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Abstract: In this paper we present a technique for shape reconstruction based on the topological and shape gradients. The shape in consideration is a solution of an inverse conductivity problem. To solve such a problem numerically, we compute the topological gradient of a Kohn–Vogelius-type cost function when the domain under consideration is perturbed by the introduction of a small inclusion instead of a hole. The reconstruction is done by considering the shape as a superposition of very thin elliptic inclusions to get a first approximation. Then, we use a gradient-type algorithm to perform a good reconstruction. Various numerical experiments of single and multiple inclusions demonstrate the viability of the designed algorithm. [Copyright &y& Elsevier]

Abstract: This paper introduces a fast algorithm for computing multilinear integrals which are defined through Fourier multipliers. The algorithm is based on generating a hierarchical decomposition of the summation domain into squares, constructing a low-rank approximation for the multiplier function within each square, and applying an FFT based fast convolution algorithm for the computation associated with each square. The resulting algorithm is accurate and has a linear complexity, up to logarithmic factors, with respect to the number of the unknowns in the input functions. Numerical results are presented to demonstrate the properties of this algorithm. [Copyright &y& Elsevier]

Abstract: In this paper we describe how an idea centered on the concept of self-saturation allows several improvements in the computation of Gröbner bases via Buchberger’s Algorithm. In a nutshell, the idea is to extend the advantages of computing with homogeneous polynomials or vectors to the general case. When the input data are not homogeneous, we use as a main tool the procedure of a self-saturating Buchberger’s Algorithm. Another strictly related topic is treated later when a mathematical foundation is given to the sugar trick which is nowadays widely used in most of the implementations of Buchberger’s Algorithm. A special emphasis is also given to the case of a single grading, and subsequently some timings and indicators showing the practical merits of our approach. [Copyright &y& Elsevier]

Abstract: A recurring task in image processing, approximation theory, and the numerical solution of partial differential equations is to reconstruct a piecewise-smooth real-valued function , where , from its truncated Fourier transform (its truncated spectrum). An essential step is edge detection for which a variety of one-dimensional schemes have been developed over the last few decades. Most higher-dimensional edge detection algorithms consist of applying one-dimensional detectors in each component direction in order to recover the locations in where is singular (the singular support). In this paper, we present a multidimensional algorithm which identifies the wavefront of a function from spectral data. The wavefront of f is the set of points which encode both the location of the singular points of a function and the orientation of the singularities. (Here denotes the unit sphere in N dimensions.) More precisely, is the direction of the normal line to the curve or surface of discontinuity at x. Note that the singular support is simply the projection of the wavefront onto its x-component. In one dimension, the wavefront is a subset of , and it coincides with the singular support. In higher dimensions, geometry comes into play and they are distinct. We discuss the advantages of wavefront reconstruction and indicate how it can be used for segmentation in magnetic resonance imaging (MRI). [ABSTRACT FROM AUTHOR]

Abstract: This work is concerned with several properties of solutions of stochastic differential equations arising from hybrid switching diffusions. The word “hybrid” highlights the coexistence of continuous dynamics and discrete events. The underlying process has two components. One component describes the continuous dynamics, whereas the other is a switching process representing discrete events. One of the main features is the switching component depending on the continuous dynamics. In this paper, weak continuity is proved first. Then continuous and smooth dependence on initial data are demonstrated. In addition, it is shown that certain functions of the solutions verify a system of Kolmogorov''s backward differential equations. Moreover, rates of convergence of numerical approximation algorithms are dealt with. [Copyright &y& Elsevier]

Abstract: The Buchberger–Möller algorithm is a well-known efficient tool for computing the vanishing ideal of a finite set of points. If the coordinates of the points are (imprecise) measured data, the resulting Gröbner basis is numerically unstable. In this paper we introduce a numerically stable Approximate Vanishing Ideal (AVI) Algorithm which computes a set of polynomials that almost vanish at the given points and almost form a border basis. Moreover, we provide a modification of this algorithm which produces a Macaulay basis of an approximate vanishing ideal. We also generalize the Border Basis Algorithm ([Kehrein, A., Kreuzer, M., 2006. Computing border bases. J. Pure Appl. Algebra 205, 279–295]) to the approximate setting and study the approximate membership problem for zero-dimensional polynomial ideals. The algorithms are then applied to actual industrial problems. [Copyright &y& Elsevier]

Abstract: Regularized classifiers (a leading example is support vector machine) are known to be a kind of kernel-based classification methods generated from Tikhonov regularization schemes, and the polynomial kernels are the original and also probably the most important kernels used in them. In this paper, we provide an error analysis for the regularized classifiers using multivariate polynomial kernels. We introduce Bernstein–Durrmeyer polynomials, whose reproducing kernel Hilbert space norms and approximation properties in space play a key role in the analysis of regularization error. We also introduce the standard estimation of sample error, and derive explicit learning rates for these algorithms. [Copyright &y& Elsevier]