Model adaptivity for finite element analysis of thin or thick plates based on equilibrated boundary stress resultants

Purpose – The purpose of this paper is to address error-controlled adaptive finite element (FE) method for thin and thick plates. A procedure is presented for determining the most suitable plate model (among available hierarchical plate models) for each particular FE of the selected mesh, that is provided as the final output of the mesh adaptivity procedure. \ud \ud Design/methodology/approach – The model adaptivity procedure can be seen as an appropriate extension to model adaptivity for linear elastic plates of so-called equilibrated boundary traction approach error estimates, previously proposed for 2D/3D linear elasticity. Model error indicator is based on a posteriori element-wise computation of improved (continuous) equilibrated boundary stress resultants, and on a set of hierarchical plate models. The paper illustrates the details of proposed model adaptivity procedure for choosing between two most frequently used plate models: the one of Kirchhoff and the other of Reissner-Mindlin. The implementation details are provided for a particular case of the discrete Kirchhoff quadrilateral four-node plate FE and the corresponding Reissner-Mindlin quadrilateral with the same number of nodes. The key feature for those elements that they both provide the same quality of the discretization space (and thus the same discretization error) is the one which justifies uncoupling of the proposed model adaptivity from the mesh adaptivity. \ud \ud Findings – Several numerical examples are presented in order to illustrate a very satisfying performance of the proposed methodology in guiding the final choice of the optimal model and mesh in analysis of complex plate structures. \ud \ud Originality/value – The paper confirms that one can make an automatic selection of the most appropriate plate model for thin and thick plates on the basis of proposed model adaptivity procedure.\u

A finite element method with mesh adaptivity for computing vortex states in fast-rotating Bose-Einstein condensates

Numerical computations of stationary states of fast-rotating Bose-Einstein condensates require high spatial resolution due to the presence of a large number of quantized vortices. In this paper we propose a low-order finite element method with mesh adaptivity by metric control, as an alternative approach to the commonly used high order (finite difference or spectral) approximation methods. The mesh adaptivity is used with two different numerical algorithms to compute stationary vortex states: an imaginary time propagation method and a Sobolev gradient descent method. We first address the basic issue of the choice of the variable used to compute new metrics for the mesh adaptivity and show that simultaneously refinement using the real and imaginary part of the solution is successful. Mesh refinement using only the modulus of the solution as adaptivity variable fails for complicated test cases. Then we suggest an optimized algorithm for adapting the mesh during the evolution of the solution towards the equilibrium state. Considerable computational time saving is obtained compared to uniform mesh computations. The new method is applied to compute difficult cases relevant for physical experiments (large nonlinear interaction constant and high rotation rates).Comment: to appear in J. Computational Physic

An Adaptive Strategy for Active Learning with Smooth Decision Boundary

We present the first adaptive strategy for active learning in the setting of classification with smooth decision boundary. The problem of adaptivity (to unknown distributional parameters) has remained opened since the seminal work of Castro and Nowak (2007), which first established (active learning) rates for this setting. While some recent advances on this problem establish adaptive rates in the case of univariate data, adaptivity in the more practical setting of multivariate data has so far remained elusive. Combining insights from various recent works, we show that, for the multivariate case, a careful reduction to univariate-adaptive strategies yield near-optimal rates without prior knowledge of distributional parameters

A model of adaptive decision making from representation of information environment by quantum fields

We present the mathematical model of decision making (DM) of agents acting in a complex and uncertain environment (combining huge variety of economical, financial, behavioral, and geo-political factors). To describe interaction of agents with it, we apply the formalism of quantum field theory (QTF). Quantum fields are of the purely informational nature. The QFT-model can be treated as a far relative of the expected utility theory, where the role of utility is played by adaptivity to an environment (bath). However, this sort of utility-adaptivity cannot be represented simply as a numerical function. The operator representation in Hilbert space is used and adaptivity is described as in quantum dynamics. We are especially interested in stabilization of solutions for sufficiently large time. The outputs of this stabilization process, probabilities for possible choices, are treated in the framework of classical DM. To connect classical and quantum DM, we appeal to Quantum Bayesianism (QBism). We demonstrate the quantum-like interference effect in DM which is exhibited as a violation of the formula of total probability and hence the classical Bayesian inference scheme.Comment: in press in Philosophical Transactions