arXiv subjects

Evgueni Dinvay

Publications and source records attributed to Evgueni Dinvay.

At least 19 recordsLinked to original sources

arXiv2026

A Regularized Shallow Water System

The shallow-water system is a standard model for long waves in shallow water. The system is hyperbolic and, for a large class of initial data, solutions develop steep gradients leading to shock formation in finite time. Since such singularities violate the long-wave assumptions underlying the model, their appearance limits the regime of validity of the equations. In this work, we introduce a regularized shallow-water system in which the nonlinear terms are modified by a bounded operator. This regularization removes the standard derivative-steepening mechanism responsible for shock formation in the classical system while remaining consistent with the long-wave regime. We establish local well-posedness and small-data global well-posedness in Sobolev spaces that exclude singularity formation. In addition, numerical simulations indicate that the system admits solitary-wave solutions.

math.AP↗

arXiv2026

A Fixed-Grid Affine-Constrained Multiwavelet Coefficient Method for Buckley--Leverett Shock Capturing

We present a fixed-grid conservative affine-constrained modal/multiwavelet coefficient method for one-dimensional Buckley--Leverett saturation transport. The saturation is evolved directly in a local orthonormal coefficient basis with a mean/detail structure: the first mode carries the conservative cell average, whereas higher modes carry zero-mean local details. The hyperbolic inflow condition is imposed as a linear trace constraint on the coefficient vector and enforced by affine lifting. For $(p>1)$, the boundary reprojection is applied in the detail subspace of the inflow cell, so that the prescribed trace is restored without modifying the conservative cell-average update. The transport operator is discretized in conservative weak form with monotone numerical fluxes, and shock-induced oscillations are controlled by a troubled-cell limiter acting on modal details. The method is validated on a Berea-core waterflood benchmark against an independent \texttt{pywaterflood} reference solution using the same Corey fractional-flow closure, physical parameters, and pore-volume-injected scaling. The affine-constrained coefficient solver reproduces the reference breakthrough curve and saturation profiles, preserves the imposed inflow trace to roundoff accuracy, controls saturation bounds through mean-preserving detail rescaling, and gives small accumulated global mass-balance defects. Mesh-refinement, flux-comparison, and modal-order studies show that $(p=2)$, corresponding to a piecewise-linear local representation, provides the most favorable accuracy--cost compromise among the tested orders for this shock-dominated benchmark.

physics.flu-dyn↗

arXiv2026

Newton optimization for the Multiconfiguration Self Consistent Field method at the basis set limit: closed-shell two-electron systems

The multiconfiguration self-consistent field (MCSCF) method is revisited with a specific focus on two-electron systems for simplicity. The wave function is represented as a linear combination of Slater determinants. Both the orbitals and the coefficients of this configuration interaction expansion are optimized according to the variational principle within the Lagrangian formalism, using a Newton optimization scheme. This reduces the MCSCF problem to solving a particular differential Newton system, which can be discretized with multiwavelets and solved iteratively.

physics.chem-ph↗

arXiv2026

Riemannian gradient descent for Hartree-Fock theory

We present a Riemannian optimization framework for Hartree-Fock theory formulated directly in the Sobolev space $H^1$. The orthonormality constraints are interpreted geometrically via infinite-dimensional Stiefel and Grassmann manifolds endowed with the embedded $H^1$ metric. Explicit expressions for Euclidean and Riemannian gradients, tangent-space projections, and retractions are derived using resolvent operators, avoiding distributional formulations. The resulting algorithms include Riemannian steepest descent and a preconditioned nonlinear conjugate gradient method equipped with Armijo backtracking and Powell-type restarts. Particular attention is given to physically motivated preconditioning based on inversion of the kinetic energy operator. The framework is naturally compatible with adaptive multiwavelet discretizations, where Coulomb-type convolutions can be evaluated efficiently. Numerical experiments demonstrate robust convergence and competitive performance compared to conventional SCF-DIIS schemes. In addition, for small molecules the gradient descent method converges from random initial guesses. The proposed formulation provides a geometrically consistent and discretization-independent perspective on electronic structure optimization and offers a foundation for further developments in infinite-dimensional Riemannian methods for quantum chemistry.

quant-ph↗

arXiv2025

Wavefunction optimization at the complete basis set limit with Multiwavelets and DMRG

The density matrix renormalization group (DMRG) is a powerful numerical technique to solve strongly correlated quantum systems: it deals well with systems which are not dominated by a single configuration (unlike Coupled Cluster) and it converges rapidly to the Full Configuration Interaction (FCI) limit (unlike truncated Configuration Interaction (CI) expansions). In this work, we develop an algorithm integrating DMRG within the multiwavelet-based multiresolution analysis (MRA). Unlike fixed basis sets, multiwavelets offer an adaptive and hierarchical representation of functions, approaching the complete basis set limit to a specified precision. As a result, this combined technique leverages the multireference capability of DMRG and the complete basis set limit of MRA and multiwavelets. More specifically, we adopt a pre-existing Lagrangian optimization algorithm for orbitals represented in the MRA domain and improve its computational efficiency by replacing the original CI calculations with DMRG. Additionally, we substitute the reduced density matrices computation with the direct extraction of energy gradients from the DMRG tensors. We apply our method to small systems such H2, He, HeH2, BeH2 and N2. The results demonstrate that our approach reduces the final energy while keeping the number of orbitals low compared to FCI calculations on an atomic orbital basis set.

physics.chem-ph↗

arXiv2025

Heat semigroup representation of Laplacian

This work introduces novel numerical algorithms for computational quantum mechanics, grounded in a representation of the Laplace operator -- frequently used to model kinetic energy in quantum systems -- via the heat semigroup. The key advantage of this approach lies in its computational efficiency, particularly within the framework of multiresolution analysis, where the heat equation can be solved rapidly. This results in an effective and scalable method for evaluating kinetic energy in quantum systems. The proposed multiwavelet-based Laplacian approximation is tested through two fundamental quantum chemistry applications: self-consistent field calculations and the time evolution of Schrödinger-type equations.

quant-ph↗

arXiv2024

Multiresolution of the one dimensional free-particle propagator. Part 1: Construction

The free-particle propagator, a key operator in various algorithms for simulating the time evolution of the Schrödinger equation, is studied. A multiscale approximation of this propagator is constructed, representing the semigroup associated with the free-particle Schrödinger operator in a multiwavelet basis. This representation involves integrals of highly oscillatory functions. These integrals are efficiently discretized using a contour deformation technique, which addresses the challenges posed by earlier discretization methods.

physics.comp-ph↗

arXiv2024

A conservative stochastic Dirac-Klein-Gordon system

Considered herein is a particular nonlinear dispersive stochastic system consisting of Dirac and Klein-Gordon equations. They are coupled by nonlinear terms due to the Yukawa interaction. We consider a case of homogeneous multiplicative noise that seems to be very natural from the perspective of the least action formalism. We are able to show existence and uniqueness of a corresponding Cauchy problem in Bourgain spaces. Moreover, the regarded model implies charge conservation, known for the deterministic analogue of the system, and this is used to prove a global existence result for suitable initial data.

math.AP↗

arXiv2024

VAMPyR -- A High-Level Python Library for Mathematical Operations in a Multiwavelets Representation

Wavelets and Multiwavelets have lately been adopted in Quantum Chemistry to overcome challenges presented by the two main families of basis sets: Gaussian atomic orbitals and plane waves. In addition to their numerical advantages (high precision, locality, fast algorithms for operator application, linear scaling with respect to system size, to mention a few) they provide a framework which narrows the gap between the theoretical formalism of the fundamental equations and the practical implementation in a working code. This realization led us to the development of the Python library called VAMPyR, (Very Accurate Multiresolution Python Routines). VAMPyR encodes the binding to a C++ library for Multiwavelet calculations (algebra, integral and differential operator application) and exposes the required functionality to write simple Python code to solve among others, the Hartree--Fock equations, the generalized Poisson Equation, the Dirac equation and the time-dependent Schrödinger equation up to any predefined precision. In this contribution we will outline the main features of Multiresolution Analysis using multiwavelets and we will describe the design of the code. A few illustrative examples will show the code capabilities and its interoperability with other software platforms.

physics.chem-ph↗

arXiv2022

A stochastic Benjamin-Bona-Mahony type equation

Considered herein is a particular nonlinear dispersive stochastic equation. It was introduced recently in [3], as a model describing surface water waves under location uncertainty. The corresponding noise term is introduced through a Hamiltonian formulation, which guarantees the energy conservation of the flow. Here the initial-value problem is studied.

math.AP↗

arXiv2022

Hamiltonian formulation of the stochastic surface wave problem

We devise a stochastic Hamiltonian formulation of the water wave problem. This stochastic representation is built within the framework of the modelling under location uncertainty. Starting from restriction to the free surface of the general stochastic fluid motion equations, we show how one can naturally deduce Hamiltonian structure under a small noise assumption. Moreover, as in the classical water wave theory, the non-local Dirichlet-Neumann operator appears explicitly in the energy functional. This, in particular, allows us, in the same way as in deterministic setting, to conduct systematic approximations of the Dirichlet-Neumann operator and to infer different simplified wave models including noise in a natural way.

math.AP↗

arXiv2022

Travelling waves in the Boussinesq type systems

Considered herein are a number of variants of the Boussinesq type systems modeling surface water waves. Such equations were derived by different authors to describe the two-way propagation of long gravity waves. A question of existence of special solutions, the so called solitary waves, is of a particular interest. There are a number of studies relying on a variational approach and a concentration-compactness argument. These proofs are technically very demanding and may vary significantly from one system to another. Our approach is based on the implicit function theorem, which makes the treatment easier and more unified.

math.AP↗

arXiv2021

Fully dispersive Boussinesq models with uneven bathymetry

Three weakly nonlinear but fully dispersive Whitham-Boussinesq systems for uneven bathymetry are studied. The derivation and discretization of one system is presented. The numerical solutions of all three are compared with wave gauge measurements from a series of laboratory experiments conducted by Dingemans. The results show that although the models are mathematically similar, their accuracy varies dramatically.

physics.flu-dyn↗

arXiv2021

Solitary wave solutions of a Whitham-Boussinesq system

The travelling wave problem for a particular bidirectional Whitham system modelling surface water waves is under consideration. This system firstly appeared in [Dinvay, Dutykh, Kalisch 2018], where it was numerically shown to be stable and a good approximation to the incompressible Euler equations. In subsequent papers [Dinvay 2018], [Dinvay, Selberg, Tesfahun 2019] the initial-value problem was studied and well-posedness in classical Sobolev spaces was proved. Here we prove existence of solitary wave solutions and provide their asymptotic description. Our proof relies on a variational approach and a concentration-compactness argument. The main difficulties stem from the fact that in the considered Euler-Lagrange equation we have a non-local operator of positive order appearing both in the linear and non-linear parts. Our approach allows us to obtain solitary waves for a particular Boussinesq system as well.

math.AP↗

arXiv2020

Well-posedness for a Whitham-Boussinesq system with surface tension

We regard the Cauchy problem for a particular Whitham-Boussinesq system modelling surface waves of an inviscid incompressible fluid layer. The system can be seen as a weak nonlocal dispersive perturbation of the shallow water system. The proof of well-posedness relies on energy estimates. However, due to the symmetry lack of the nonlinear part, in order to close the a priori estimates one has to modify the traditional energy norm in use. Hamiltonian conservation provides with global well-posedness at least for small initial data in the one dimensional settings.

math.AP↗

arXiv2020

The Whitham Equation with Surface Tension

The viability of the Whitham equation as a nonlocal model for capillary-gravity waves at the surface of an inviscid incompressible fluid is under study. A nonlocal Hamiltonian system of model equations is derived using the Hamiltonian structure of the free surface water wave problem and the Dirichlet-Neumann operator. The system features gravitational and capillary effects, and when restricted to one-way propagation, the system reduces to the capillary Whitham equation. It is shown numerically that in various scaling regimes the Whitham equation gives a more accurate approximation of the free-surface problem for the Euler system than other models like the KdV, and Kawahara equation. In the case of relatively strong capillarity considered here, the KdV and Kawahara equations outperform the Whitham equation with surface tension only for very long waves with negative polarity.

physics.flu-dyn↗

arXiv2019

Well-posedness for a dispersive system of the Whitham-Boussinesq type

We regard the Cauchy problem for a particular Whitham-Boussinesq system modelling surface waves of an inviscid incompressible fluid layer. We are interested in well-posedness at a very low level of regularity. We derive dispersive and Strichartz estimates, and implement them together with a fixed point argument to solve the problem locally. Hamiltonian conservation guarantees global well-posedness for small initial data in the one dimensional settings.

math.AP↗

arXiv2019

Modified Babenko's equation for periodic gravity waves on water of finite depth

A new operator equation for periodic gravity waves on water of finite depth is derived and investigated; it is equivalent to Babenko's equation considered in \cite{KD}. Both operators in the proposed equation are nonlinear and depend on the parameter equal to the mean depth of water, whereas each solution defines a parametric representation for a symmetric free surface profile. The latter is a component of a solution of the two-dimensional, nonlinear problem describing steady waves propagating in the absence of surface tension. Bifurcation curves (including a branching one) are obtained numerically for solutions of the new equation; they are compared with known results.

math-ph↗