A Priori and A Posteriori: A Bootstrapping Relationship

The distinction between a priori and a posteriori knowledge has been the subject of an enormous amount of discussion, but the literature is biased against recognizing the intimate relationship between these forms of knowledge. For instance, it seems to be almost impossible to find a sample of pure a priori or a posteriori knowledge. In this paper, it will be suggested that distinguishing between a priori and a posteriori is more problematic than is often suggested, and that a priori and a posteriori resources are in fact used in parallel. We will define this relationship between a priori and a posteriori knowledge as the bootstrapping relationship. As we will see, this relationship gives us reasons to seek for an altogether novel definition of a priori and a posteriori knowledge. Specifically, we will have to analyse the relationship between a priori knowledge and a priori reasoning, and it will be suggested that the latter serves as a more promising starting point for the analysis of aprioricity. We will also analyse a number of examples from the natural sciences and consider the role of a priori reasoning in these examples. The focus of this paper is the analysis of the concepts of a priori and a posteriori knowledge rather than the epistemic domain of a posteriori and a priori justification.     

A New Definition of A Priori Knowledge: In Search of a Modal Basis

In this paper I will offer a novel understanding of a priori knowledge. My claim is that the sharp distinction that is usually made between a priori and a posteriori knowledge is groundless. It will be argued that a plausible understanding of a priori and a posteriori knowledge has to acknowledge that they are in a constant bootstrapping relationship. It is also crucial that we distinguish between a priori propositions that hold in the actual world and merely possible, non-actual a priori propositions, as we will see when considering cases like Euclidean geometry. Furthermore, contrary to what Kripke seems to suggest, a priori knowledge is intimately connected with metaphysical modality, indeed, grounded in it. The task of a priori reasoning, according to this account, is to delimit the space of metaphysically possible worlds in order for us to be able to determine what is actual.     

A particularist epistemology: ‘Affectual intuitionism’

Jonathan Dancy has developed a very refined theory called ethical particularism. He has argued extensively for the metaphysical part of his position. However, the accompanying epistemology is not yet clear. In this paper I will sketch a particularist epistemology that is consistent with Dancy’s particularist metaphysics, although my approach differs in certain respects from epistemological claims Dancy has made. I will defend an epistemology that states: 1. that moral knowledge is based on intuitions and 2. that we need emotions in order to have moral knowledge. I will call this approach ‘affectual intuitionism’. Dancy rejects both claims, but I will argue that his arguments against these claims are not convincing.     

Sensitivity Statistical Estimates for Local A Posteriori Inference Matrix-Vector Equations in Algebraic Bayesian Networks over Quantum Propositions

An approach to the sensitivity analysis of local a posteriori inference equations in algebraic Bayesian networks is proposed in this paper. Some basic definitions and formulations are briefly given and the development of the matrix-vector a posteriori inference approach is considered. Some cases of the propagation of deterministic and stochastic evidence in a knowledge pattern with scalar estimates of component truth probabilities over quantum propositions are described. For each of the considered cases, the necessary metrics are introduced, and some transformations resulting in four linear programming problems are performed. The solution of these problems gives the required estimates. In addition, two theorems postulating the covering estimates for the considered parameters are formulated. The results obtained in this work prove the correct application of models and create a basis for the sensitivity analysis of local and global probabilistic-logic inference equations.     

Particularism, perception and judgement

According to the most detailed articulation and defence of moral particularism, it is a metaphysical doctrine about the nature of reasons. This paper addresses aspects of particularist epistemology. In rejecting the existence and efficacy of principles in moral thinking and reasoning particularists typically appeal to a theory of moral knowledge which operates with a ‘perceptual’ metaphor. This is problematic. Holism about valence can give rise to a moral epistemology that is a metaethical variety of atomistic empiricism. To avoid what could be called the Myth of the Moral Given, particularism has made use of a judgement-centred account of moral epistemology. This paper critically examines that account with reference to a proposed analogy between our moral knowledge and our knowledge of similarities.     

A Regularized Iterative Scheme for Solving Nonlinear Ill-Posed Problems

In this article, we consider a regularized iterative scheme for solving nonlinear ill-posed problems. The convergence analysis and error estimates are derived by choosing the regularization parameter according to both a priori and a posteriori methods. The iterative scheme is stopped using an a posteriori stopping rule, and we prove that the scheme converges to the solution of the well-known Lavrentiev scheme. The salient features of the proposed scheme are: (i) convergence and error estimate analysis require only weaker assumptions compared to standard assumptions followed in literature, and (ii) consideration of an adaptive a posteriori stopping rule and a parameter choice strategy that gives the same convergence rate as that of an a priori method without using the smallness assumption, the source condition. The above features are very useful from theory and application points of view. We also supply the numerical results to illustrate that the method is adaptable. Further, we compare the numerical result of the proposed method with the standard approach to demonstrate that our scheme is stable and achieves good computational output.     

Numerical Analysis of a Nonlinear Singularly Perturbed Delay Volterra Integro-Differential Equation on an Adaptive Grid

In this paper, we study a nonlinear first-order singularly perturbed Volterra integro-differential equation with delay. This equation is discretized by the backward Euler for differential part and the composite numerical quadrature formula for integral part for which both an a priori and an a posteriori error analysis in the maximum norm are derived. Based on the a priori error bound and mesh equidistribution principle, we prove that there exists a mesh gives optimal first order convergence which is robust with respect to the perturbation parameter. The a posteriori error bound is used to choose a suitable monitor function and design a corresponding adaptive grid generation algorithm. Furthermore, we extend our presented adaptive grid algorithm to a class of second-order nonlinear singularly perturbed delay differential equations. Numerical results are provided to demonstrate the effectiveness of our presented monitor function. Meanwhile, it is shown that the standard arc-length monitor function is unsuitable for this type of singularly perturbed delay differential equations with a turning point.     

Reflective equilibrium and underdetermination in epistemology

The basic aim of Alvin Goldman’s approach to epistemology, and the tradition it represents, is naturalistic; that is, epistemological theories in this tradition aim to identify the naturalistic, nonnormative criteria on which justified belief supervenes (Goldman, 1986; Markie, 1997). The basic method of Goldman’s epistemology, and the tradition it represents, is the reflective equilibrium test; that is, epistemological theories in this tradition are tested against our intuitions about cases of justified and unjustified belief (Goldman, 1986; Markie, 1997). I will argue that the prospect of having to reject their standard methodology is one epistemologists have to take very seriously; and I will do this by arguing that some current rival theories of epistemic justification are in fact in reflective equilibrium with our intuitions about cases of justified and unjustified belief. That is, I will argue that intuition underdetermines theory choice in epistemology, in much the way that observation underdetermines theory choices in empirical sciences. If reflective equilibrium leads to the underdetermination problem I say it leads to, then it cannot satisfy the aims of contemporary epistemology, and so cannot serve as its standard methodology.     

On a Kurchatov's method of linear interpolation for solving nonlinear equations

We apply the method of linear interpolation of Kurchatov to solve nonlinear operator equations in Banach spaces. Using the principle of majorants of L. V. Kantorovich, we study the semilocal convergence of method of Kurchatov. Quadratic order of convergence of this method is determined. A priori and a posteriori estimations of method's error are received. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Guaranteed error bounds for finite element approximations of noncoercive elliptic problems and their applications

In this paper, we discuss with guaranteed a priori and a posteriori error estimates of finite element approximations for not necessarily coercive linear second order Dirichlet problems. Here, ‘guaranteed’ means we can get the error bounds in which all constants included are explicitly given or represented as a numerically computable form. Using the invertibility condition of concerning elliptic operator, guaranteed a priori and a posteriori error estimates are formulated. This kind of estimates plays essential and important roles in the numerical verification of solutions for nonlinear elliptic problems. Several numerical examples that confirm the actual effectiveness of the method are presented.     

A modified Landweber iteration for general sideways parabolic equations

In this paper,we introduce a modified Landweber iteration to solve the sideways parabolic equation,which is an inverse heat conduction problem(IHCP) in the quarter plane and is severely ill-posed.We shall show that our method is of optimal order under both a priori and a posteriori stopping rule.Furthermore,if we use the discrepancy principle we can avoid the selection of the a priori bound.Numerical examples show that the computation effect is satisfactory.     

Quasi-norm a priori and a posteriori error estimates for the nonconforming approximation of p-Laplacian

Summary. In this paper, we derive quasi-norm a priori and a posteriori error estimates for the Crouzeix-Raviart type finite element approximation of the p-Laplacian. Sharper a priori upper error bounds are obtained. For instance, for sufficiently regular solutions we prove optimal a priori error bounds on the discretization error in an energy norm when . We also show that the new a posteriori error estimates provide improved upper and lower bounds on the discretization error. For sufficiently regular solutions, the a posteriori error estimates are further shown to be equivalent on the discretization error in a quasi-norm. Received January 25, 1999 / Revised version received June 5, 2000 Published online March 20, 2001     

A nonparametric variable step-size NLMS algorithm for transversal filters

A nonparametric adaptive filtering approach is proposed in this paper. The algorithm is obtained by exploiting a time-varying step size in the traditional NLMS weight update equation. The step size is adjusted according to the square of a time-averaging estimate of the autocorrelation of a priori and a posteriori error. Therefore, the new algorithm has more effective sense proximity to the optimum solution independent of uncorrelated measurement noise. Moreover, this algorithm has fast convergence at the early stages of adaptation and small final misadjustment at steady-state process. It works reliably and is easy to implement since the update function is nonparametric. Furthermore, the experimental results in system identification applications are presented to illustrate the principle and efficiency of the proposed algorithm.     

Uniform analysis of a stabilized hybrid finite element method for Reissner-Mindlin plates

This paper presents a low order stabilized hybrid quadrilateral finite element method for ReissnerMindlin plates based on Hellinger-Reissner variational principle,which includes variables of displacements,shear stresses and bending moments.The approach uses continuous piecewise isoparametric bilinear interpolations for the approximations of the transverse displacement and rotation.The stabilization achieved by adding a stabilization term of least-squares to the original hybrid scheme,allows independent approximations of the stresses and moments.The stress approximation adopts a piecewise independent 4-parameter mode satisfying an accuracy-enhanced condition.The approximation of moments employs a piecewise-independent 5-parameter mode.This method can be viewed as a stabilized version of the hybrid finite element scheme proposed in [Carstensen C,Xie X,Yu G,et al.A priori and a posteriori analysis for a locking-free low order quadrilateral hybrid finite element for Reissner-Mindlin plates.Comput Methods Appl Mech Engrg,2011,200:1161-1175],where the approximations of stresses and moments are required to satisfy an equilibrium criterion.A priori error analysis shows that the method is uniform with respect to the plate thickness t.Numerical experiments confirm the theoretical results.     

Linear estimates of accuracy for approximate solutions of inverse problems

In this paper, we explore the question of which non-linear inverse problems, which are solved by a selected regularization method, may have so-called linear a priori accuracy estimates – that is, the accuracy of corresponding approximate solutions linearly depends on the error level of the data. In particular, we prove that if such a linear estimate exists, then the inverse problem under consideration is well posed, according to Tikhonov. For linear inverse problems, we find that the existence of linear estimates lead to, under some assumptions, the well-posedness (according to Tikhonov) on the whole space of solutions. Moreover, we consider a method for solving inverse problems with guaranteed linear estimates, called the residual method on the correctness set (RMCS). The linear a priori estimates of absolute and relative accuracy for the RMCS are presented, as well as analogous a posteriori estimates. A numerical illustration of obtaining linear a priori estimates for appropriate parametric sets of solutions using RMCS is given in comparison with Tikhonov regularization. The a posteriori estimates are calculated on these parametric sets as well.     

A Knowledge-Based Approach To Response Surface Modelling in Multifidelity Optimization

This paper is concerned with approximations for expensive function evaluation – the expensive functions arising in an engineering design context. The problem of reducing the computational cost of generating sufficient learning samples is addressed. Several approaches of using a priori knowledge to achieve computational economy are presented. In all these, the results of a cheap model are treated as knowledge to be incorporated in the training process. Several approaches are described here: in particular, we focus on neural based systems. This approach is then developed as a new knowledge-based kriging model which is shown to be as accurate as neural based alternatives while being much easier to train. Examples from the domain of structural optimization are given to demonstrate the approach.     

A posteriori error estimates for the Crank-Nicolson method for parabolic equations

We derive optimal order a posteriori error estimates for time discretizations by both the Crank-Nicolson and the Crank-Nicolson-Galerkin methods for linear and nonlinear parabolic equations. We examine both smooth and rough initial data. Our basic tool for deriving a posteriori estimates are second-order Crank-Nicolson reconstructions of the piecewise linear approximate solutions. These functions satisfy two fundamental properties: (i) they are explicitly computable and thus their difference to the numerical solution is controlled a posteriori, and (ii) they lead to optimal order residuals as well as to appropriate pointwise representations of the error equation of the same form as the underlying evolution equation. The resulting estimators are shown to be of optimal order by deriving upper and lower bounds for them depending only on the discretization parameters and the data of our problem. As a consequence we provide alternative proofs for known a priori rates of convergence for the Crank-Nicolson method.

     

Using error bounds to compare aggregated generalized transportation models

A comparative study of aggregation error bounds for the generalized transportation problem is presented. A priori and a posteriori error bounds were derived and a computational study was performed to (a) test the correlation between the a priori, the a posteriori, and the actual error and (b) quantify the difference of the error bounds from the actual error. Based on the results we conclude that calculating the a priori error bound can be considered as a useful strategy to select the appropriate aggregation level. The a posteriori error bound provides a good quantitative measure of the actual error.