Stefano De Marchi

Last updated

Stefano De Marchi (born 17 December 1962 in Candiana, Padua) is an Italian mathematician who works in numerical analysis and is a professor at the University of Padua. He is managing editor of the open access journal Dolomites Research Notes on Approximation published by the Padua University Press, coordinator of the Constructive Approximation and Applications Research Group, coordinator of the Research Italian network on Approximation, and responsible for the Unione Matematica Italiana Thematic Group on "Approximation Theory and Applications (A.T.A.)".

Contents

His scientific interests deal mainly with interpolation and approximation of functions and data by polynomials and radial basis functions (RBFs)).

Education and career

Stefano De Marchi studied Bachelor's degree of Mathematics in 1981-1987, Master in Applied Mathematics in 1991 at the University of Padua, and received his doctorate in Computational Mathematics, Consorzio Nord-Orgientale, VI ciclo, University of Padua under Maria Morandi Cecchi and Larry Lee Schumaker supervisions (dissertation: Approssimazione e Interpolazione su "Simplices": Caratterizzazioni, Metodi ed Estensioni)

He habilitated in 2017 and became a Full Professor of Numerical Analysis at the Department of Mathematics “Tullio Levi-Civita”, University of Padua in 2022.

Recognition

Stefano has made many important contributions to approximation theory such as Weakly Admissible Meshes, Barycentric rational interpolation, Stability issues and greedy algorithms in RBF theory, Rational RBF approximation, Medical image reconstruction, and Fake nodes. [1] He is one of the discoverers of the so called Padua points, which are the only set of quasi-optimal interpolation points explicitly known on the square, for polynomial interpolation of total degree. Their name is due to the University of Padua, where they were originally discovered. [2] [3] He is also author of the books: ′′Funzioni Splines Univariate″, [4]  ′′Appunti di Calcolo Numerico″, [5] ′′Meshfree Approximation for Multi-Asset European and American Option Problems″ [6] and the Lecture notes: ′′Four lectures on radial basis functions″ and '′Lectures on multivariate polynomial interpolation″. [7]

Related Research Articles

<span class="mw-page-title-main">Interpolation</span> Method for estimating new data within known data points

In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points.

<span class="mw-page-title-main">B-spline</span> Spline function

In the mathematical subfield of numerical analysis, a B-spline or basis spline is a spline function that has minimal support with respect to a given degree, smoothness, and domain partition. Any spline function of given degree can be expressed as a linear combination of B-splines of that degree. Cardinal B-splines have knots that are equidistant from each other. B-splines can be used for curve-fitting and numerical differentiation of experimental data.

In mathematics, a polynomial is a mathematical expression consisting of indeterminates and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate x is x2 − 4x + 7. An example with three indeterminates is x3 + 2xyz2yz + 1.

<span class="mw-page-title-main">Linear interpolation</span> Method of curve fitting to construct new data points within the range of known data points

In mathematics, linear interpolation is a method of curve fitting using linear polynomials to construct new data points within the range of a discrete set of known data points.

<span class="mw-page-title-main">Runge's phenomenon</span> Failure of convergence in interpolation

In the mathematical field of numerical analysis, Runge's phenomenon is a problem of oscillation at the edges of an interval that occurs when using polynomial interpolation with polynomials of high degree over a set of equispaced interpolation points. It was discovered by Carl David Tolmé Runge (1901) when exploring the behavior of errors when using polynomial interpolation to approximate certain functions. The discovery shows that going to higher degrees does not always improve accuracy. The phenomenon is similar to the Gibbs phenomenon in Fourier series approximations.

<span class="mw-page-title-main">Time series</span> Sequence of data points over time

In mathematics, a time series is a series of data points indexed in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Examples of time series are heights of ocean tides, counts of sunspots, and the daily closing value of the Dow Jones Industrial Average.

<span class="mw-page-title-main">Isaac Jacob Schoenberg</span> Romanian-American mathematician (1903–1990)

Isaac Jacob Schoenberg was a Romanian-American mathematician, known for his invention of splines.

<span class="mw-page-title-main">Guido Castelnuovo</span> Italian mathematician (1865–1952)

Guido Castelnuovo was an Italian mathematician. He is best known for his contributions to the field of algebraic geometry, though his contributions to the study of statistics and probability theory are also significant.

In mathematics a radial basis function (RBF) is a real-valued function whose value depends only on the distance between the input and some fixed point, either the origin, so that , or some other fixed point , called a center, so that . Any function that satisfies the property is a radial function. The distance is usually Euclidean distance, although other metrics are sometimes used. They are often used as a collection which forms a basis for some function space of interest, hence the name.

In applied mathematics, polyharmonic splines are used for function approximation and data interpolation. They are very useful for interpolating and fitting scattered data in many dimensions. Special cases include thin plate splines and natural cubic splines in one dimension.

<span class="mw-page-title-main">Meshfree methods</span> Methods in numerical analysis not requiring knowledge of neighboring points

In the field of numerical analysis, meshfree methods are those that do not require connection between nodes of the simulation domain, i.e. a mesh, but are rather based on interaction of each node with all its neighbors. As a consequence, original extensive properties such as mass or kinetic energy are no longer assigned to mesh elements but rather to the single nodes. Meshfree methods enable the simulation of some otherwise difficult types of problems, at the cost of extra computing time and programming effort. The absence of a mesh allows Lagrangian simulations, in which the nodes can move according to the velocity field.

In numerical analysis, multivariate interpolation is interpolation on functions of more than one variable ; when the variates are spatial coordinates, it is also known as spatial interpolation.

In polynomial interpolation of two variables, the Padua points are the first known example of a unisolvent point set with minimal growth of their Lebesgue constant, proven to be . Their name is due to the University of Padua, where they were originally discovered.

<span class="mw-page-title-main">Joseph L. Walsh</span> American mathematician

Joseph Leonard Walsh was an American mathematician who worked mainly in the field of analysis. The Walsh function and the Walsh–Hadamard code are named after him. The Grace–Walsh–Szegő coincidence theorem is important in the study of the location of the zeros of multivariate polynomials.

Stephen R. Hilbert is an American mathematician best known as co-author of the Bramble–Hilbert lemma, which he published with James H. Bramble in 1970. Hilbert's area of specialty is numerical analysis. He has been a professor of mathematics at Ithaca College since 1968. Additionally, he taught mathematics at Cornell University as a visiting program professor during the 2003–2004 academic year.

The Kansa method is a computer method used to solve partial differential equations. Its main advantage is it is very easy to understand and program on a computer. It is much less complicated than the finite element method. Another advantage is it works well on multi variable problems. The finite element method is complicated when working with more than 3 space variables and time.

<span class="mw-page-title-main">Edward B. Saff</span> American mathematician

Edward Barry Saff is an American mathematician, specializing in complex analysis, approximation theory, numerical analysis, and potential theory.

Radial basis function (RBF) interpolation is an advanced method in approximation theory for constructing high-order accurate interpolants of unstructured data, possibly in high-dimensional spaces. The interpolant takes the form of a weighted sum of radial basis functions. RBF interpolation is a mesh-free method, meaning the nodes need not lie on a structured grid, and does not require the formation of a mesh. It is often spectrally accurate and stable for large numbers of nodes even in high dimensions.

Charles Anthony Micchelli is an American mathematician, with an international reputation in numerical analysis, approximation theory, and machine learning.

References

  1. Bos., Len; Sommariva, Alvise; Perracchione, Emma; Santin, Gabriele; Marchetti, Francesco; Erb, Wolfgang; Dell'Accio, Francesco; Elefante, Giacomo; Poggiali, Davide (December 2022). "Special Issue dedicated to Stefano De Marchi on the occasion of his 60th birthday". Dolomites Research Notes on Approximation. 15 (4).
  2. Bos, Len; Caliari, Marco; De Marchi, Stefano; Vianello, Marco; Xu, Yuan (2006). "Bivariate Lagrange interpolation at the Padua points: The generating curve approach". Journal of Approximation Theory. 143 (1): 15–25. doi:10.1016/j.jat.2006.03.008. hdl: 11577/2481660 .
  3. Caliari, Marco; De Marchi, Stefano; Vianello, Marco (2005). "Bivariate polynomial interpolation on the square at new nodal sets". Applied Mathematics and Computation. 165 (2): 261–274. doi:10.1016/j.amc.2004.07.001.
  4. De Marchi, Stefano (2000). Funzioni Spline Univariate (in Italian) (2nd ed.). Udine: Universitaria FORUM. p. 106.
  5. De Marchi, Stefano (2016). Appunti di Calcolo Numerico con codici in Matlab/Octave (in Italian) (2nd ed.). Bologna: Esculapio. p. 260. ISBN   9788874889396.
  6. De Marchi; Mandarà; Viero, Stefano; Maddalena; Anna (2012). Meshfree Approximation for Multi-Asset European and American Option Problems (1st ed.). ARACNE EDITRICE SRL. p. 92.{{cite book}}: CS1 maint: multiple names: authors list (link)
  7. De Marchi, Stefano (2013). Four lectures on radial basis functions. Department of Mathematics, University of Padua.