Octavian **Agratini** (ICTP and Babeș-Bolyai University, Cluj-Napoca)

Uniform approximation by classes of linear processes

Sebastian **Anița** (Al. I. Cuza University, Iași)

Regional control of an alien predator population

Constantin **Băcuță** (University of Delaware, USA)

Discretization and Preconditioning for Singularly Perturbed Problems

Lori **Badea** (IMAR – Romanian Academy, Bucharest)

Convergence of the Damped Additive Schwarz Methods and the Subdomain Coloring

Radu **Bălan** (University of Maryland, USA)

Among infinitely many factorizations \(A=V\cdot V^\ast\) of a psd matrix \(A\), we seek the factor \(V\) that has the smallest \((1,2)\) norm.

In this talk we review the origin of this problem as well as existing results regarding the optimal value.

We discuss also the conjecture that the squared \((1,2)\) norm of \(V\) is equivalent to the \((1,1)\) norm of \(A\).

Beniamin **Bogoșel** (École Polytechnique, Palaiseau, France)

It has been conjectured by Pólya and Szegö in 1951 that among \(n\)-gons with fixed area the regular one minimizes the first eigenvalue of the Dirichlet-Laplace operator. Despite its apparent simplicity, this result has only been proved for triangles and quadrilaterals. In this work we show that the proof of the conjecture can be reduced to finitely many certified numerical computations. Moreover, the local minimality of the regular polygon is reduced to a single validated numerical computation.

The steps of the proof strategy include the analytic computation of the Hessian matrix of the first eigenvalue, the stability of the Hessian with respect to vertex perturbations and analytic upper bounds for the diameter of an optimal set. Explicit a priori error estimates are given for the finite element computation of the eigenvalues of the Hessian matrix of the first eigenvalue associated to the regular polygon.

Results presented are obtained in collaboration with Dorin Bucur.

Liliana **Borcea** (University of Michigan, USA)

A data driven reduced order model approach to inverse scattering with sensor arrays

Imre **Boros** (ICTP, Cluj-Napoca)

On the Mathematics of Neural Networks

Radu **Boţ** (University of Vienna, Austria)

Fast Optimistic Gradient Descent Ascent method in continuous and discrete time

Renata **Bunoiu** (University of Lorraine, France)

We present a method for the localization of solutions for a class of nonlinear problems arising in periodic homogenization. This method combines concepts and results from the linear theory of PDEs, linear periodic homogenization theory, and nonlinear functional analysis. In particular, we use the Moser-Harnack inequality, arguments of fixed point theory and Ekeland’s variational principle. A significant gain in the homogenization theory of nonlinear problems is that our method makes possible the emergence of finitely or infinitely many solutions. This study is motivated by real-world applications in physics, engineering and biology.

Emil **Cătinaş** (ICTP, Cluj-Napoca)

On the C-order of convergence

Teodora **Cătinaş** (Babeş-Bolyai University, Cluj-Napoca)

On some operators for interpolating scattered data

Radu **Cîmpeanu** (University of Warwick, UK)

The canonical framework of drop impact provides excellent opportunities to co-develop experimental, analytical and computational techniques in a rich multi-scale context. The talk will represent a journey across parameter space, as we address beautiful phenomena such as bouncing, coalescence and splashing, with a particular focus on scientific computing aspects and associated numerical methods.

We then shift gears towards the much more violent regime of high-speed impact resulting in splashing, where a combination of matched asymptotic expansions grounded in Wagner theory and DNS allow us to produce theoretical predictions for the location and velocity of the ejected liquid jet, as well as its thickness (Cimpeanu and Moore, JFM 856, 2019). While the early-time analytical methodology neglects effects such as surface tension or viscosity (focusing on inertia instead), corrections and adaptations of the technique (Moore et al., JFM 882, 2020) will also be presented and validated against an associated computational framework, bringing us even closer to efficiently providing information of interest for applications such as inkjet printing and pesticide distribution.

Nicolae **Cîndea** (University of Clermont Auvergne, France)

We study the boundary controllability of the linear elasticity system reformulated as a first-order system in both space and time. Using the observability inequality known for the usual second-order elasticity system, we deduce an equivalent observability inequality for the associated first-order system. Then, the control of minimal \(L^2\)-norm can be found as the solution to a space-time mixed formulation. This first-order framework is particularly interesting from a numerical perspective since it is possible to solve the space-time mixed formulation using only piecewise linear \(C^0\)-finite elements. Numerical simulations illustrate the theoretical results.

These results are obtained in a joint work with Arthur Bottois.

Maria **Crăciun** (ICTP, Cluj-Napoca)

On the analysis and modelling of some time series from astronomy and astrophysics

Dan **Crișan** (Imperial College, UK)

This talk covers some recent work on developing particle filters based data assimilation methodology for high dimensional fluid dynamics models.

The algorithm presented here is a particle filter with a so-called ”nudging” mechanism. The nudging procedure is used in the prediction step. In the absence of nudging, the particles have trajectories that are independent solutions of the model equations. The nudging presented here consists in adding a drift to the trajectories of the particles with the aim of maximising the likelihood of their positions given the observation data. This introduces a bias in the system that is corrected during the resampling step. The nudging procedure is theoretically justified through a standard convergence argument.

The corresponding Data Assimilation algorithm presented gives an asymptotically (as the number of particles increases) consistent approximation of the posterior distribution of the state given the data. The methodology is tested on a two-layer quasi-geostrophic model for a beta-plane channel flow with \({\mathcal O}(10^6)\) degrees of freedom out of which only a minute fraction are noisily observed. I will present the effect of the nudging procedure on the performance of the data assimilation procedure for a reduced model in terms of the accuracy and uncertainty of the results. The results presented here are incorporated in [1] and [2]. The talk is based on the papers:

[1] C Cotter, D Crisan, D Holm, W Pan, I Shevchenko, Data assimilation for a quasi-geostrophic model with circulation-preserving stochastic transport noise, Journal of Statistical Physics, 1-36, 2020.

[2] D Crisan, I Shevchenko, Particle filters with nudging, work in progress.

Victoriţa **Dolean Maini** (University of Strathclyde, UK)

Robust solvers for time-harmonic wave propagation problems

Ionuț **Farcaș** (University of Texas at Austin, USA)

We present a method for enhancing data-driven reduced-order modeling with a preprocessing step in which the training data are filtered prior to training the reduced model. Filtering the data prior to training has a number of benefits for data-driven modeling: it attenuates (or even eliminates) wavenumber or frequency content that would otherwise be difficult or impossible to capture with the reduced model, it smoothens discontinuities in the data that would be difficult to capture in a low-dimensional representation, and it reduces noise in the data. This makes the reduced modeling learning task numerically better conditioned, less sensitive to numerical errors in the training data, and less prone to overfitting when the amount of training data is limited.

We first illustrate the effects of filtering in one-dimensional advection and inviscid Burgers’ equations. We then consider large-scale rotating detonation rocket engine simulations with millions of spatial degrees of freedom for which only a few hundred down-sampled training snapshots are available. A reduced-order model is derived from these snapshots using operator inference. Our results indicate the potential benefits of filtering to reduce overfitting, which is particularly important for complex physical systems where the amount of training data is limited.

Distribution Statement A: Approved for Public Release; Distribution is Unlimited. PA# AFRL-2022-4312

Silviu **Filip** (INRIA Rennes, France)

Sorin **Gal**, Ionuț **Iancu **(University of Oradea)

The main aim of this talk is to introduce mixed operators between Choquet integral operators and max-product operators of Bernstein-Kantorovich type and to study their quantitative approximation properties.

We show that for large classes of functions, these nonlinear max-product Kantorovich-Choquet type operators give essentially better approximation orders than their linear classical correspondents.

Călin-Ioan **Gheorghiu** (ICTP, Cluj-Napoca)

Chebyshev Collocation Solutions to the Liouville-Bratu-Gelfand Problem

Ion Victor **Goșea** (Max Planck Institute, Magdeburg, Germany)

Data-driven discovery of reduced-order dynamical models from frequency-response measurements

Sever **Hîrștoagă** (INRIA Paris, France)

We solve numerically multi-scale in time Vlasov-type models, by using a specific version of the parareal algorithm. More precisely we use for the coarse solving reduced models obtained from the two-scale asymptotic expansion method. The reduced models are useful to approximate the original Vlasov model at a low computational cost since they are free of high oscillations. We illustrate this strategy with numerical experiments based on long time simulations of charged particle beams in an electromagnetic field. We provide an analysis of the efficiency of the parareal algorithm in terms of speedup.

Liviu **Ignat** (IMAR – Romanian Academy, Bucharest)

Large-time asymptotics of some numerical schemes for Burgers’ like equations

Traian **Iliescu** (Virginia Tech, USA)

Reduced Order Models for Turbulent Flows

Stelian **Ion** (ISMMA – Romanian Academy, Bucharest)

Large and Noisy Data Interpolation

Petru **Jebelean** (West University of Timișoara)

Odd perturbations of the relativistic operator

Nataliia **Kolun** (Odessa Military Academy, Ukraine & Babeș-Bolyai University)

Localization of positive solutions for stationary equations and systems of Kirchhoff type

Cătălin **Lefter** (Al.I. Cuza University and Octav Mayer Institute, Iași)

We consider systems of parabolic equations coupled in zero order terms.

Controllability problems with controls acting only in part of the equations of the system, as well as inverse source problems through observations only on some components of the system use as essential ingredients appropriate Carleman estimates.

In our talk we discuss various aspects concerning the deduction and the use of these estimates.

Daniel **Loghin** (University of Birmingham, UK)

Large scale problems are commonly solved using iterative methods. These methods are usually combined with preconditioning techniques aimed at rendering the solver performance optimal, i.e., independent of problem size, possibly also independent of other problem parameters. In the case of problems arising from the discretisation of PDE, the design of an efficient preconditioner is linked to the choice of partial differential operator. In particular, a suitable inclusion of the boundary operator in the preconditioning technique is essential. While this is well understood for simple (scalar) PDE, for complex applications this is not always a straightforward task. This is the case of, for example, boundary control problems or problems coupled at an interface.

In this talk I will discuss some classes of problems where a suitable choice of boundary preconditioner ensures the optimal performance of the iterative solver. Analysis will be presented, together with validating numerical experiments.

Andra **Malina** (ICTP, Cluj-Napoca)

New Shepard operators in the univariate case

Liviu **Marin** (University of Bucharest and ISMMA – Romanian Academy)

Stable numerical reconstruction of solutions to the Cauchy problem in anisotropic heat conduction with non-smooth coefficients

Sorin **Micu** (University of Craiova and ISMMA – Romanian Academy)

Control and its approximation

Cornel **Murea** (Université de Haute Alsace, France)

This is a joint work with Dan Tiba (Institute of Mathematics, Romanian Academy, dan.tiba@imar.ro).

We study the penalized steady Navier-Stokes with Neumann boundary conditions system in a holdall domain, its approximation properties (with error estimates) and the uniqueness of the solution that is obtained in a non standard manner. Numerical tests are presented.

Mihai **Nechita** (ICTP, Cluj-Napoca)

We consider the unique continuation problem for a stationary convection–diffusion equation, with data given in an interior subset of the domain and no boundary conditions.

For this ill-posed problem, we first discuss conditional stability estimates that are explicit in the physical parameters.

Casting the problem as PDE-constrained optimisation, we present a finite element method based on a discretise-then-regularise approach. The regularisation is based on penalising the jumps of the gradient across the interior faces of the finite element triangulation.

When diffusion dominates, we prove convergence rates by applying the continuum stability estimates to the approximation error and controlling the residual through stabilisation.

When convection dominates, we perform a local analysis and obtain weighted error estimates with quasi-optimal convergence along the characteristics of the convective field through the data set.

The talk is based on the papers:

[1] E. Burman, M. Nechita, L. Oksanen, *A stabilized finite element method for inverse problems subject to the convection–diffusion equation. I: diffusion-dominated regime*, Numer. Math., 144:451–477, 2020.

[2] E. Burman, M. Nechita, L. Oksanen, *A stabilized finite element method for inverse problems subject to the convection–diffusion equation. II: convection-dominated regime*, Numer. Math., 150:769–801, 2022.

Ion **Necoară** (Politehnica University, Bucharest)

Efficiency of coordinate descent beyond separability and smoothness

Claudia **Negulescu** (Université Paul Sabatier, Toulouse, France)

Fokker-Planck multi-species equations in the adiabatic asymptotics

Maria **Neuss-Radu** (Friedrich Alexander Universitaet, Erlangen-Nuernberg, Germany)

We derive an effective model for transport processes in periodically perforated elastic media, taking into account, e.g., cyclic elastic deformations as they occur in lung tissue due to respiratory movement. The underlying microscopic problem couples the deformation of the domain with a diffusion process within a mixed Lagrangian/Eulerian formulation. After a transformation of the diffusion problem onto the fixed domain, we use the formal method of two-scale asymptotic expansion to derive the upscaled model, which is nonlinearly coupled through effective coefficients.

The effective model is implemented and validated using an application-inspired model problem. Numerical solutions for both, cell problems and macroscopic equations, are investigated and interpreted. We use simulations to qualitatively determine the effect of the deformation on the transport process.

This research is supported by SCIDATOS (Scientific Computing for Improved Detection and Therapy of Sepsis), a collaborative project funded by the Klaus Tschira Foundation, Germany. The results are obtained in collaboration with J. Knoch and N. Neuss (FAU Erlangen-Nürnberg) and M. Gahn (University of Heidelberg).

Constantin P. **Niculescu** (University of Craiova)

The Hornich-Hlawka functional inequality in the framework of n-convex functions

Diana **Otrocol** (ICTP and Technical University of Cluj-Napoca)

Applications of fibre contraction principle

Cosmin **Petra** (Lawrence Livermore National Laboratory, USA)

Mathematical optimization of complex energy systems using high-performance computing

Sorin **Pop** (Hasselt University, Belgium)

Dimensionality reduction for unsaturated flow in fractured porous media

Radu **Precup** (ICTP and Babeș-Bolyai University, Cluj-Napoca)

An algorithm of solving general control problems with application to a medical model

Florin **Radu** (University of Bergen, Norway)

Iterative schemes for variational phase-field models of brittle fracture

Vicenţiu **Rădulescu** (University of Craiova and IMAR – Romanian Academy)

In this talk, I shall report on some recent results obtained jointly with N. Papageorgiou, C. Alves and D. Repovs. In the first part, I will discuss an interesting discontinuity property of the spectrum in the case of isotropic double phase equations with Dirichlet boundary condition. Next, I shall consider the anisotropic setting and I will discuss three sufficient conditions for the existence of solutions in the case of double phase with multiple “subcritical-critical-supercritical” regimes.

Adrian **Sandu** (Virginia Tech, USA)

Numerical integration of differential equations modeling multiphysics phenomena

Andrei **Stan** (ICTP, Cluj-Napoca)

On approximating the convergence orders

Nicolae **Suciu** (ICTP, Cluj-Napoca)

Three issues of numerical modeling of reactive transport in porous media

Alexandru **Tămășan** (University of Central Florida, Orlando, Florida, USA)

The range of the X-ray transform of symmetric tensors play an important role in Computerized Tomography (0-order tensors) for data denoising, data completion, or hardware failure and calibration problems. The X-ray for higher order tensors also appears in Doppler Tomography (1st-order tensors) and Seismology (2nd-order tensors). With lines parametrized as points on the torus, the X-ray data can be understood and a function on the torus.

I will present new necessary and sufficient constraints of the X-ray data in terms of the Fourier coefficients on the pairs of integers lattice. The derivation uses the theory of A-analytic maps.