The starting point is a generalization of Szász-Mirakyan operators in the Durrmeyer sense. Over time, various polynomial weights have been used such as Jacobi or Appell functions. The last class is comprehensive containing as special classes a large number of sequences as for example Bernoulli, Euler, Hermite polynomials. Following a construction defined in 2025 by P. G. Patel, in this talk we study a family of operators that depends on two parameters and involves associated Laguerre – type kernels ꞷ_{α,β}(t) = e^{βt}t^α, α > -1, β < 0. The weight ensures that the sequence of operators converges for a wide class of functions acting on [0, ∞) and preserves the positivity of the operators. We investigate in detail approximation properties of the new class. The main objectives aimed to establish the moments of these operators, to derive their recurrence relations in differential form and to investigate their convergence in suitable weighted spaces. Both quantitative estimate on compact intervals and weighted approximation for unbounded functions having a polynomial growth are highlighted. A distinct section is focused on numerical application. The experiments are carried out for various functions confirming that the new operators provide stable and convergent approximations with errors consistently decreasing as the discretization parameter increases.
Joint work with Maria Crăciun.

For a model convection-diffusion problem, we address the presence of oscillatory discrete solutions using the relation between the exact solution and the two solutions of the related transport problems. We identify difficulties in recovering standard approximation results for the exact solution in global energy norms. We justify the presence of non-physical oscillations and propose ways to eliminate oscillations. One important idea is that oscillation free discretization is related to approximation of the exact solution in the discrete infinity norm or the closeness property. We present a bubble upwinding discretization of a two dimensional convection diffusion problem using the efficient discretization of the corresponding one dimensional problem along each stream line. Our results are essential to building new and robust discretizations for multi-dimensional convection dominated problems.

We study finite sets in dimension three realizing the maximum number of diameters. These sets provide a framework for constructing a dense family of finite parameter constant width shapes that are useful for the study of shape optimization problems. In this context it is natural to optimize functionals depending on extremal diameter graphs in dimension three, which we develop in this project. We provide a numerical framework together with computational results and conjectures that provide a rich basis for future work. In particular the three dimensional Blaschke-Lebesgue problem enters into this framework.

The aim of this talk is to present a mathematical model of gas-bubble formation in magmas. As a first step, we identify a binodal curve and then solve a degenerate elliptic system. Numerical simulations are presented to illustrate the model.

ASTERIX is a software package based on an integrated mathematical model that combines an extended Saint-Venant system of equations for water flow with a Hairsine–Rose-type model for erosion processes on plant-covered soil surfaces. Despite the complexity of the non-linear hyperbolic system of PDEs on which ASTERIX is based, we identify several configurations of soil-surface geometry and vegetation-density distributions for which analytical solutions can be obtained. These solutions provide useful benchmarks for assessing the performance of the numerical model implemented in ASTERIX. We present a series of numerical experiments aimed at validating both the continuous mathematical model and its discrete numerical formulation, and illustrate the capabilities of ASTERIX for simulating water flow and soil erosion under different vegetation and surface configurations.

We investigate in detail approximation properties of the new class. The main objectives aimed to establish the moments of these operators, to derive their recurrence relations in differential form and to investigate their convergence in suitable weighted spaces. Both quantitative estimate on compact intervals and weighted approximation for unbounded functions having a polynomial growth are highlighted. A distinct section is focused on numerical application. The experiments are carried out for various functions confirming that the new operators provide stable and convergent approximations with errors consistently decreasing as the discretization parameter increases.
Joint work with Octavian Agratini

Joint work with Alexandru Pop (Astronomical Observatory Cluj-Napoca, Astronomical Institute of the Romanian Academy, Romania).

Introduced by Polyak in 1966, the class of strongly quasiconvex functions includes some interesting nonconvex members, like the square root of the Euclidean norm or ratios with a nonnegative strongly convex numerator and a concave and positive denominator. In this talk we survey the most relevant examples of strongly quasiconvex functions and results involving them available in the literature at the moment. In particular, we recall some recent algorithms for minimizing such functions, and hint toward some directions where additional investigations would be welcome.

In this talk we consider some well known quotients related with either eigenvalue problems, Sobolev or Hardy’s inequality. We consider the infimum of these quotients and their discrete analogous in a finite element subspace. We estimate the difference between the best constants above as the discretization parameter goes to zero and obtain sharp convergence rates. This is joint work with Andreea Dima and Enrique Zuazua.

Spatial filters have played a central role in large eddy simulation and reduced order model (ROM) stabilization. We propose a novel strategy that is fundamentally different from current filter-based stabilizations and closures: The traditional spatial filters are replaced with a novel data-driven stabilization operator (StabOp) that yields the most accurate results for a given resolution, quantity of interest, and stabilization strategy. Although the new StabOp is general, for clarity, it is investigated for the Leray ROM (L-ROM) stabilization. To build StabOp, its model form is first postulated to be a linear mapping, a quadratic mapping, or a nonlinear neural network mapping, and then a PDE-constrained optimization problem is solved to minimize the given loss function. Using the resulting StabOp in the given L-ROM yields a new stabilized ROM, StabOp-L-ROM. The new StabOp-L-ROM is compared with the L-ROM in the numerical simulation of under-resolved, convection-dominated flows, such as the minimal channel flow at Re=5000. The numerical results show that StabOp-L-ROM can be orders-of-magnitude more accurate than the classical L-ROM tuned with the optimal filter radius in the predictive regime. Furthermore, while StabOp smooths the input flow fields, its smoothing mechanism is entirely different from those of classical spatial filters.

We explain how to obtain good approximations of the fundamental solution of a (uniformly) parabolic PDEs and how to use it to devise fast numerical methods for parabolic equations.

Sequential Quadratic Programming is the standard Newton-type approach to solving a constraind optimization problem. In order to avoid the cost of solving a quadratic program, we propose an alternative formulation, based on replacing quadratic programs with a simple iterative linear equation solver. The proposed algorithm is based on a splitting scheme, which first produces an affine approximation of the constraints, and then uses an unconstrained Newton method to optimize the objective over this lower dimensional space.

We consider a general Bazykin-type model that describes the interaction of two populations. We provide sufficient conditions for the existence of a positive periodic solution where each population is localized independently within a distinct conical annular region. We illustrate how our conditions can be verified for certain particular cases and support these results with numerical simulations.

Training generative models on prior synthetic data leads to model collapse. We analyze finite-budget training pipelines and compare “total replacement” protocols (which cause rapid collapse) with “fixed-budget” sampling from a growing pool of real and synthetic data. Experiments on synthetic and image benchmarks (MNIST, Fashion-MNIST, CIFAR-10) reveal that while replacement causes severe degradation, fixed-budget sampling results in partial degradation: fragile features are lost, but robust features are preserved. Stochastic recursion analysis confirms these dynamics, challenging the inevitability of collapse and suggesting sustainable reuse of synthetic data is feasible.