arXiv subjects

Arghir Zarnescu

Publications and source records attributed to Arghir Zarnescu.

At least 19 recordsLinked to original sources

arXiv2026

Complexity reduction of physical models: An equation-free approach by means of scaling

The description of complex physical phenomena often involves sophisticated models that rely on a large number of parameters, with many dimensions and scales. One practical way to simplify those kinds of models is to discard some of the parameters, or terms of underlying equations, thus giving rise to reduced models. Here, we propose a general approach to obtaining such reduced models. The method is independent of the model in use, i.e., equation-free, depends only on the interplay between the scales and dimensions involved in the description of the phenomena, and controls over-parametrization. It also quantifies conditions for asymptotic models by providing explicitly computable thresholds on values of parameters that allow for reducing complexity of a model, while preserving essential predictive properties. Although our focus is on complexity reduction, this approach may also help with calibration by mitigating the risks of over-parameterization and instability in parameter estimation. The benefits of this approach are discussed in the context of the classical projectile model.

cond-mat.soft↗

arXiv2026

From Monte Carlo to neural networks approximations of boundary value problems

In this paper we study probabilistic and neural network approximations for solutions to Poisson equation subject to Holder data in general bounded domains of $\mathbb{R}^d$. We aim at two fundamental goals. The first, and the most important, we show that the solution to Poisson equation can be numerically approximated in the sup-norm by Monte Carlo methods, and that this can be done highly efficiently if we use a modified version of the walk on spheres algorithm as an acceleration method. This provides estimates which are efficient with respect to the prescribed approximation error and with polynomial complexity in the dimension and the reciprocal of the error. A crucial feature is that the overall number of samples does not not depend on the point at which the approximation is performed. As a second goal, we show that the obtained Monte Carlo solver renders in a constructive way ReLU deep neural network (DNN) solutions to Poisson problem, whose sizes depend at most polynomialy in the dimension $d$ and in the desired error. In fact we show that the random DNN provides with high probability a small approximation error and low polynomial complexity in the dimension.

math.PR↗

arXiv2026

Extended Walk-on-Spheres Algorithm for Linear and Nonlinear Elliptic Problems of Divergence-type

The Walk-on-Spheres algorithm, introduced by M. E. Muller in 1956, is a well known Monte Carlo method that leverages Brownian exit distributions from spheres to solve the Laplace equation with Dirichlet boundary conditions. Its mesh-free nature, robustness on complex geometries, favorable scaling with dimension, and intrinsic parallelism distinguish it from mesh-based solvers. However, its efficient applicability has been essentially limited to operators that admit explicit probabilistic exit laws, excluding most variable-coefficient and nonlinear elliptic operators. We propose a general framework that aims to overcome this limitation by using the classical Dirichlet Laplacian and harmonic extension as universal building blocks. Rather than seeking a custom stochastic representation for each operator, we employ Walk-on-Spheres to precompute a reusable numerical operator toolbox that approximates the inverse Dirichlet Laplacian, the harmonic extension operator, and their gradients. These precomputed operators are then used to represent candidate solutions and to transform arbitrary Dirichlet boundary value problems into a finite-dimensional algebraic system/optimization problem for an unknown source term. Solving the resulting algebraic system/optimization problem and substituting back yields an approximate solution to the original PDE. Even more, for a general linear second order elliptic operator, the above mentioned precomputed toolbox can be directly used to obtain not just an approximation of a certain solution of the corresponding generalized Dirichlet problem, but an estimator of both the Green's integral operator and the elliptic measure operator. Numerical experiments on a range of benchmarks, including non-symmetric and anisotropic linear elliptic equations, semilinear and quasilinear problems, demonstrate the method's flexibility and efficiency.

math.NA↗

arXiv2026

Variational principles for the interaction of liquid crystals and electric fields in the Oseen--Frank model

We develop a rigorous variational framework for uniaxial nematic liquid crystals interacting with an external electric field in the one-constant Oseen--Frank approximation. Equilibrium configurations are governed by a nonlocal, nonlinear energy with a challenging min-max saddle-point structure. Our first main result reformulates this problem as a pure double-minimization problem. Using convex duality and the Hodge decomposition, we replace the scalar electrostatic potential with a vector potential, yielding a closed-form dual functional with a unique minimizer. This direct energy-minimization principle is advantageous for both theoretical analysis and numerical simulation. Our second result rigorously quantifies the decoupling of the electrostatic back-reaction in the limit of small dielectric anisotropy. We establish a uniform quadratic energy bound between the exact nonlocal energy and its standard local approximation, formally justifying the widespread physics heuristic of neglecting induced depolarization fields. Finally, under a strict coercivity condition, we combine $Γ$-convergence, uniform Sobolev regularity, and a perturbative coercivity transfer to prove that physical minimizers converge to limiting harmonic maps at an optimal, quantitative linear rate.

math.AP↗

arXiv2026

On the impact of clusters of rigid balls on the motion of a viscous fluid

We develop a new approach to the problem of the motion of a large number of rigid bodies immersed in a viscous fluid. The leading idea is the concept of cluster - a collection of individual rigid objects that may be grouped or even connected in such a way that their collective impact on the bulk motion of the system is similar to that of a single body. The applications of the new approach include: 1. Improving the critical value of the number of balls of small radius such that their cloud has no impact on the limit system represented by the incompressible Navier--Stokes equations. 2. The balls follow the fluid flow in the asymptotic limit of vanishing radius and increasing number even if a gravitational force is imposed.

math.AP↗

arXiv2026

The Ginzburg-Landau system with general potential: maximum principle and gradient estimates

We study critical points of the Ginzburg-Landau energy functional $$\mathcal{F}_\varepsilon[u] = \int_Ω\Big[ \frac{1}{2}|\nabla u|^2 + \frac{1}{2\varepsilon^2} W(1 - |u|^2)\Big]\,dx, \quad u \in H^1(Ω, \mathbb{R}^N),$$ with $Ω\subset \mathbb{R}^M$, $\varepsilon>0$, $M,N \geq 2$ and general conditions on the non-negative potential $W$ allowing for super-quadratic behaviour near its zero set. Under a Dirichlet boundary data of unit-length on $\partial Ω$, we prove the following maximum principle: every critical point $u_\varepsilon$ satisfies the global uniform bound $|u_\varepsilon| \leq 1$ in $Ω$. Furthermore, if a family of critical points $(u_\varepsilon)$ converges (in energy) to a smooth $\mathbb{S}^{N-1}$-valued harmonic map in the limit $\varepsilon \to 0$, then we prove global uniform bounds for $(Δu_\varepsilon)_{\varepsilon>0}$ in $Ω$ and, in particular, global Hölder convergence of the gradients $(\nabla u_\varepsilon)$ in $Ω$ as $\varepsilon \to 0$.

math.AP↗

arXiv2026

On the long time behaviour of a system of several rigid bodies immersed in a viscous fluid

We consider several rigid bodies immersed in a viscous Newtonian fluid contained in a bounded domain in $R^3$. We introduce a new concept of dissipative weak solution of the problem based on a combination of the approach proposed by Judakov with a suitable form of energy inequality. We show that global--in--time dissipative solutions always exist as long as the rigid bodies are connected compact sets. In addition, in the absence of external driving forces, the system always tends to a static equilibrium as time goes to infinity. The results hold independently of possible collisions of rigid bodies and for any finite energy initial data.

math.AP↗

arXiv2026

A not-so-strange term coming from somewhere

We consider Laplace's equation in a periodically perforated domain with Robin boundary conditions on the holes, where the Robin coefficient is scaled proportionally to the inverse total surface area of the performations. We identify a regime in which surface and bulk effects contribute at the same order and show that the homogenised equation contains an additional zeroth-order term depending nonlinearly on the Robin parameter. This term is characterised via a Steklov-type spectral problem in which the spectral parameter appears both in the equation and in the boundary condition. The resulting term interpolates continuously between the Neumann and Dirichlet limits, recovering the classical capacitary strange term in the strong-coupling limit.

math.AP↗

arXiv2026

Variational Dual Solutions for Incompressible Fluids

We consider a construction proposed in \cite{acharyaQAM} that builds on the notion of weak solutions for incompressible fluids to provide a scheme that generates variationally a certain type of dual solutions. If these dual solutions are regular enough one can use them to recover standard solutions. The scheme provides a generalisation of a construction of Y$.$Brenier for the Euler equations. We rigorously analyze the scheme, extending the work of Y$.$Brenier for Euler, and also provide an extension of it to the case of the Navier-Stokes equations. Furthermore we obtain the inviscid limit of Navier-Stokes to Euler as a $Γ$-limit.

math.AP↗

arXiv2025

Quantitative boundary Hölder estimates for the inhomogeneous Poisson problem through a probabilistic approach

In this paper we derive quantitative boundary Hölder estimates, with explicit constants, for the inhomogeneous Poisson problem in a bounded open set $D\subset \mathbb{R}^d$. Our approach has two main steps: firstly, we consider an arbitrary $D$ as above and prove that the boundary $α$-Hölder regularity of the solution the Poisson equation is controlled, with explicit constants, by the Hölder seminorm of the boundary data, the $L^ γ$-norm of the forcing term with $γ>d/2$, and the $α/2$-moment of the exit time from $D$ of the Brownian motion. Secondly, we derive explicit estimates for the $α/2$-moment of the exit time in terms of the distance to the boundary, the regularity of the domain $D$, and $α$. Using this approach, we derive explicit estimates for the same problem in domains satisfying exterior ball conditions, respectively exterior cone/wedge conditions, in terms of simple geometric features. As a consequence we also obtain explicit constants for pointwise estimates for the Green function and for the gradient of the solution. The obtained estimates can be employed to bypass the curse of high dimensions when aiming to approximate the solution of the Poisson problem using neural networks, obtaining polynomial scaling with dimension, which in some cases can be shown to be optimal.

math.PR↗

arXiv2025

Patterns in a Smoluchowski Equation

We analyze the dynamics of concentrated polymer solutions modeled by a 2D Smoluchowski equation. We describe the long time behavior of the polymer suspensions in a fluid. When the flow influence is neglected the equation has a gradient structure. The presence of a simple flow introduces significant structural changes in the dynamics. We study the case of an externally imposed flow with homogeneous gradient. We show that the equation is still dissipative but new phenomena appear.The dynamics depend on both the concentration intensity and the structure of the flow. In certain limit cases the equation has a gradient structure, in an appropriate reference frame, and the solutions evolve to either a steady state or a tumbling wave. For small perturbations of the gradient structure we show that for small concentrations the solutions evolve in the long time limit to a steady state. However for high concentrations there is a rigidity phenomenon for the tumbling wave.

math.AP↗

arXiv2025

Interaction energies in paranematic colloids

We consider a system of colloidal particles embedded in a paranematic -- an isotropic phase of a nematogenic medium above the temperature of the nematic-to-isotropic transition. In this state, the nematic order is induced by the boundary conditions in a narrow band around each particle and it decays exponentially in the bulk. We develop rigorous asymptotics of the linearization of the appropriate variational model that allow us to describe weak far-field interactions between the colloidal particles in two dimensional paranematic suspensions. We demonstrate analytically that decay rates of solutions to the full nonlinear and linear problems are similar and verify numerically that the interactions between the particles in these problems have similar dependence on the distance between the particles. Finally, we perform Monte-Carlo simulations for a system of colloidal particles in a paranematic and describe the statistical properties of this system.

math.AP↗

arXiv2024

On hydrostatic limit of Beris-Edwards system in a thin strip

In this paper we consider the 3D co-rotational Beris-Edwards system modeling the hydrodynamic motion of nematic liquid crystals in a thin strip. The system contains the incompressible Navier-Stokes, coupled with a parabolic system for matrix-valued functions, the $Q$-tensors. We show that under a suitable scaling, corresponding, in the Navier-Stokes part, to the hydrostatic scaling, one obtains in the limit a partly decoupled system. For the fluid part we obtain the Prandtl system while for the $Q$-tensors we obtain a non-standard system, involving fluids components and a non-standard combination of partly dissipative equations and algebraic constraints. We prove the convergence of the rescaled system and the well-posedness of the limit in Sobolev spaces.

math.AP↗

arXiv2024

On the collective effect of a large system of heavy particles immersed in a Newtonian fluid

We consider the motion of a large number of heavy particles in a Newtonian fluid occupying a bounded spatial domain. When we say "heavy", we mean a particle with a mass density that approaches infinity at an appropriate rate as its radius vanishes. We show that the collective effect of heavy particles on the fluid motion is similar to the Brinkman perturbation of the Navier-Stokes system identified in the homogenization process.

math.AP↗

arXiv2024

On the effect of a large cloud of rigid particles on the motion of an incompressible non--Newtonian fluid

We show that the collective effect of $N$ rigid bodies $(\mathcal{S}_{n,N})_{n=1}^N$ of diameters $(r_{n,N})_{n=1}^N$ immersed in an incompressible non--Newtonian fluid is negligible in the asymptotic limit $N \to \infty$ as long as their total packing volume $\sum_{n=1}^N r_{n,N}^d$, $d=2,3$ tends to zero exponentially -- $\sum_{n=1}^N r_{n,N}^d \approx A^{-N}$ -- for a certain constant $A > 1$. The result is rather surprising and in a sharp contrast with the associated homogenization problem, where the same number of obstacles can completely stop the fluid motion in the case of shear thickening viscosity. A large class of non--Newtonian fluids is included, for which the viscous stress is a subdifferential of a convex potential.

math.AP↗

arXiv2024

Colloidal Homogenisation for the Hydrodynamics of Nematic Liquid Crystals

This paper analytically explores a simplified model for the hydrodynamics of nematic liquid crystal colloids. We integrate a Stokes equation for the velocity field with a Ginzburg-Landau transported heat flow for the director field. The study focuses on a bounded spatial domain containing periodically distributed colloidal particles, which impose no-anchoring conditions on the nematic liquid crystal. By progressively reducing the particle size to zero and simultaneously increasing the number of particles, we delve into the associated homogenisation problem. Our analysis uncovers a form of decoupling where the velocity field asymptotically satisfies a Darcy equation, independent of the director, while the director follows a gradient flow, unaffected by the velocity field. One of the most intricate aspects of the homogenisation process is the absence of an extension operator for the director field that preserves the uniform estimates related to the system's energy. We address this challenge with a novel variation of the Aubin-Lions lemma, specifically adapted for homogenisation problems.

math.AP↗

arXiv2024

Reducing model complexity by means of the Optimal Scaling: Population Balance Model for latex particles morphology formation

Rational computer-aided design of multiphase polymer materials is vital for rapid progress in many important applications, such as: diagnostic tests, drug delivery, coatings, additives for constructing materials, cosmetics, etc. Several property predictive models, including the prospective Population Balance Model for Latex Particles Morphology Formation (LPMF PBM), have already been developed for such materials. However, they lack computational efficiency, and the accurate prediction of materials' properties still remains a great challenge. To enhance performance of the LPMF PBM, we explore the feasibility of reducing its complexity through disregard of the aggregation terms of the model. The introduced nondimensionalization approach, which we call Optimal Scaling with Constraints, suggests a quantitative criterion for locating regions of slow and fast aggregation and helps to derive a family of dimensionless LPMF PBM of reduced complexity. The mathematical analysis of this new family is also provided. When compared with the original LPMF PBM, the resulting models demonstrate several orders of magnitude better computational efficiency.

cond-mat.soft↗

arXiv2023

Sufficient conditions for the existence of minimizing harmonic maps with axial symmetry in the small-average regime

The paper concerns the analysis of global minimizers of a Dirichlet-type energy functional defined on the space of vector fields $H^1(S,T)$, where $S$ and $T$ are surfaces of revolution. The energy functional we consider is closely related to a reduced model in the variational theory of micromagnetism for the analysis of observable magnetization states in curved thin films. We show that axially symmetric minimizers always exist, and if the target surface $T$ is never flat, then any coexisting minimizer must have line symmetry. Thus, the minimization problem reduces to the computation of an optimal one-dimensional profile. We also provide a necessary and sufficient condition for energy minimizers to be axially symmetric.

math.AP↗