**Preface to "Numerical Linear Algebra and the Applications"**

This Special Issue named Numerical Linear Algebra with Applications is celebrating the **98th** birthday of the Greek mathematician Mr. Constantin M. Petridi, wishing him a long and happy life. The aim of this issue in the journal *Mathematics* was to invite some colleagues to submit their new and high-quality work related to numerical linear algebra and applications in different and modern fields such as machine learning and others.

This Special Issue was edited by Professors Marilena Mitrouli from the National and Kapodistrian University of Athens in Greece and Jbilou from the University of Littoral d'Opale in France.

As this Special Issue was dedicated to Mr. Constantin M. Petridi, we also present, here, a short CV written by Mr. Constantin M. Petridi himself.

#### **CV of Mr. Constantin M. Petridi**

Born 1923, Athens, Greece.

Father: Milton C. Petridi, heir to an internationally known tobacco company founded in 1848 in Constantinople (now Istanbul).

Mother: Nina C. Petridi, born Fauqier, stemming from a British colony, Corfu, Greece.

1952 Married Lisa Skouze, daughter of Dimitri Skouzes and Athina Skouze. Have one son, Milton C.Petridi. 1955–1972 I was consul for Sweden at Kavala, Greece, the location of my family's tobacco business. Languages: Fluent in English, French, German, and Swedish.

## **My Mathematical Life**

After graduating from Athens College, Athens, Greece, I studied Mathematics at Stockholm University, graduating with an M.Sc.

My professors were


1947–1949: I was an Assistant to both chairs of Mathematics at the Royal Technical University of Stockholm. My best friend, at that time, was fellow student Tord Ganelius.

1948: A Swedish business of my father, who knew the world renowned Hungarian Mathematician Prof. Marcel Riesz, invited the latter to examine my mathematical background and research interests. The result: Marcel Riesz invited me to become his student at Lund University, Sweden.

For better or for worse, I cannot say, I decided to remain in Stockholm.

End of 1948: I gave a manuscript to Prof. Carlson, titled *Mathematical Structures defined by Identities*. Carlson sent it to Prof. Trygve Nagell, Oslo University, as a greater expert on such matters. In reply, Nagell, in a long letter, said that my "Basic ideas undoubtedly opened new perspectives

for Mathematics". He suggested, for example, that I axiomatize trigonometry (Fourier coefficient formulas).

Early 1949: I returned to Greece to take up the business of my father, who, in the meantime, had died.

A several-decades-long "Mathematical" interruption followed. I continued, however, my mathematical research, whenever I had time, and read mathematical reviews for recent developments.

1982: I left the tobacco business to devote myself, completely again, to mathematics.

1996: My old friend Tord Galenius, who, by then, was a permanent secretary of the Swedish Academy of Sciences (Crafoord Prizes), introduced me to Prof. Torsten Ekedahl, University of Stockholm. I had Q4 papers, on which I had written that the number In of irreducible identities (algebras) was equal to

$$\begin{aligned} I\_n &= \left[ \frac{n+1}{2} \right]\_{-1}^{-1} \\ I\_n &= \sum\_{k=1}^{\infty} \quad (-1)^{k-1} \binom{n-k+1}{k} S\_{n-k}^2, \end{aligned}$$

where Si := are the Catalan numbers 1 i + 1 2i i .

Ekedahl praised my findings, telling me to send him the LaTeX manuscript, after my return to Greece. Unfortunately, my hopes were dashed, as Ekedahl died some time later, aged 54. The Mathematical Community acclaimed his achievements, including Jean-Pierre Serre.

2001: I started to publish my hitherto-discovered mathematical findings as ArXiv preprints. Peter Krikelis, Assistant, Athens University, wrote them in LaTeX form. The number of downloads (reads) was great.

My interest in mathematics started early in childhood, which I think is due to the **unexpected results it yields and its elegance**. I remember, for example, how awe-stricken I was, when reading, as a child, that the discriminant in a formula of the Fibonacci numbers is an integer.

Besides the above, I am, however, also interested in the work of other mathematicians, looking up in Wikipedia, etc., their published results.

In conclusion, as many of my papers are on number theory, I quote one of the giants of the past: Carl Fridrich Gauss: *Mathematics is the Queen of Sciences and Number Theory is the Queen of Mathematics*.

> **Khalide Jbilou, Marilena Mitrouli** *Editors*

## *Article* **Solving High-Dimensional Problems in Statistical Modelling: A Comparative Study †**

**Stamatis Choudalakis ‡, Marilena Mitrouli ‡, Athanasios Polychronou ‡ and Paraskevi Roupa \*,‡**

> Department of Mathematics, National and Kapodistrian University of Athens, Panepistimiopolis, 15784 Athens, Greece; stchoud@math.uoa.gr (S.C.); mmitroul@math.uoa.gr (M.M.); apolychronou@math.uoa.gr(A.P.)

**\*** Correspondence: parask\_roupa@math.uoa.gr

 † This paper is dedicated to Mr. Constantin M. Petridi.

‡ These authors contributed equally to this work.

**Abstract:** In this work, we present numerical methods appropriate for parameter estimation in high-dimensional statistical modelling. The solution of these problems is not unique and a crucial question arises regarding the way that a solution can be found. A common choice is to keep the corresponding solution with the minimum norm. There are cases in which this solution is not adequate and regularisation techniques have to be considered. We classify specific cases for which regularisation is required or not. We present a thorough comparison among existing methods for both estimating the coefficients of the model which corresponds to design matrices with correlated covariates and for variable selection for supersaturated designs. An extensive analysis for the properties of design matrices with correlated covariates is given. Numerical results for simulated and real data are presented.

**Keywords:** high-dimensional; minimum norm solution; regularisation; Tikhonov; *p*-*q*; variable selection
