Abstraction was investigated by examining extrapolation behavior in a function-learning task. During training, participants associated stimulus and response magnitudes (in the form of horizontal bar lengths) that covaried according to a linear, exponential, or quadratic function. After training, novel stimulus magnitudes were presented as tests of extrapolation and interpolation. Participants e...

A static Partial Differential Equation (PDE) approach is presented for multidimensional extrapolation under the assumption that a level set function exists which separates the region of known values from the region to be extrapolated. Arbitrary orders of polynomial extrapolation can be obtained through solutions of a series of static linear PDEs. Fast sweeping methods of first and second orders...

We consider the variable coefficient Poisson equation with Dirichlet boundary conditions on irregular domains. We present numerical evidence for the accuracy of the solution and its gradients for different treatments at the interface using the Ghost Fluid Method for Poisson problems of Gibou et al. (J. Comput. Phys. 176:205–227, 2002; 202:577–601, 2005). This paper is therefore intended as a gu...

Humans are constantly challenged to make use of internal models to fill in missing sensory information. We measured human performance in a simple motion extrapolation task where no feedback was provided in order to elucidate the models of object motion incorporated into observers' extrapolation strategies. There was no "right" model for extrapolation in this task. Observers consistently adopted...

In this paper, we consider a class of new accelerated multilevel aggregation methods using two polynomial-type vector extrapolation methods, namely the reduced rank extrapolation (RRE) and the generalization of quadratic extrapolation (GQE) methods. We show how to combine the multilevel aggregation methods with the RRE and GQE algorithms on the finest level in order to speed up the numerical co...

Critical Load Levels (CLLs) are load levels at which a new binding or non-binding constraint occurs. Successful prediction of CLLs is very useful for identifying congestion and price change patterns. This paper extends the existing work to completely solve the problem of predicting the Previous CLL and Next CLL around the present operating point for an AC Optimal Power Flow (ACOPF) framework. F...

Splitting extrapolation is an efficient technique for solving large scale scientific and engineering problems in parallel. This article discusses a finite element splitting extrapolation for second order hyperbolic equations with time-dependent coefficients. This method possesses a higher degree of parallelism, less computational complexity, and more flexibility than Richardson extrapolation wh...

Abstract: We consider the minimization of an objective function given access to unbiased estimates of its gradient through stochastic gradient descent (SGD) with constant step-size. While the detailed analysis was only performed for quadratic functions, we provide an explicit asymptotic expansion of the moments of the averaged SGD iterates that outlines the dependence on initial conditions, the...

Recent efforts in differentiable non-linear programming have been focused on interior point methods, akin to penalty and barrier algorithms. In this paper, we address the classical equality constrained program solved using the simple quadratic loss penalty function/algorithm. The suggestion to use extrapolations to track the differentiable trajectory associated with penalized subproblems goes b...

Aitken extrapolation, applied to certain sequences, yields the evennumbered subsequence of the original. We prove that this is true for sequences generated by iterating a linear fractional transformation, and for some sequences of convergents of the regular continued fractions of certain quadratic irrational numbers.