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.     

Particularism and the contingent a priori

Particularism renders the options for a sound moral epistemology few and the prospects dim. One leading approach treats basic knowledge of particular cases as derivable from an a priori moral principle and a posteriori knowledge of the contingent non-moral facts to which the principle applies. Particularists must forgo this approach because it requires principles. Yet a purely a posteriori moral epistemology is also implausible, especially when combined with particularism. Particularists such as Jonathan Dancy are thus led to the view that our basic moral knowledge is a priori knowledge of contingent moral facts. We argue that this epistemology is unsound. While some cases of a priori knowledge of (even deeply) contingent facts may be defensible, they are not sufficient for particularist purposes. Moreover, neither Dancy’s appeal to the distinction between positive and negative dependence nor his discussion of intuitive examples provides sufficient support for this epistemology.     

On the Convergence of Adaptive Finite Element Methods

State of the art simulations in computational mechanics aim reliability and efficiency via adaptive finite element methods (AFEMs) with a posteriori error control. The a priori convergence of finite element methods is justified by the density property of the sequence of finite element spaces which essentially assumes a quasi‐uniform mesh‐refining. The advantage is guaranteed convergence for a large class of data and solutions; the disadvantage is a global mesh refinement everywhere accompanied by large computational costs. AFEMs automatically refine exclusively wherever the refinement indication suggests to do so and so violate the density property on purpose. Then, the a priori convergence of AFEMs is not guaranteed automatically and, in fact, crucially depends on algorithmic details. The advantage of AFEMs is a more effective mesh accompanied by smaller computational costs in many practical examples; the disadvantage is that the desirable error reduction property is not always guaranteed a priori. Efficient error estimators can justify a numerical approximation a posteriori and so achieve reliability. But it is not clear from the start that the adaptive mesh‐refinement will generate an accurate solution at all. This paper discusses particular versions of an AFEMs and their analyses for error reduction, energy reduction, and convergence results for linear and nonlinear problems. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical analysis of the primal problem of elastoplasticity with hardening

Summary. The finite element method is a reasonable and frequently utilised tool for the spatial discretization within one time-step in an elastoplastic evolution problem. In this paper, we analyse the finite element discretization and prove a priori and a posteriori error estimates for variational inequalities corresponding to the primal formulation of (Hencky) plasticity. The finite element method of lowest order consists in minimising a convex function on a subspace of continuous piecewise linear resp. piecewise constant trial functions. An a priori error estimate is established for the fully-discrete method which shows linear convergence as the mesh-size tends to zero, provided the exact displacement field u is smooth. Near the boundary of the plastic domain, which is unknown a priori, it is most likely that u is non-smooth. In this situation, automatic mesh-refinement strategies are believed to improve the quality of the finite element approximation. We suggest such an adaptive algorithm on the basis of a computable a posteriori error estimate. This estimate is reliable and efficient in the sense that the quotient of the error by the estimate and its inverse are bounded from above. The constants depend on the hardening involved and become larger for decreasing hardening. Received May 7, 1997 / Revised version received August 31, 1998     

Effective implementation of the ε-constraint method in Multi-Objective Mathematical Programming problems

As indicated by the most widely accepted classification, the Multi-Objective Mathematical Programming (MOMP) methods can be classified as a priori, interactive and a posteriori, according to the decision stage in which the decision maker expresses his/her preferences. Although the a priori methods are the most popular, the interactive and the a posteriori methods convey much more information to the decision maker. Especially, the a posteriori (or generation) methods give the whole picture (i.e. the Pareto set) to the decision maker, before his/her final choice, reinforcing thus, his/her confidence to the final decision. However, the generation methods are the less popular due to their computational effort and the lack of widely available software. The present work is an effort to effectively implement the ε-constraint method for producing the Pareto optimal solutions in a MOMP. We propose a novel version of the method (augmented ε-constraint method - AUGMECON) that avoids the production of weakly Pareto optimal solutions and accelerates the whole process by avoiding redundant iterations. The method AUGMECON has been implemented in GAMS, a widely used modelling language, and has already been used in some applications. Finally, an interactive approach that is based on AUGMECON and eventually results in the most preferred Pareto optimal solution is also proposed in the paper.     

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.     

Three-level warranty service contract among manufacturer,agent and customer: A game-theoretical approach

A warranty is a service contract between a manufacturer and a customer which plays a vital role in many businesses and legal transactions. In this paper, various three-level service contracts will be presented among the following three participants; a manufacturer, an agent, and a customer. In order to obtain a better result, the interaction between the aforementioned participants will be modeled using the game theory approach. Under non-cooperative and semi-cooperative games, the optimal sale price, warranty period and warranty price for the manufacturer and the optimal maintenance cost or repair cost for the agent are obtained by maximizing their profits. The satisfaction of the customer is also maximized by being able to choose one of the suggested options from the manufacturer and the agent, based on the risk parameter. Several numerical examples and managerial insights are presented and used to illustrate the models presented in this paper.     

A posteriori error estimators for the first-order least-squares finite element method

In this paper, we propose a posteriori error estimators for certain quantities of interest for a first-order least-squares finite element method. In particular, we propose an a posteriori error estimator for when one is interested in where . Our a posteriori error estimators are obtained by assigning proper weight (in terms of local mesh size h_T) to the terms of the least-squares functional. An a posteriori error analysis yields reliable and efficient estimates based on residuals. Numerical examples are presented to show the effectivity of our error estimators.     

Identifying an unknown source in the Poisson equation by the method of Tikhonov regularization in Hilbert scales

In this paper, we consider the problem for identifying the unknown source in the Poisson equation. The Tikhonov regularization method in Hilbert scales is extended to deal with illposedness of the problem and error estimates are obtained with an a priori strategy and an a posteriori choice rule to find the regularization parameter. The user does not need to estimate the smoothness parameter and the a priori bound of the exact solution when the a posteriori choice rule is used. Numerical examples show that the proposed method is effective and stable.     

Error estimation and adaptive discretization for the discrete stochastic Hamilton–Jacobi–Bellman equation

Summary. Generalizing an idea from deterministic optimal control, we construct a posteriori error estimates for the spatial discretization error of the stochastic dynamic programming method based on a discrete Hamilton–Jacobi–Bellman equation. These error estimates are shown to be efficient and reliable, furthermore, a priori bounds on the estimates depending on the regularity of the approximate solution are derived. Based on these error estimates we propose an adaptive space discretization scheme whose performance is illustrated by two numerical examples.Mathematics Subject Classification (2000): 93E20, 65N50, 49L20, 49M25, 65N15Acknowledgments. This research was supported by the Center for Empirical Macroeconomics, University of Bielefeld. The support is gratefully acknowledged. I would also like to thank an anonymous referee who suggested several improvements for the paper.     

The symbiosis mechanism for effective knowledge transfer

Knowledge transfer is widely emphasized as a strategic issue for sustainable competitive advantage of an organization. To date, it seems that the soft and hard mechanisms are being employed for knowledge transfer. Although there is considerable variation in the researchers’ views about the role of the two mechanisms of knowledge transfer of an organization, this paper suggests a symbiotic strategy that recognizes the interplay between the soft and hard mechanisms. It is argued that the symbiosis mechanism falls somewhere between the rather divergent mechanisms. Drawing from the analogy of knowledge as an iceberg, the perceived explicitness of knowledge is contended to be the essential determinant for the proposed symbiosis mechanism. The paper then proceeds to analyse the tradeoffs of implementing the symbiosis mechanism. It is found that when an organization has similar unit costs of transferring explicit and tacit knowledge, it can ideally minimize its total knowledge transfer cost, aligning individual's effort with the organization's. The proposed symbiosis approach to knowledge transfer will provide powerful arguments for a more holistic view which is crucial for the effective knowledge transfer. However, it is to be noted that the model is a conceptual one, not an operational one in which the mathematics only explain a structure but do not guide the practitioner to find optimal solutions.     

A MODIFIED TIKHONOV REGULARIZATION METHOD FOR THE CAUCHY PROBLEM OF LAPLACE EQUATION

In this paper,we consider the Cauchy problem for the Laplace equation,which is severely ill-posed in the sense that the solution does not depend continuously on the data.A modified Tikhonov regularization method is proposed to solve this problem.An error estimate for the a priori parameter choice between the exact solution and its regularized approximation is obtained.Moreover,an a posteriori parameter choice rule is proposed and a stable error estimate is also obtained.Numerical examples illustrate the validity and effectiveness of this method.     

A cut-free sequent system for two-dimensional modal logic,and why it matters

The two-dimensional modal logic of Davies and Humberstone (1980) [3] is an important aid to our understanding the relationship between actuality, necessity and a priori knowability. I show how a cut-free hypersequent calculus for 2D modal logic not only captures the logic precisely, but may be used to address issues in the epistemology and metaphysics of our modal concepts. I will explain how the use of our concepts motivates the inference rules of the sequent calculus, and then show that the completeness of the calculus for Davies–Humberstone models explains why those concepts have the structure described by those models. The result is yet another application of the completeness theorem.     

Recent developments in the qualitative approach to inverse electromagnetic scattering theory

We consider the inverse scattering problem of determining both the shape and some of the physical properties of the scattering object from a knowledge of the (measured) electric and magnetic fields due to the scattering of an incident time-harmonic electromagnetic wave at fixed frequency. We shall discuss the linear sampling method for solving the inverse scattering problem which does not require any a priori knowledge of the geometry and the physical properties of the scatterer. Included in our discussion is the case of partially coated objects and inhomogeneous background. We give references for numerical examples for each problem discussed in this paper.     

Representations of small braid groups in Temperley-Lieb algebra

In this paper we consider the question of faithfulness of the Jones' representation of braid group Bn into the Temperley-Lieb algebra TLn. The obvious motivation to study this problem is that any non-trivial element in the kernel of this representation (for any n) would almost certainly yield a non-trivial knot with trivial Jones polynomial (see [S. Bigelow, Does the Jones polynomial detect the unknot? J. Knot Theory Ramifications 11 (4) (2002) 493-505], we will explain it in more detail in Section 1). As one of the two main results we prove Theorem 1 in which we present a method to obtain non-trivial elements in the kernel of the representation of B₆ into TL9,2—to the authors' knowledge the first such examples in the second gradation of the Temperley-Lieb algebra. Theorem 2 which is a refinement of Theorem 1 may be used to produce smaller examples of the same kind. We also show briefly how some braids that are used in Section 4 to construct specific examples were generated with a computer program.     

A posteriori error estimates for hp finite element solutions of convex optimal control problems

In this paper, we present a posteriori error analysis for hp finite element approximation of convex optimal control problems. We derive a new quasi-interpolation operator of Clément type and a new quasi-interpolation operator of Scott-Zhang type that preserves homogeneous boundary condition. The Scott-Zhang type quasi-interpolation is suitable for an application in bounding the errors in L²-norm. Then hp a posteriori error estimators are obtained for the coupled state and control approximations. Such estimators can be used to construct reliable adaptive finite elements for the control problems.     

IDENTIFYING AN UNKNOWN SOURCE IN SPACE-FRACTIONAL DIFFUSION EQUATION

In this paper, we identify a space-dependent source for a fractional diffusion equation. This problem is ill-posed, i.e., the solution （if it exists） does not depend continuously on the data. The generalized Tikhonov regularization method is proposed to solve this problem. An a priori error estimate between the exact solution and its regularized approximation is obtained. Moreover, an a posteriori parameter choice rule is proposed and a stable error estimate is also obtained, Numerical examples are presented to illustrate the validity and effectiveness of this method.     

An iterative method for backward time‐fractional diffusion problem

The aim of this work is to solve the backward problem for a time‐fractional diffusion equation with variable coefficients in a general bounded domain. The problem is ill‐posed in L ² norm sense. An iteration scheme is proposed to obtain a regularized solution. Two kinds of convergence rates are obtained using an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule. Numerical examples in one‐dimensional and two‐dimensional cases are provided to show the effectiveness of the proposed methods. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 2029–2041, 2014     

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.