Bivariate Product Cubature Using Peano Kernels for Local Error Estimates

Scientific Computing with Automatic Result Verification

In addition to the well-known and widely-used adaptive strategies for region subdivision and the choice of local quadrature rules, there still exists a third possibility for doing numerical integration adaptively, when interval computation is considered. Based on the Peano's kernels theorem and interval arithmetic, we are able to estimate the truncation error of a quadrature rule by means of different derivatives of the integrand, where the available orders of the derivatives depend on the degree of smoothness of the integrand and the exactness degree of the underlying quadrature rule. We classify the methods as the adaptive orders strategies, if they make use of the derivatives of different orders to improve each single local error estimation. In this paper, alternatives for adaptive error estimation are discussed. Moreover, a practical way for realization of the optimal error estimation is suggested. Numerical results for integrands of different classes as well as numerical comparisons of different methods are given.

Such soil structures as river dikes, high way embankments and earth dams frequently have been damaged during major earth- quakes, often due to liquefaction of embankments and foundation soils. In most cases, unacceptable, large, permanent deformation has occurred due to liquefaction of the supporting, loose, cohesion- less foundation soil. The seismic design of soil structures in these liquefiable soils poses very difficult problems for analysis and design. The possibility of liquefaction necessitates further investi- gation not only experimentally but by numerical analysis. ABSTRACT The focus of this paper is on 3-D adaptive analysis method of liquefiable soil improvement of the approximate quality of nonlinear numerical simulation of the liquefaction process with large deformation. This adaptive tech- nique was applied to the 3-D non-linear FE analysis of liquefiable soil considering large deformation, including the liquefaction phenomenon. The cyclic elasto-plastic model and updated Lagrangian formulation were adopted in the three-dimensional FE analysis. The fission procedure belonging to the h-refinement indicated by the error estimator, a posteriori error estimate procedure depending on L 2 -norm of strain and superconvergent patch recov- ery method, was used. The convergence of this error estimate scheme and the effectiveness of h-adaptive mesh refinement were shown by simple examples. The adaptive FE method was applied to the three-dimensional prac- tical seismic analysis of an embankment constructed on liquefiable soil, and its efficacy demonstrated in detail. Application of the adaptive technique to the nonlinear analysis of saturated soil, including the liquefaction process, is a valuable step in the three-dimensional seismic analysis of liquefaction.

This article investigates estimators of a weighted average of stratum-specific univariate parameters and compares them in terms of a design-based estimate of mean-squared error (MSE). The research is motivated by a stratified survey sample of Florida Medicaid beneficiaries, in which the parameters are population stratum means and the weights are known and determined by the population sampling frame. Assuming heterogeneous parameters, it is common to estimate the weighted average with the weighted sum of sample stratum means; under homogeneity, one ignores the known weights in favor of precision weighting. Adaptive estimators arise from random effects models for the parameters. We propose adaptive estimators motivated from these random effects models, but we compare their design-based performance. We further propose selecting the tuning parameter to minimize a design-based estimate of mean-squared error. This differs from the model-based approach of selecting the tuning parameter to accurately represent the heterogeneity of stratum means. Our design-based approach effectively downweights strata with small weights in the assessment of homogeneity, which can lead to a smaller MSE. We compare the standard random effects model with identically distributed parameters to a novel alternative, which models the variances of the parameters as inversely proportional to the known weights. We also present theoretical and computational details for estimators based on a general class of random effects models. The methods are applied to estimate average satisfaction with health plan and care among Florida beneficiaries just prior to Medicaid reform.

This article pertains to interpolation of Sobolev functions at shrinking lattices $h\mathbb{Z}^d$ from $L_p$ shift-invariant spaces associated with cardinal functions related to general multiquadrics, $\phi_{\alpha,c}(x):=(|x|^2+c^2)^\alpha$. The relation between the shift-invariant spaces generated by the cardinal functions and those generated by the multiquadrics themselves is considered. Additionally, $L_p$ error estimates in terms of the dilation $h$ are considered for the associated cardinal interpolation scheme. This analysis expands the range of $\alpha$ values which were previously known to give such convergence rates (i.e. $O(h^k)$ for functions with derivatives of order up to $k$ in $L_p$, $1<p<\infty$). Additionally, the analysis here demonstrates that some known best approximation rates for multiquadric approximation are obtained by their cardinal interpolants.

The subdiffusion equation with a Caputo fractional derivative of order � 2 (0, 1) in time arises in a wide variety of practical applications, and it is often adopted to model anomalous subdiffusion processes in heterogeneous media. The L1 scheme is one of the most popular and successful numerical methods for dis- cretizing the Caputo fractional derivative in time. The scheme was analyzed earlier independently by Lin and Xu (2007) and Sun and Wu (2006), and an O(� 2−� ) convergence rate was established, under the assumption that the solution is twice continuously differentiable in time. However, in view of the smoothing property of the subdiffusion equation, this regularity condition is restrictive, since it does not hold even for the homogeneous problem with a smooth initial data. In this work, we revisit the error analysis of the scheme, and establish an O(�) convergence rate for both smooth and nonsmooth initial data. The analysis is valid for more gen- eral sectorial operators. In particular, the L1 scheme is applied to one-dimensional space-time fractional diffusion equations, which involves also a Riemann-Liouville derivative of order � 2 (3/2,2) in space, and error estimates are provided for the fully discrete scheme. Numerical experiments are provided to verify the sharpness of the error estimates, and robustness of the scheme with respect to data regularity.

This paper describes a fast multi-scale vessel enhancement filter in 3D medical images. For efficient review of the vascular information, clinicians need rendering the 3D vascular information as a 2D image. Generally, the maximum intensity projection (MIP) is a useful and widely used technique for producing a 2D image from the 3D vascular data. However, the MIP algorithm reduces the conspicuousness for small and faint vessels owing to the overlap of non-vascular structures. To overcome this invisibility, researchers have examined the multi-scale vessel enhancement filter based on a combination of the eigenvalues of the 3D Hessian matrix. This multi-scale vessel enhancement filter produces higher contrast. However, it is time-consuming and requires high cost computation due to large volume of data and complex 3D convolution. For fast vessel enhancement, we propose a novel multi-scale vessel enhancement filter using 3D integral images and 3D approximated Gaussian kernel. This approximated kernel looks like cube but it is not exact cube. Each layer of kernel is approximated 2D Gaussian second order derivative by dividing it into three rectangular regions whose sum is integer. 3D approximated kernel is a pile of these 2D box kernels which are normalized by Frobenius norm. Its size fits to vessel width in order to achieve better visualization of the small vessel. Proposed method is approximately five times faster and produces comparable results with previous multi-scale vessel enhancement filter.

In this thesis, we introduce and examine four new temporal logic formalisms that can be used as specification languages for the automated verification of the reliability of hardware and software designs with respect to a desired behavior. The work is organized in two parts. In the first one, we reason about two logics for computations, GCTL* and MCTL*, which are useful to describe a correct execution of monolithic closed systems. In the second one, instead, we focus on two logics for strategies, SL and mATL*, which are useful to formalize several interesting properties about interactive plays in multi-entities systems modeled as multi-agent games. In the “Logics for Computations” part, we first study the immersion of the idea of graded quantifications into the temporal-logic framework. In first order logic, existential and universal quantifiers express the concepts of the existence of at least one individual object satisfying a formula, or that all individual objects satisfy a formula. In other logics, these quantifiers have been generalized to express that, for a given non-negative integer n, at least n or all but n individuals satisfy a particular formula. Here, we consider GCTL, a temporal logic with graded path quantifiers, which allows to describe properties like “there exist at least n different classes of computational fluxes in which a system reaches a predetermined state”, where the classes over paths are computed by means of a predetermined equivalence relation. More precisely, we uniformly extend the classic concept of graded quantifiers from states to paths, through the use of a concept of path equivalence with respect to a given path formula. About this logic, in particular, we study the expressiveness and succinctness relationships with respect to GµCalculus and the complexity of the satisfiability problem, which results to be ExpTime-Complete. This research is partially based on the works [BMM09] “Graded Computation Tree Logic” and [BMM10] “Graded Computation Tree Logic with Binary Coding” published, respectively, in the proceedings of the “IEEE Symposium on Logic in Computer Science, 2009” and “EACSL Annual Conference on Computer Science Logic, 2010”. Preliminary results can be also found in [Mog07]. Furthermore, we consider special quantifiers over substructures, which allow to select, using parametric criteria, small critical parts of a system to be successively verified. In literature, there are some attempts to define a logic that allows to modify the underlying structure under exam and then to verify on it some assigned property. However, as far as we know, none of them is able to select minimal submodels of a given property describing the criteria on which then execute the verification process. Here, we base our work on the search of a new operator that merges the concept of quantifiers on structures with that one derived by a generalization of the concept of pruning. The results of this work, is a class of three different extensions of CTL* with minimal model quantifiers, which we name MCTL*. Regarding these logics, we study several reductions among them, as well as the satisfiability problem that we prove to be highly undecidability, i.e., Σ_1^1-Hard, for two out of the three cases. This research is partially based on the work [MM09] “Branching-Time Temporal Logics with Minimal Model Quantifiers” published in the proceedings of the “International Conference on Developments in Language Theory, 2009”. In the “Logics for Strategies” part, we first study the problem of defining a new specification language through which it is possible to express several important properties of multi-entities systems that are neither expressible using classical monolithic temporal logics, such as CTL*, nor using two-agent-teams temporal logics, such as ATL*. In literature, we can found some proposal of logics that try to achieve this goal, but unfortunately, none of them succeeds completely on all the aspects. Among them, one of the most important attempts is CHP-SL, a logics in which one can use variables over strategies. However, this logic has a deep weakness, since it does not allow to describe games with more than two players and even two-players concurrent games. Here, we introduce SL, a logic with a syntax similar in some aspects to the first order logic, in which the strategies of the agent building the game are treated as first order objects on which we can quantify. This logic generalizes CHP-SL, by allowing the specification of the correct behavior of multi-agent concurrent games. In SL, for example, we are able to express very complex but useful Nash equilibria that are not expressible with CHP-SL. We enlighten that Nash equilibrium is one of the most important concepts in game theory. For the introduced logic, we solve two problems left open in the work on CHP-SL. Precisely, we show that the related model-checking problem is 2ExpTime-Complete, thus not harder of the same problem for several subsumed logics, while we prove that its satisfiability problem is highly undecidable, i.e., Σ_1^1-Hard. This research is partially based on the work [MMV10a] “Reasoning About Strategies” published in the proceedings of the “IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, 2010”. Finally, we consider the concept of relentful strategic reasoning, i.e., a formalism that expresses the ability of a strategy to be used not only to achieve a first given goal, but also to change its final goal in dependence of the history of the play. In the context of planning, memoryful quantification, i.e., quantification over computations that does not lose information about the past along the time, is one of the principal way to express the fact that a system is able to achieve a desired result, and shift to a different goal if some event happens. However, this kind of quantification was not considered before in the context of multi-agent planning. Here, we introduce mATL*, a fusion of the classic alternating temporal logic ATL* with memoryful quantification, with the aim of covering the previous idea. About this logics, we prove that, although it is equivalent to ATL*, it is exponentially more succinct. Nevertheless, we prove that both the model-checking and the satisfiability problems remain 2ExpTime-Complete, as for ATL*. This research is partially based on the work [MMV10b] “Relentful Strategic Reasoning in Alternating-Time Temporal Logic” published in the proceedings of the “International Conference on Logic for Programming Artificial Intelligence and Reasoning, 2010”.

Photodynamic therapy (PDT) of cancer is an alternative treatment for tumors resistant to chemo- and radiotherapy. It induces cancer cell death mainly through generation of reactive oxygen species by a laser light-activated photosensitizer. It has been suggested that the p53 tumor suppressor protein sensitizes some human cancer cells to PDT. However, there is still no direct evidence for this. We have demonstrated here for the first time that the photosensitizer protoporphyrin IX (PpIX) binds to p53 and disrupts the interaction between p53 tumor suppressor protein and its negative regulator HDM2 in vitro and in cells. Moreover, HCT116 colon cancer cells exhibited a p53-dependent sensitivity to PpIX in a dose-dependent manner, as was demonstrated using 3-(4,5-dimethylthiazol-2-yl)-2,5-diphenyltetrazolium bromide assay and fluorescence-activated cell sorter (FACS) analysis of cell cycle profiles. We have also observed induction of p53 target pro-apoptotic genes, e.g. puma (p53-up-regulated modulator of apoptosis), and bak in PpIX-treated cells. In addition, p53-independent growth suppression by PpIX was detected in p53-negative cells. PDT treatment (2 J/cm2) of HCT116 cells induced p53-dependent activation of pro-apoptotic gene expression followed by growth suppression and induction of apoptosis.

In traditional thinking, when the elastic problems are solved, we need to repeatedly plot element grids and analyze computing results according to diverse precision requirement. Against the malpractice exists in the above process, a new method of error estimation was suggested on H-R adaptive boundary element method in this paper. Based on the discrete meshes that are generated for the process of H-R adaptive refinement, the solution error was estimated by the interpolation residue. In addition, this method is easy to programming, which is carried out in the program by automatically creating new adaptive data files. Then a great deal of fore-disposal and post-disposal can be saved. Its validity and effectiveness have been confirmed by numerical example.

The low stress design of welded plates using the self-designing structures approach

In this paper, a finite volume element (FVE) method is considered for spatial approximations of time fractional diffusion equations involving a Riemann-Liouville fractional derivative of order α ∈ (0, 1) in time. Improving upon earlier results [Karaa et al., IMA J. Numer. Anal. 37 (2017) 945–964], error estimates in L2 (Ω)- and H1 (Ω)-norms for the semidiscrete problem with smooth and mildly smooth initial data, i.e., v ∈ H2(Ω) ∩ H01(Ω) and v ∈ H01(Ω) are established. For nonsmooth data, that is, v ∈ L2 (Ω), the optimal L2 (Ω)-error estimate is shown to hold only under an additional assumption on the triangulation, which is known to be satisfied for symmetric triangulations. Superconvergence result is also proved and as a consequence, a quasi-optimal error estimate is established in the L∞ (Ω)-norm. Further, two fully discrete schemes using convolution quadrature in time generated by the backward Euler and the second-order backward difference methods are analyzed, and error estimates are derived for both smooth and nonsmooth initial data. Based on a comparison of the standard Galerkin finite element solution with the FVE solution and exploiting tools for Laplace transforms with semigroup type properties of the FVE solution operator, our analysis is then extended in a unified manner to several time fractional order evolution problems. Finally, several numerical experiments are conducted to confirm our theoretical findings.

&lt;p style='text-indent:20px;'&gt;The paper deals with the approximation of first order derivative of a function by the first order derivative of Szász-type operators based on Charlier polynomials introduced by Varma and Taşdelen [&lt;xref ref-type="bibr" rid="b20"&gt;20&lt;/xref&gt;]. The uniform convergence theorem, Voronovskaja type asymptotic theorem and an estimate of error in terms of the second order modulus of continuity of the derivative of the function are investigated. Further, it is shown that linear combinations of the derivative of the above operators converge to the derivative of function at a faster rate. Finally, an estimate of error in the approximation is obtained in terms of the &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ (2k+2)th $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; order modulus of continuity using Steklov mean.&lt;/p&gt;

The application of the backward differentiation formulas to the solution of dynamic systems which are modeled by nonlinear functions with a finite number of continuous derivatives is studied. It is shown that if the functions have <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</tex> continuous derivatives, the derivative of order <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q + 1</tex> being discontinuous, the local truncation error of a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> -step backward differentiation formula applied over a discontinuity will be of order <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q + 1</tex> for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k \geq q</tex> . For a finite number of discontinuities the order of convergence is <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q +1</tex> for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k &gt; q</tex> . For piecewise-linear (pwl) systems the order of the formulas is therefore restricted to 2. Practical expressions for computing estimates of the local truncation error when a second-order formula is applied to pwl systems are given.

Numerical methods for initial value problems in ordinary differential equations : S.O. Fatunla: (Volume in Computer Science and Scientific Computing). Academic Press, San Diego, 1988, 295 pp., U.S. $44.50, ISBN 0-12-249930-1