if the Lagrangian is G-invariant and coisotropic then the extremal curves can be found by quadratures. Our proof is constructive and relies on the reduction theory for coisotropic optimal control problems. This gives a unified explanation of the integrability of several classical variational problems such as the total squared curvature functional, the projective, conformal and pseudo-conformal arc-length functionals, the Delaunay and the Poincaré variational problems.     

Stochastic variational derivations of the Navier-Stokes equation

Stochastic calculus of variations in terms of sample paths is shown to be equivalent to ordinary calculus of variations in terms of local drift fields when applied to Markov processes. This equivalence enables us to derive the Navier-Stokes equation directly through a variational principle. Partially supported by the Swiss National Science Foundation.     

On the variational formulation of systems with non‐holonomic constraints

In a previous article by the author, it was shown that one could effectively give a variational formulation to non‐conservative mechanical systems by starting with the first variation functional instead of an action functional. In this article, it is shown that this same approach will also allow one to give a variational formulation to systems with non‐holonomic constraints. The key is to use an adapted anholonomic local frame field in the formulation, which then implies the replacement of ordinary derivatives with covariant ones. The method is then applied to the case of a vertical disc rolling without slipping or friction on a plane.     

Analysis of skew plate problems with various constraints

This paper presents the application of the modified Rayleigh-Ritz method with Lagrange multipliers to analyze skew plate problems with various constraints. By this procedure one can satisfy both geometric and natural boundary conditions of skew plates. To demonstrate the accuracy and versatility of the method, several examples of bending, vibration and buckling of skew plates are solved, and results are compared with those obtained by other approximate methods.     

A variational Bayesian method to inverse problems with impulsive noise

We propose a novel numerical method for solving inverse problems subject to impulsive noises which possibly contain a large number of outliers. The approach is of Bayesian type, and it exploits a heavy-tailed t distribution for data noise to achieve robustness with respect to outliers. A hierarchical model with all hyper-parameters automatically determined from the given data is described. An algorithm of variational type by minimizing the Kullback–Leibler divergence between the true posteriori distribution and a separable approximation is developed. The numerical method is illustrated on several one- and two-dimensional linear and nonlinear inverse problems arising from heat conduction, including estimating boundary temperature, heat flux and heat transfer coefficient. The results show its robustness to outliers and the fast and steady convergence of the algorithm.     

A variational approach to stochastic nonlinear problems

A variational principle is formulated which enables the mean value and higher moments of the solution of a stochastic nonlinear differential equation to be expressed as stationary values of certain quantities. Approximations are generated by using suitable trial functions in this variational principle and some of these are investigated numerically for the case of a Bernoulli oscillator driven by white noise. Comparison with exact data available for this system shows that the variational approach to such problems can be quite effective.     

A variational approach to interfacial stability problems

A variational approach to interfacial stability problems of stagnant fluids is shown to give rise to an eigenvalue problem. The sign of the eigenvalues determines stability and their value is proportional to the perturbation growth rate.     

Counting symmetry-breaking solutions to symmetric variational problems

By combining Michel's geometric theory of symmetry breaking and classical results from variational analysis, we obtain a lower bound on the number of critical points with given symmetryH G of a potential symmetric underG. The result is obtained by applying the Ljusternik-Schnirelman category in the group orbit space, and can be extended along the same lines to more general situations.     

Active-space N-representability constraints for variational two-particle reduced density matrix calculations

The ground-state energy of a system of fermions can be calculated by minimizing a linear functional of the two-particle reduced density matrix (2-RDM) if an accurate set of N-representability conditions is applied. In this Letter we introduce a class of linear N-representability conditions based on exact calculations on a reduced active space. Unlike wave-function-based approaches, the 2-RDM methodology allows us to combine information from calculations on different active spaces. By adding active-space constraints, we can iteratively improve our estimate for the ground-state energy. Applying our methodology to a 1D Hubbard model yields a significant improvement over traditional 2-positivity constraints with the same computational scaling.     

The variational problems for classical N-vector models

We prove existence and regularity properties of solutions of the variational problems introduced in the previous paper [1] for classical lattice N-vector models. These results form a basis of our renormalization group approach to low temperature expansions for the considered models.The work has been partially supported by the NSF Grant DMS-9102639.     

Stochastic eigenvalue problems for differential equations

We consider an eigenvalue problem for stochastic ordinary differential operators with stochastic boundary conditions. By replacing the random variables and stochastic processes by their mean values one gets the so-called averaged problem corresponding to the given stochastic problem. Let λ_i(ω) denote the stochastic eigenvalues and μ_i the eigenvalues of the averaged problem. In this paper the differences 〈λ〉?μ_i are calculated. The motive of such investigations was the effect of “bowing” in mixed crystal theory.     

Two variational approaches for domain decomposition problems solved by SGBEM with nonconforming discretizations

Two original approaches for the solution of elastic boundary value problems with domain decomposition (DDBVP) using the symmetric Galerkin boundary element method (SGBEM) are presented. Each approach is based on a variational principle, a difference between them consisting in the treatment of the coupling conditions which connect the solutions through an interface. The computer codes developed are able to deal with curved interfaces in a domain decomposition problem discretized by nonmatching meshes of linear elements along the interfaces. The effectiveness of the methods is documented by a numerically solved example. The text was submitted by the authors in English.     

Hierarchical Bayesian inference for Ill-posed problems via variational method

This paper investigates a novel approximate Bayesian inference procedure for numerically solving inverse problems. A hierarchical formulation which determines automatically the regularization parameter and the noise level together with the inverse solution is adopted. The framework is of variational type, and it can deliver the inverse solution and regularization parameter together with their uncertainties calibrated. It approximates the posteriori probability distribution by separable distributions based on Kullback–Leibler divergence. Two approximations are derived within the framework, and some theoretical properties, e.g. variance estimate and consistency, are also provided. Algorithms for their efficient numerical realization are described, and their convergence properties are also discussed. Extensions to nonquadratic regularization/nonlinear forward models are also briefly studied. Numerical results for linear and nonlinear Cauchy-type problems arising in heat conduction with both smooth and nonsmooth solutions are presented for the proposed method, and compared with that by Markov chain Monte Carlo. The results illustrate that the variational method can faithfully capture the posteriori distribution in a computationally efficient way.