52,652 research outputs found
On the existence of weak variational solutions to stochastic differential equations
We study the existence of weak variational solutions in a Gelfand triplet of real separable Hilbert spaces, under continuity, growth, and coercivity conditions on the coefficients of the stochastic differential equation. The laws of finite dimensional approximations are proved to weakly converge to the limit which is identified as a weak solution. The solution is an H– valued continuous process in L2 (Ω, C([0, T], H)) ∩ L2([0, T] × Ω, V ). Under the assumption of monotonicity the solution is strong and unique
Continuity properties of the inf-sup constant for the divergence
The inf-sup constant for the divergence, or LBB constant, is explicitly known
for only few domains. For other domains, upper and lower estimates are known.
If more precise values are required, one can try to compute a numerical
approximation. This involves, in general, approximation of the domain and then
the computation of a discrete LBB constant that can be obtained from the
numerical solution of an eigenvalue problem for the Stokes system. This
eigenvalue problem does not fall into a class for which standard results about
numerical approximations can be applied. Indeed, many reasonable finite element
methods do not yield a convergent approximation. In this article, we show that
under fairly weak conditions on the approximation of the domain, the LBB
constant is an upper semi-continuous shape functional, and we give more
restrictive sufficient conditions for its continuity with respect to the
domain. For numerical approximations based on variational formulations of the
Stokes eigenvalue problem, we also show upper semi-continuity under weak
approximation properties, and we give stronger conditions that are sufficient
for convergence of the discrete LBB constant towards the continuous LBB
constant. Numerical examples show that our conditions are, while not quite
optimal, not very far from necessary
Variational analysis of free-edge stress and displacement fields in general un-symmetric and thin-ply laminates under in-plane, bending and thermal loading
A variational approach based on the minimization of complementary energy is developed to determine accurately a complete solution for both free-edge stress and displacement distributions of a laminate with arbitrary lay-ups (possibly un-symmetric and made of thin plies) under combined in-plane, bending and thermal loading. The key idea is partitioning the total stresses/displacements in a laminate with free edges into unperturbed (without free edges) and unknown perturbation stresses/displacements caused by the presence of free edges. It enables the theory of variational stress-transfer to deal easily with both applied traction and displacement boundary conditions. A methodology is introduced to obtain displacement fields for a stress-based variational approach. The resulting stress and displacement fields exactly satisfy local equilibrium equations, strain-displacement relations together with all traction/displacement boundary and continuity conditions. By comparing the results with those obtained from the finite element method, the accuracy and computational efficiency of the developed model, is confirmed
Skeleton Integral Equations for Acoustic Transmission Problems with Varying Coefficients
In this paper we will derive an non-local (``integral'') equation which
transforms a three-dimensional acoustic transmission problem with
\emph{variable} coefficients, non-zero absorption, and mixed boundary
conditions to a non-local equation on a ``skeleton'' of the domain
, where ``skeleton'' stands for the union of the
interfaces and boundaries of a Lipschitz partition of . To that end, we
introduce and analyze abstract layer potentials as solutions of auxiliary
coercive full space variational problems and derive jump conditions across
domain interfaces. This allows us to formulate the non-local skeleton equation
as a \emph{direct method} for the unknown Cauchy data of the solution of the
original partial differential equation. We establish coercivity and continuity
of the variational form of the skeleton equation based on auxiliary full space
variational problems. Explicit expressions for Green's functions is not
required and all our estimates are \emph{explicit} in the complex wave number
On Hölder calmness of minimizing sets
We present conditions for Hölder calmness and upper Hölder continuity of optimal solution sets to perturbed optimization problems in finite dimensions. Studies on Hölder type stability were a popular subject in variational analysis already in the 1980s and 1990s, and have become a revived interest in the last decade. In this paper, we focus on conditions for Hölder calmness of the argmin mapping in the case of non-isolated minima. We recall known ideas and results in this context for general as well as special parametric programs, refine them and discuss particular settings, including nonlinear programs and convex semi-infinite optimization problems
A Variational Assimilation Method for Satellite and Conventional Data: a Revised Basic Model 2B
A variational objective analysis technique that modifies observations of temperature, height, and wind on the cyclone scale to satisfy the five 'primitive' model forecast equations is presented. This analysis method overcomes all of the problems that hindered previous versions, such as over-determination, time consistency, solution method, and constraint decoupling. A preliminary evaluation of the method shows that it converges rapidly, the divergent part of the wind is strongly coupled in the solution, fields of height and temperature are well-preserved, and derivative quantities such as vorticity and divergence are improved. Problem areas are systematic increases in the horizontal velocity components, and large magnitudes of the local tendencies of the horizontal velocity components. The preliminary evaluation makes note of these problems but detailed evaluations required to determine the origin of these problems await future research
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