Hamiltonian Differential Equations Research Articles

A conservative exponential integrators method for fractional conservative differential equations

<abstract><p>The paper constructs a conservative Fourier pseudo-spectral scheme for some conservative fractional partial differential equations. The scheme is obtained by using the exponential time difference averaged vector field method to approximate the time direction and applying the Fourier pseudo-spectral method to discretize the fractional Laplacian operator so that the FFT technique can be used to reduce the computational complexity in long-time simulations. In addition, the developed scheme can be applied to solve fractional Hamiltonian differential equations because the scheme constructed is built upon the general Hamiltonian form of the equations. The conservation and accuracy of the scheme are demonstrated by solving the fractional Schrödinger equation.</p></abstract>

Highly efficient invariant-conserving explicit Runge-Kutta schemes for nonlinear Hamiltonian differential equations

A unified framework of invariant-conserving explicit Runge-Kutta schemes for the nonlinear Hamiltonian ODEs and PDEs are proposed by utilizing the invariant energy quadratization technique. First, the nonlinear Hamiltonian differential equation is transformed into an equivalent reformulation which admits a quadratic energy invariant. For the nonlinear ODE, the reformulation is then discretized using a class of relaxation Runge-Kutta schemes. For the nonlinear PDE, the reformulation is first discretized in the space direction by adopting the Fourier pseudo-spectral discretization which preserves the semi-discrete quadratic conservation laws. Then the semi-discrete system is integrated in the time direction using the explicit relaxation Runge-Kutta method. The obtained method can conserve different quadratic conservation laws to machine accuracy, which improves the numerical stability during long time computations. Besides, the proposed method keeps the same convergence rate of the standard Runge-Kutta scheme when computed with time step size rescaling, and gives a rate of convergence reduced by at most one when computed without step size rescaling. Numerical experiments for several ODEs and PDEs are provided to illustrate the advantages of the proposed algorithms over long time and verify the theoretical analysis.

An iterated graph construction and periodic orbits of Hamiltonian delay equations

According to the Arnold conjectures and Floer's proofs, there are non-trivial lower bounds for the number of periodic solutions of Hamiltonian differential equations on a closed symplectic manifold whose symplectic form vanishes on spheres. We use an iterated graph construction and Lagrangian Floer homology to show that these lower bounds also hold for certain Hamiltonian delay equations.

Integrable structure of products of finite complex Ginibre random matrices

We consider the squared singular values of the product of M standard complex Gaussian matrices. Since the squared singular values form a determinantal point process with a particular Meijer G-function kernel, the gap probabilities are given by a Fredholm determinant based on this kernel. It was shown by Strahov (2014) that a hard edge scaling limit of the gap probabilities is described by Hamiltonian differential equations which can be formulated as an isomonodromic deformation system similar to the theory of the Kyoto school. We generalize this result to the case of finite matrices by first finding a representation of the finite kernel in integrable form. As a result we obtain the Hamiltonian structure for finite size matrices and formulate it in terms of a (M+1)×(M+1) matrix Schlesinger system. The case M=1 reproduces the Tracy and Widom theory which results in the Painlevé V equation for the (0,s) gap probability. Some integrals of motion for M=2 are identified, and a coupled system of differential equations in two unknowns is presented which uniquely determines the corresponding (0,s) gap probability.

Degenerate variational integrators for magnetic field line flow and guiding center trajectories

Symplectic integrators offer many benefits for numerically approximating solutions to Hamiltonian differential equations, including bounded energy error and the preservation of invariant sets. Two important Hamiltonian systems encountered in plasma physics—the flow of magnetic field lines and the guiding center motion of magnetized charged particles—resist symplectic integration by conventional means because the dynamics are most naturally formulated in non-canonical coordinates. New algorithms were recently developed using the variational integration formalism; however, those integrators were found to admit parasitic mode instabilities due to their multistep character. This work eliminates the multistep character, and therefore the parasitic mode instabilities via an adaptation of the variational integration formalism that we deem “degenerate variational integration.” Both the magnetic field line and guiding center Lagrangians are degenerate in the sense that the resultant Euler-Lagrange equations are systems of first-order ordinary differential equations. We show that retaining the same degree of degeneracy when constructing discrete Lagrangians yields one-step variational integrators preserving a non-canonical symplectic structure. Numerical examples demonstrate the benefits of the new algorithms, including superior stability relative to the existing variational integrators for these systems and superior qualitative behavior relative to non-conservative algorithms.

Geometric-integration tools for the simulation of musical sounds

During the last decade, much attention has been given to sound rendering and the simulation of acoustic phenomena by solving appropriate models described by Hamiltonian partial differential equations. In this contribution, we introduce a procedure to develop appropriate tools inspired from geometric integration in order to simulate musical sounds. Geometric integrators are numerical integrators of excellent quality that are designed exclusively for Hamiltonian ordinary differential equations. The introduced procedure is a combination of two techniques in geometric integration: the semi-discretization method by Celledoni et al. (J Comput Phys 231:6770–6789, 2012) and symplectic partitioned Runge–Kutta methods. This combination turns out to be a right procedure that derives numerical schemes that are effective and suitable for computation of musical sounds. By using this procedure we derive a series of explicit integration algorithms for a simple model describing piano sounds as a representative example for virtual instruments. We demonstrate the advantage of the numerical methods by evaluating a variety of numerical test cases.

SPATIAL HAMILTONIAN IDENTITIES FOR NONLOCALLY COUPLED SYSTEMS

We consider a broad class of systems of nonlinear integro-differential equations posed on the real line that arise as Euler–Lagrange equations to energies involving nonlinear nonlocal interactions. Although these equations are not readily cast as dynamical systems, we develop a calculus that yields a natural Hamiltonian formalism. In particular, we formulate Noether’s theorem in this context, identify a degenerate symplectic structure, and derive Hamiltonian differential equations on finite-dimensional center manifolds when those exist. Our formalism yields new natural conserved quantities. For Euler–Lagrange equations arising as traveling-wave equations in gradient flows, we identify Lyapunov functions. We provide several applications to pattern-forming systems including neural field and phase separation problems.

Long-term analysis of numerical integrators for oscillatory Hamiltonian systems under minimal non-resonance conditions

For trigonometric and modified trigonometric integrators applied to oscillatory Hamiltonian differential equations with one or several constant high frequencies, near-conservation of the total and oscillatory energies are shown over time scales that cover arbitrary negative powers of the step size. This requires non-resonance conditions between the step size and the frequencies, but in contrast to previous results the results do not require any non-resonance conditions among the frequencies. The proof uses modulated Fourier expansions with appropriately modified frequencies.

Lattice Differential Equations

The workshop focused on recent advances in the analysis of lattice differential equations such as discrete Klein-Gordon and nonlinear Schrödinger equations as well as the Fermi-Pasta-Ulam lattice. Lattice differential equations play an important role in emergent directions of modern science. These equations are fascinating subjects for mathematicians because they exhibit phenomena, which are not encountered in classical partial differential equations, on one hand, but they may present toy problems for understanding more complicated Hamiltonian differential equations, on the other hand.

Analysis for parareal algorithms applied to Hamiltonian differential equations

Long-time integrations are an important issue in the numerical solution of Hamiltonian systems. They are time consuming and it is natural to consider the use of parallel architectures for reasons of efficiency. In this context the parareal algorithm has been proposed by several authors.The present work is a theoretical study of the parareal algorithm when it is applied to Hamiltonian differential equations. The idea of backward error analysis is employed to get insight into the long-time behavior of numerical approximations. One of the main results is that convergence of the parareal iterations restricts the length of the time window. For nearly integrable systems its length is bounded by the square root of the inverse of the accuracy of the coarse integrator. The theoretical bounds are confirmed by numerical experiments.

Delineation of shallow resistivity structure around Malvan, Konkan region, Maharashtra by neural network inversion using vertical electrical sounding measurements

Modeling resistivity profiles, especially from hard rock areas, is of specific relevance for groundwater exploration. A method based on Bayesian neural network (BNN) theory using a Hybrid Monte Carlo (HMC) simulation scheme is applied to model and interpret direct current vertical electrical sounding measurements from 28 locations around the Malvan region, in the Sindhudurg district, southwest India. The modeling procedure revolves around optimizing the objective function using the HMC based sampling technique which is followed by updating each trajectory by integrating the Hamiltonian differential equations via a second order leapfrog discretization scheme. The inversion results suggest a high resistivity structure in the north-western part of the area, which correlates well with the presence of laterites. In the south-western part, a very high conductive zone is observed near the coast indicating an extensive influence of saltwater intrusion. Our results also show that the effect of intrusion of saline water diminishes from the south-western part to the north-eastern part of the region. Two dimensional modeling of four resistivity profiles shows that the groundwater flow is partly controlled by existing lineaments, fractures, and major joints. Groundwater occurs at a weathered/semi-weathered layer of laterite/clayey sand and the interface of overburden and crystalline basement. The presence of conduits is identified at a depth between 10 and 15 m along the Dhamapur–Kudal and Parule–Oros profiles, which seems to be potential zone for groundwater exploration. The NW–SE trending major lineaments and its criss-cross sections are indentified from the apparent and true resistivity surface map. The pseudo-section at different depths in the western part of the area, near Parule, shows extensive influence of saltwater intrusion and its impact reaching up to a depth of 50 m from the surface along the coastal area. Further, the deduced true electrical resistivity section against depth correlates well with available borehole lithology in the area. Present analyses suggest that HMC-based BNN method is robust for modeling resistivity data especially in hard rock terrains. These results are useful for interpreting fractures, major joints, and lineaments and crystalline basement rock and also for constraining the higher dimensional models.

On conjugate symplecticity of B-series integrators

The long-time integration of Hamiltonian differential equations requires special numerical methods. Symplectic integrators are an excellent choice, but there are situations (e.g., multistep schemes or energy-preserving methods), where symplecticity is not possible. It is then of interest to study whether the methods are conjugate symplectic and thus have the same long-time behaviour as symplectic methods. This question is addressed in this work for the class of B-series integrators. Algebraic criteria for conjugate symplecticity up to a certain order are presented in terms of the coefficients of the B-series. The effect of simplifying assumptions is investigated. These criteria are then applied to characterize the conjugate symplecticity of implicit Runge–Kutta methods (Lobatto IIIA and Lobatto IIIB) and of energy-preserving collocation methods.

Quantum theory of open systems in the representation of effective Hamiltonian and stochastic differential equations

An original general approach to the description of open quantum systems in the Markovian approximation was formulated for the example of the dynamics of high-Q cavity mode photons in the presence of nonresonant cavity atoms bound with an external vacuum electromagnetic field.

Inversion of Schlumberger resistivity sounding data from the critically dynamic Koyna region using the Hybrid Monte Carlo-based neural network approach

Abstract. Koyna region is well-known for its triggered seismic activities since the hazardous earthquake of M=6.3 occurred around the Koyna reservoir on 10 December 1967. Understanding the shallow distribution of resistivity pattern in such a seismically critical area is vital for mapping faults, fractures and lineaments. However, deducing true resistivity distribution from the apparent resistivity data lacks precise information due to intrinsic non-linearity in the data structures. Here we present a new technique based on the Bayesian neural network (BNN) theory using the concept of Hybrid Monte Carlo (HMC)/Markov Chain Monte Carlo (MCMC) simulation scheme. The new method is applied to invert one and two-dimensional Direct Current (DC) vertical electrical sounding (VES) data acquired around the Koyna region in India. Prior to apply the method on actual resistivity data, the new method was tested for simulating synthetic signal. In this approach the objective/cost function is optimized following the Hybrid Monte Carlo (HMC)/Markov Chain Monte Carlo (MCMC) sampling based algorithm and each trajectory was updated by approximating the Hamiltonian differential equations through a leapfrog discretization scheme. The stability of the new inversion technique was tested in presence of correlated red noise and uncertainty of the result was estimated using the BNN code. The estimated true resistivity distribution was compared with the results of singular value decomposition (SVD)-based conventional resistivity inversion results. Comparative results based on the HMC-based Bayesian Neural Network are in good agreement with the existing model results, however in some cases, it also provides more detail and precise results, which appears to be justified with local geological and structural details. The new BNN approach based on HMC is faster and proved to be a promising inversion scheme to interpret complex and non-linear resistivity problems. The HMC-based BNN results are quite useful for the interpretation of fractures and lineaments in seismically active region.

Error propagation in numerical approximations near relative equilibria

We study the propagation of errors in the numerical integration of perturbations of relative equilibrium solutions of Hamiltonian differential equations with symmetries. First it is shown that taking an initial perturbation of a relative equilibrium, the corresponding solution is related, in a first approximation, to another relative equilibrium, with the parameters perturbed from the original. Then, this is used to prove that, for stable relative equilibria, error growth with respect to the perturbed solution is in general quadratic, but only linear for schemes that preserve the invariant quantities of the problem. In this sense, the conclusion is similar to the one obtained when integrating unperturbed relative equilibria. Numerical experiments illustrate the results.

A Hybrid Monte Carlo Method Based Artificial Neural Networks Approach for Rock Boundaries Identification: A Case Study from the KTB Bore Hole

Identification of rock boundaries and structural features from well log response is a fundamental problem in geological field studies. However, in a complex geologic situation, such as in the presence of crystalline rocks where metamorphisms lead to facies changes, it is not easy to discern accurate information from well log data using conventional artificial neural network (ANN) methods. Moreover inferences drawn by such methods are also found to be ambiguous because of the strong overlapping of well log signals, which are generally tainted with deceptive noise. Here, we have developed an alternative ANN approach based on Bayesian statistics using the concept of Hybrid Monte Carlo (HMC)/Markov Chain Monte Carlo (MCMC) inversion scheme for modeling the German Continental Deep Drilling Program (KTB) well log data. MCMC algorithm draws an independent and identically distributed (i.i.d) sample by Markov Chain simulation technique from posterior probability distribution using the principle of statistical mechanics in Hamiltonian dynamics. In this algorithm, each trajectory is updated by approximating the Hamiltonian differential equations through a leapfrog discrimination scheme. We examined the stability and efficiency of the HMC-based approach on “noisy” data assorted with different levels of colored noise. We also perform uncertainty analysis by estimating standard deviation (STD) error map of a posteriori covariance matrix at the network output of three types of lithofacies over the entire length of the litho section of KTB. Our analyses demonstrate that the HMC-based approach renders robust means for classification of complex lithofacies successions from the KTB borehole noisy signals, and hence may provide a useful guide for understanding the crustal inhomogeneity and structural discontinuity in many other tectonically critical and complex regions.

Important aspects of geometric numerical integration

At the example of Hamiltonian differential equations, geometric properties of the flow are discussed that are only preserved by special numerical integrators (such as symplectic and/or symmetric methods). In the ‘non-stiff’ situation the long-time behaviour of these methods is well-understood and can be explained with the help of a backward error analysis. In the highly oscillatory (‘stiff’) case this theory breaks down. Using a modulated Fourier expansion, much insight can be gained for methods applied to problems where the high oscillations stem from a linear part of the vector field and where only one (or a few) high frequencies are present. This paper terminates with numerical experiments at space discretizations of the sine-Gordon equation, where a whole spectrum of frequencies is present.

Important Aspects of Geometric Numerical Integration

At the example of Hamiltonian differential equations, geometric properties of the flow are discussed that are only preserved by special numerical integrators (such as symplectic and/or symmetric methods). In the `non-stiff' situation the long-time behaviour of these methods is well-understood and can be explained with the help of a backward error analysis. In the highly oscillatory (`stiff') case this theory breaks down. Using a modulated Fourier expansion, much insight can be gained for methods applied to problems where the high oscillations stem from a linear part of the vector field and where only one (or a few) high frequencies are present. This paper terminates with numerical experiments at space discretizations of the sine-Gordon equation, where a whole spectrum of frequencies is present

Remarks on a Lie Theorem on the Integrability of Differential Equations in Closed Form

is integrable by quadratures [1] (for details, see also [2, 3]). More precisely, all of its solutions can be found by “algebraic operations” (including inversion of functions) and “quadratures,” that is, computation of integrals of known functions of one variable. This definition has a local nature. The algebra g generates an n-dimensional solvable Lie group G that acts freely on R. Relations (1) imply that the transformations in G take the trajectories of system (2) to trajectories of the same system. In other words, the solvable group G “permutes” the trajectories of the differential equation (2). Lie treated his theorem as an analog of Galois theory for ordinary differential equations. However, there is a more natural analog of Galois theory, namely, Picard–Vessiot theory, which studies extensions of differential fields by solutions of linear differential equations (e.g., see [4]). Nevertheless, the Lie theorem is one of the key results in the problem of closed-form integrability of nonlinear differential equations. It suffices to note that, in the simplest case, this theorem readily implies the well-known Liouville theorem on the integrability of Hamiltonian differential equations with a complete set of independent first integrals in involution [3]. The following assertion is the main result of the present paper.

A Technique to Improve the Error Propagation when Integrating Relative Equilibria

We study and perform a way to improve the propagation of errors in the numerical integration of relative equilibria solutions of Hamiltonian differential equations with symmetries. It is well known that in this case and for stable equilibria, the error growth is typically quadratic for ‘general’ schemes and linear for schemes that preserve the invariant quantities of the problem. Here we present a technique to construct methods whose error propagation in time integration of stable relative equilibria is bounded. Numerical results are presented.