arXiv subjects

Alexander Bihlo

Publications and source records attributed to Alexander Bihlo.

At least 19 recordsLinked to original sources

arXiv2022

Extended symmetry analysis of remarkable (1+2)-dimensional Fokker-Planck equation

We carry out the extended symmetry analysis of an ultraparabolic Fokker-Planck equation with three independent variables, which is also called the Kolmogorov equation and is singled out within the class of such Fokker-Planck equations by its remarkable symmetry properties. In particular, its essential Lie invariance algebra is eight-dimensional, which is the maximum dimension within the above class. We compute the complete point symmetry pseudogroup of the Fokker-Planck equation using the direct method, analyze its structure and single out its essential subgroup. After listing inequivalent one- and two-dimensional subalgebras of the essential and maximal Lie invariance algebras of this equation, we exhaustively classify its Lie reductions, carry out its peculiar generalized reductions and relate the latter reductions to generating solutions with iterative action of Lie-symmetry operators. As a result, we construct wide families of exact solutions of the Fokker-Planck equation, in particular, those parameterized by an arbitrary finite number of arbitrary solutions of the (1+1)-dimensional linear heat equation. We also establish the point similarity of the Fokker-Planck equation to the (1+2)-dimensional Kramers equations whose essential Lie invariance algebras are eight-dimensional, which allows us to find wide families of exact solutions of these Kramers equations in an easy way.

math.AP↗

arXiv2021

Selective decay for the rotating shallow-water equations with a structure-preserving discretization

Numerical models of weather and climate critically depend on long-term stability of integrators for systems of hyperbolic conservation laws. While such stability is often obtained from (physical or numerical) dissipation terms, physical fidelity of such simulations also depends on properly preserving conserved quantities, such as energy, of the system. To address this apparent paradox, we develop a variational integrator for the shallow water equations that conserves energy, but dissipates potential enstrophy. Our approach follows the continuous selective decay framework [F. Gay-Balmaz and D. Holm. Selective decay by Casimir dissipation in inviscid fluids. Nonlinearity, 26(2):495, 2013], which enables dissipating an otherwise conserved quantity while conserving the total energy. We use this in combination with the variational discretization method [D. Pavlov, P. Mullen, Y. Tong, E. Kanso, J. Marsden and M. Desbrun. Structure-preserving discretization of incompressible fluids. Physica D: Nonlinear Phenomena, 240(6):443-458, 2011] to obtain a discrete selective decay framework. This is applied to the shallow water equations, both in the plane and on the sphere, to dissipate the potential enstrophy. The resulting scheme significantly improves the quality of the approximate solutions, enabling long-term integrations to be carried out.

math.NA↗

arXiv2021

Conservative Integrators for Many-body Problems

Conservative symmetric second-order one-step schemes are derived for dynamical systems describing various many-body systems using the Discrete Multiplier Method. This includes conservative schemes for the $n$-species Lotka-Volterra system, the $n$-body problem with radially symmetric potential and the $n$-point vortex models in the plane and on the sphere. In particular, we recover Greenspan-Labudde's conservative schemes for the $n$-body problem. Numerical experiments are shown verifying the conservative property of the schemes and second-order accuracy.

math.NA↗

arXiv2020

Lie symmetries of two-dimensional shallow water equations with variable bottom topography

We carry out the group classification of the class of two-dimensional shallow water equations with variable bottom topography using an optimized version of the method of furcate splitting. The equivalence group of this class is found by the algebraic method. Using algebraic techniques, we construct additional point equivalences between some of the listed cases of Lie-symmetry extensions, which are inequivalent up to transformations from the equivalence group.

nlin.SI↗

arXiv2020

Generalization of the algebraic method of group classification with application to nonlinear wave and elliptic equations

Enhancing and essentially generalizing previous results on a class of (1+1)-dimensional nonlinear wave and elliptic equations, we apply several new techniques to classify admissible point transformations within this class up to the equivalence generated by its equivalence group. This gives an exhaustive description of its equivalence groupoid. After extending the algebraic method of group classification to non-normalized classes of differential equations, we solve the complete group classification problem for the class under study up to both usual and general point equivalences. The solution includes the complete preliminary group classification of the class and the construction of singular Lie-symmetry extensions, which are not related to subalgebras of the equivalence algebra. The complete preliminary group classification is based on classifying appropriate subalgebras of the entire infinite-dimensional equivalence algebra whose projections are qualified as maximal extensions of the kernel invariance algebra. The results obtained can be used to construct exact solutions of nonlinear wave and elliptic equations.

math-ph↗

arXiv2020

Generalized symmetries, conservation laws and Hamiltonian structures of an isothermal no-slip drift flux model

We study the hydrodynamic-type system of differential equations modeling isothermal no-slip drift flux. Using the facts that the system is partially coupled and its subsystem reduces to the (1+1)-dimensional Klein--Gordon equation, we exhaustively describe generalized symmetries, cosymmetries and local conservation laws of this system. A generating set of local conservation laws under the action of generalized symmetries is proved to consist of two zeroth-order conservation laws. The subspace of translation-invariant conservation laws is singled out from the entire space of local conservation laws. We also find broad families of local recursion operators and a nonlocal recursion operator, and construct an infinite family of Hamiltonian structures involving an arbitrary function of a single argument. For each of the constructed Hamiltonian operators, we obtain the associated algebra of Hamiltonian symmetries.

math.AP↗

arXiv2020

A generative adversarial network approach to (ensemble) weather prediction

We use a conditional deep convolutional generative adversarial network to predict the geopotential height of the 500 hPa pressure level, the two-meter temperature and the total precipitation for the next 24 hours over Europe. The proposed models are trained on 4 years of ERA5 reanalysis data from 2015-2018 with the goal to predict the associated meteorological fields in 2019. The forecasts show a good qualitative and quantitative agreement with the true reanalysis data for the geopotential height and two-meter temperature, while failing for total precipitation, thus indicating that weather forecasts based on data alone may be possible for specific meteorological parameters. We further use Monte-Carlo dropout to develop an ensemble weather prediction system based purely on deep learning strategies, which is computationally cheap and further improves the skill of the forecasting model, by allowing to quantify the uncertainty in the current weather forecast as learned by the model.

physics.ao-ph↗

arXiv2020

Zeroth-order conservation laws of two-dimensional shallow water equations with variable bottom topography

We classify zeroth-order conservation laws of systems from the class of two-dimensional shallow water equations with variable bottom topography using an optimized version of the method of furcate splitting. The classification is carried out up to equivalence generated by the equivalence group of this class. We find additional point equivalences between some of the listed cases of extensions of the space of zeroth-order conservation laws, which are inequivalent up to transformations from the equivalence group. Hamiltonian structures of systems of shallow water equations are used for relating the classification of zeroth-order conservation laws of these systems to the classification of their Lie symmetries. We also construct generating sets of such conservation laws under action of Lie symmetries.

math-ph↗

arXiv2020

Equivalence groupoid and group classification of a class of variable-coefficient Burgers equations

We study admissible transformations and Lie symmetries for a class of variable-coefficient Burgers equations. We combine the advanced methods of splitting into normalized subclasses and of mappings between classes that are generated by families of point transformations parameterized by arbitrary elements of the original classes. A nontrivial differential constraint on the arbitrary elements of the class of variable-coefficient Burgers equations leads to its partition into two subclasses, which are related to normalized classes via families of point transformations parameterized by subclasses' arbitrary elements. One of the mapped classes is proved to be normalized in the extended generalized sense, and its effective extended generalized equivalence group is found. Using the mappings between classes and the algebraic method of group classification, we carry out the group classification of the initial class with respect to its equivalence groupoid.

math-ph↗

arXiv2019

Extended symmetry analysis of isothermal no-slip drift flux model

We perform extended group analysis for a system of differential equations modeling an isothermal no-slip drift flux. The maximal Lie invariance algebra of this system is proved to be infinite-dimensional. We also find the complete point symmetry group of this system, including discrete symmetries, using the megaideal-based version of the algebraic method. Optimal lists of one- and two-dimensional subalgebras of the maximal Lie invariance algebra in question are constructed and employed for obtaining reductions of the system under study. Since this system contains a subsystem of two equations that involves only two of three dependent variables, we also perform group analysis of this subsystem. The latter can be linearized by a composition of a fiber-preserving point transformation with a two-dimensional hodograph transformation to the Klein-Gordon equation. We also employ both the linearization and the generalized hodograph method for constructing the general solution of the entire system under study. We find inter alia genuinely generalized symmetries for this system and present the connection between them and the Lie symmetries of the subsystem we mentioned earlier. Hydrodynamic conservation laws and their generalizations are also constructed.

math.AP↗

arXiv2019

Differential invariants for a class of diffusion equations

We find the complete equivalence group of a class of (1+1)-dimensional second-order evolution equations, which is infinite-dimensional. The equivariant moving frame methodology is invoked to construct, in the regular case of the normalization procedure, a moving frame for a group related to the equivalence group in the context of equivalence transformations among equations of the class under consideration. Using the moving frame constructed, we describe the algebra of differential invariants of the former group by obtaining a minimum generating set of differential invariants and a complete set of independent operators of invariant differentiation.

math-ph↗

arXiv2019

Invariant parameterization of geostrophic eddies in the ocean

The framework of invariant parameterization is extended to higher-order closure schemes. We also define, for the first time, generalized invariant parameterization schemes, where symmetries of the corresponding original model are preserved as equivalence transformations of related classes of closed system of differential equations. As a particular problem, we consider invariant parameterization schemes for geostrophic eddies in a barotropic ocean. Here the initial model is the barotropic vorticity equation, which is equivalent to the system of incompressible inviscid two-dimensional Euler equations on a midlatitude beta-plane. The maximal Lie invariance algebra of this model is infinite-dimensional, and we intend to preserve it in the course of invariant parameterization, at least partially. The parameterizations proposed for the eddy vorticity flux and the energy flux are of order one and a half since we explicitly consider the equation for the turbulent kinetic energy in the closure models. These parameterizations are therefore the first examples of invariant higher-order closure schemes. Numerical experiments are carried out to assess the performance of these invariant schemes in studies of freely decaying two-dimensional turbulence, and it is verified that the invariant parameterization schemes give, on average, better results than the standard non-invariant closure models do.

physics.ao-ph↗

arXiv2019

Inverse problem on conservation laws

The explicit formulation of the general inverse problem on conservation laws is presented for the first time. In this problem one aims to derive the general form of systems of differential equations that admit a prescribed set of conservation laws. The particular cases of the inverse problem on first integrals of ordinary differential equations and on conservation laws for evolution equations are studied. We also solve the inverse problem on conservation laws for differential equations admitting an infinite dimensional space of zeroth-order conservation-law characteristics. This particular case is further studied in the context of conservative first-order parameterization schemes for the two-dimensional incompressible Euler equations. We exhaustively classify conservative first-order parameterization schemes for the eddy-vorticity flux that lead to a class of closed, averaged Euler equations possessing generalized circulation, generalized momentum and energy conservation.

math-ph↗

arXiv2019

Precipitation nowcasting using a stochastic variational frame predictor with learned prior distribution

We propose the use of a stochastic variational frame prediction deep neural network with a learned prior distribution trained on two-dimensional rain radar reflectivity maps for precipitation nowcasting with lead times of up to 2 1/2 hours. We present a comparison to a standard convolutional LSTM network and assess the evolution of the structural similarity index for both methods. Case studies are presented that illustrate that the novel methodology can yield meaningful forecasts without excessive blur for the time horizons of interest.

cs.CV↗

arXiv2018

Variational integrator for the rotating shallow-water equations on the sphere

We develop a variational integrator for the shallow-water equations on a rotating sphere. The variational integrator is built around a discretization of the continuous Euler-Poincar\'{e} reduction framework for Eulerian hydrodynamics. We describe the discretization of the continuous Euler-Poincar\'{e} equations on arbitrary simplicial meshes. Standard numerical tests are carried out to verify the accuracy and the excellent conservational properties of the discrete variational integrator.

math.NA↗

arXiv2018

Symmetry-preserving finite element schemes: An introductory investigation

Using the method of equivariant moving frames, we present a procedure for constructing symmetry-preserving finite element methods for second-order ordinary differential equations. Using the method of lines, we then indicate how our constructions can be extended to (1+1)-dimensional evolutionary partial differential equations, using Burgers' equation as an example. Numerical simulations verify that the symmetry-preserving finite element schemes constructed converge at the expected rate and that these schemes can yield better results than their non-invariant finite element counterparts.

math.NA↗

arXiv2017

Massively parallel stochastic solution of the geophysical gravity problem

In this paper, we report the advantages of using a stochastic algorithm in the context of mineral exploration based on gravity measurements. This approach has the advantage over deterministic methods in that it allows one to find the solution of the Poisson equation in specified, isolated points without the need of meshing the computational domain and solving the Poisson equation over the entire domain. Moreover, the stochastic approach is embarrassingly parallelizable and therefore suitable for an implementation on multi-core compute clusters with or without GPUs. Benchmark tests are carried out that show that the stochastic approach can yield accurate results for both the gravitational potential and the gravitational acceleration and could hence provide an alternative to existing deterministic methods used in mineral exploration.

math.NA↗

arXiv2017

Group analysis of general Burgers-Korteweg-de Vries equations

The complete group classification problem for the class of (1+1)-dimensional $r$th order general variable-coefficient Burgers-Korteweg-de Vries equations is solved for arbitrary values of $r$ greater than or equal to two. We find the equivalence groupoids of this class and its various subclasses obtained by gauging equation coefficients with equivalence transformations. Showing that this class and certain gauged subclasses are normalized in the usual sense, we reduce the complete group classification problem for the entire class to that for the selected maximally gauged subclass, and it is the latter problem that is solved efficiently using the algebraic method of group classification. Similar studies are carried out for the two subclasses of equations with coefficients depending at most on the time or space variable, respectively. Applying an original technique, we classify Lie reductions of equations from the class under consideration with respect to its equivalence group. Studying of alternative gauges for equation coefficients with equivalence transformations allows us not only to justify the choice of the most appropriate gauge for the group classification but also to construct for the first time classes of differential equations with nontrivial generalized equivalence group such that equivalence-transformation components corresponding to equation variables locally depend on nonconstant arbitrary elements of the class. For the subclass of equations with coefficients depending at most on the time variable, which is normalized in the extended generalized sense, we explicitly construct its extended generalized equivalence group in a rigorous way. The new notion of effective generalized equivalence group is introduced.

math-ph↗