Implicit Lagrangian Research Articles

A Hamilton–Jacobi formalism for higher order implicit Lagrangians

In this paper, we present a generalization of the Hamilton–Jacobi theory for higher order implicit Lagrangian systems. We propose two different backgrounds to deal with higher order implicit Lagrangian theories: the Ostrogradski approach and the Schmidt transform, which convert a higher order Lagrangian into a first order one. The Ostrogradski approach involves the addition of new independent variables to account for higher order derivatives, whilst the Schmidt transform adds gauge invariant terms to the Lagrangian function. In these two settings, the implicit character of the resulting equations will be treated in two different ways in order to provide a Hamilton–Jacobi equation. On one hand, the implicit Lagrangian system will be realized as a Lagrangian submanifold of a higher order tangent bundle that is generated by a Morse family. On the other hand, we will rely on the existence of an auxiliary section of a certain bundle that allows the construction of local vector fields, even if the differential equations are implicit. We will illustrate some examples of our proposed schemes, and discuss the applicability of the proposal.

On implicit Lagrangian twin support vector regression by Newton method

AbstractIn this work, an implicit Lagrangian for the dual twin support vector regression is proposed. Our formulation leads to determining non-parallel e –insensitive down- and up- bound functions for the unknown regressor by constructing two unconstrained quadratic programming problems of smaller size, instead of a single large one as in the standard support vector regression (SVR). The two related support vector machine type problems are solved using Newton method. Numerical experiments were performed on a number of interesting synthetic and real-world benchmark datasets and their results were compared with SVR and twin SVR. Similar or better generalization performance of the proposed method clearly illustrates its effectiveness and applicability.

MERIT FUNCTIONS FOR MATRIX CONE COMPLEMENTARITY PROBLEMS

The merit function arises from the development of the solution methods for the complementarity problems defined over the cone of non negative real vectors and has been well extended to the complementarity problems defined over the symmetric cones. In this paper, we focus on the extension of the merit functions including the gap function, the regularized gap function, the implicit Lagrangian and others to the complementarity problems defined over the nonsymmetric matrix cone. These theoretical results of this paper suggest new solution methods based on unconstrained and/or simply constrained methods to solve the matrix cone complementarity problems (MCCP).

Hamilton–Jacobi theory for degenerate Lagrangian systems with holonomic and nonholonomic constraints

We extend Hamilton–Jacobi theory to Lagrange–Dirac (or implicit Lagrangian) systems, a generalized formulation of Lagrangian mechanics that can incorporate degenerate Lagrangians as well as holonomic and nonholonomic constraints. We refer to the generalized Hamilton–Jacobi equation as the Dirac–Hamilton–Jacobi equation. For non-degenerate Lagrangian systems with nonholonomic constraints, the theory specializes to the recently developed nonholonomic Hamilton–Jacobi theory. We are particularly interested in applications to a certain class of degenerate nonholonomic Lagrangian systems with symmetries, which we refer to as weakly degenerate Chaplygin systems, that arise as simplified models of nonholonomic mechanical systems; these systems are shown to reduce to non-degenerate almost Hamiltonian systems, i.e., generalized Hamiltonian systems defined with non-closed two-forms. Accordingly, the Dirac–Hamilton–Jacobi equation reduces to a variant of the nonholonomic Hamilton–Jacobi equation associated with the reduced system. We illustrate through a few examples how the Dirac–Hamilton–Jacobi equation can be used to exactly integrate the equations of motion.

Finite Newton method for implicit Lagrangian support vector regression

In this paper a finite Newton iterative method of solution for solving the implicit Lagrangian Support Vector Regression (SVR) formulation has been proposed. Unlike solving a quadratic programming problem for the case of the standard SVR the solution

Dirac algebroids in Lagrangian and Hamiltonian mechanics

We present a unified approach to constrained implicit Lagrangian and Hamiltonian systems based on the introduced concept of Dirac algebroid. The latter is a certain almost Dirac structure associated with the Courant algebroid T E ∗ ⊕ M T ∗ E ∗ on the dual E ∗ to a vector bundle τ : E → M . If this almost Dirac structure is integrable (Dirac), we speak about a Dirac–Lie algebroid. The bundle E plays the role of the bundle of kinematic configurations (quasi-velocities), while the bundle E ∗ –the role of the phase space. This setting is totally intrinsic and does not distinguish between regular and singular Lagrangians. The constraints are part of the framework, so the general approach does not change when nonholonomic constraints are imposed, and produces the (implicit) Euler–Lagrange and Hamilton equations in an elegant geometric way. The scheme includes all important cases of Lagrangian and Hamiltonian systems, no matter if they are with or without constraints, autonomous or non-autonomous etc., as well as their reductions; in particular, constrained systems on Lie algebroids. we prove also some basic facts about the geometry of Dirac and Dirac–Lie algebroids.

Stokes-Dirac Differential and Field Implicit Lagrangian Representations

Stokes-Dirac Differential and Field Implicit Lagrangian Representations

DDA in fluid-structure problems for the study of gravity dam failure

In this paper, we present a 2D coupled fluid-structure interaction procedure to assess the stability of a cracked gravity dam against increasing water level. In this procedure the structure is assumed to be a discontinuous deformable solid containing pre-existing cracks. Thus, the Discontinuous Deformation Analysis DDA method is used to study the solid discontinuous medium, and the continuum finite element flow model to calculate the hydrostatic forces corresponding to the water level. The contact constraints between discrete blocks are imposed through an implicit augmented Lagrangian format. Numerical results show the potential of the proposed coupled model to quantify the direct influence of the nature and location of the internal cracks on the overall stability of common gravity dams.

DDA in fluid-structure problems for the study of gravity dam failure

In this paper, we present a 2D coupled fluid-structure interaction procedure to assess the stability of a cracked gravity dam against increasing water level. In this procedure the structure is assumed to be a discontinuous deformable solid containing pre-existing cracks. Thus, the Discontinuous Deformation Analysis DDA method is used to study the solid discontinuous medium, and the continuum finite element flow model to calculate the hydrostatic forces corresponding to the water level. The contact constraints between discrete blocks are imposed through an implicit augmented Lagrangian format. Numerical results show the potential of the proposed coupled model to quantify the direct influence of the nature and location of the internal cracks on the overall stability of common gravity dams.

VECTOR-VALUED IMPLICIT LAGRANGIAN FOR SYMMETRIC CONE COMPLEMENTARITY PROBLEMS

The implicit Lagrangian was first proposed by Mangasarian and Solodov as a smooth merit function for the nonnegative orthant complementarity problem. It has attracted much attention in the past ten years because of its utility in reformulating complementarity problems as unconstrained minimization problems. In this paper, exploiting the Jordan-algebraic structure, we extend it to the vector-valued implicit Lagrangian for symmetric cone complementary problem (SCCP), and show that it is a continuously differentiable complementarity function for SCCP and whose Jacobian is strongly semismooth. As an application, we develop the real-valued implicit Lagrangian and the corresponding smooth merit function for SCCP, and give a necessary and sufficient condition for the stationary point of the merit function to be a solution of SCCP. Finally, we show that this merit function can provide a global error bound for SCCP with the uniform Cartesian P-property.

Non-hydrostatic semi-elastic hybrid-coordinate SISL extension of HIRLAM. Part I: numerical scheme

Two-time-level, semi-implicit, semi-Lagrangian (SISL) scheme is applied to the non-hydrostatic pressure coordinate equations, constituting a modified Miller—Pearce—White model, in hybrid-coordinate framework. Neutral background is subtracted in the initial continuous dynamics, yielding modified equations for geopotential, temperature and logarithmic surface pressure fluctuation. Implicit Lagrangian marching formulae for single time-step are derived. A disclosure scheme is presented, which results in an uncoupled diagnostic system, consisting of 3-D Poisson equation for omega velocity and 2-D Helmholtz equation for logarithmic pressure fluctuation. The model is discretized to create a non-hydrostatic extension to numerical weather prediction model HIRLAM. The discretization schemes, trajectory computation algorithms and interpolation routines, as well as the physical parametrization package are maintained from parent hydrostatic HIRLAM. For stability investigation, the derived SISL model is linearized with respect to the initial, thermally non-equilibrium resting state. Explicit residuals of the linear model prove to be sensitive to the relative departures of temperature and static stability from the reference state. Relayed on the stability study, the semi-implicit term in the vertical momentum equation is replaced to the implicit term, which results in stability increase of the model.

On minimizing the implicit Lagrangian for nonlinear complementarity problems under H-differentiability

In this paper, we describe H-differential of the implicit Lagrangian function. We show how, under appropriate regularity conditions on an H-differential of f, minimizing the implicit Lagrangian function corresponding to f leads to a solution of the nonlinear complementarity problem. Our results give a unified treatment of such results for C 1-functions and for locally Lipschitzian functions.

Dirac structures in Lagrangian mechanics Part I: Implicit Lagrangian systems

This paper develops the notion of implicit Lagrangian systems and presents some of their basic properties in the context of Dirac structures. This setting includes degenerate Lagrangian systems and systems with both holonomic and nonholonomic constraints, as well as networks of Lagrangian mechanical systems. The definition of implicit Lagrangian systems with a configuration space Q makes use of Dirac structures on T ∗ Q that are induced from a constraint distribution on Q as well as natural symplectomorphisms between the spaces T ∗ T Q , T T ∗ Q , and T ∗ T ∗ Q . Two illustrative examples are presented; the first is a nonholonomic system, namely a vertical disk rolling on a plane, and the second is an L–C circuit, a degenerate Lagrangian system with holonomic constraints.

A physical SNOWPACK model for the Swiss avalanche warning: Part I: numerical model

The numerical formulation of a one-dimensional physical snowpack model is presented. The model is operationally employed on a day-to-day basis by avalanche warners to predict snowpack settlement, layering, surface energy exchange and mass balance. Meteorological data obtained from automatic weather stations positioned near avalanche starting zones is used as model input. In this paper, the one-dimensional equations governing the heat transfer, water transport, vapour diffusion and mechanical deformation of a phase changing snowpack are stated. New snow, wind drift and snow ablation are treated as special mass boundary conditions. Snow is modelled as a three-component (ice, water, air) porous material capable of undergoing large irreversible viscous deformations. Phase changes between the components are simulated. Snow layers are defined not only in terms of height and density, but also microstructure. That is, by the size, shape and bonding of the grains composing the ice lattice. The governing differential equations are solved using a fully implicit Lagrangian Gauss–Seidel finite-element method. Example calculations from the catastrophic avalanche winter 1999 are presented to document model performance. The overall mass balance evaluation shows that the model accurately predicts the build-up and ablation of the seasonal alpine snowcover.

10.1162/15324430152748218

An implicit Lagrangian for the dual of a simple reformulation of the standard quadratic program of a linear support vector machine proposed. This leads to the minimization of an unconstrained differentiable convex function in a space of dimensionality equal to the number of classified points. This problem solvable by an extremely simple linearly convergent Lagrangian support vector machine (LSVM) algorithm. LSVM requires the inversion at the outset of a single matrix of the order of the much smaller dimensionality of the original input space plus one. The full algorithm given in this paper in 11 lines of MATLAB code without any special optimization tools such linear or quadratic programming solvers. This LSVM code can be used as is to solve classification problems with millions of points. For example, 2 million points in 10 dimensional input space were classified by a linear surface in 82 minutes on a Pentium III 500 MHz notebook with 384 megabytes of memory (and additional swap space), and in 7 minutes on a 250 MHz UltraSPARC II processor with 2 gigabytes of memory. Other standard classification test problems were also solved. Nonlinear kernel classification can also be solved by LSVM. Although it does not scale up to very large problems, it can handle any positive semidefinite kernel and guaranteed to converge. A short MATLAB code also given for nonlinear kernels and tested on a number of problems.

Merit functions for semi-definite complemetarity problems

Merit functions such as the gap function, the regularized gap function, the implicit Lagrangian, and the norm squared of the Fischer-Burmeister function have played an important role in the solution of complementarity problems defined over the cone of nonnegative real vectors. We study the extension of these merit functions to complementarity problems defined over the cone of block-diagonal symmetric positive semi-definite real matrices. The extension suggests new solution methods for the latter problems.

Magnetorotational Mechanism of Supernova Explosions - Results of 2D Simulations

Results of 2D numerical simulation of the magneto rotational mechanism of a supernova explosion are presented. Simulation has been done for the real equations of state and neutrino energy losses have been taken into account. Simulation has been done on the basis of an Implicit Lagrangian scheme on atriangular grid with grid reconstructuring. It is shown that, due to differential rotation of the star, a toroidal component of the magnetic field appears and grows with time. Rotational momentum transfers outwards as the toroidal component grows with time. With the evolution of the process, part of the envelope of the star is ejected. The amounts of the thrown-off mass and energy are estimated. The results of the simulation could be used as a possible explanation for the supernova explosion picture.

Stationary Points of Bound Constrained Minimization Reformulations of Complementarity Problems

We consider two merit functions which can be used for solving the nonlinear complementarity problem via nonnegatively constrained minimization. One of the functions is the restricted implicit Lagrangian (Refs. 1–3), and the other appears to be new. We study the conditions under which a stationary point of the minimization problem is guaranteed to be a solution of the underlying complementarity problem. It appears that, for both formulations, the same regularity condition is needed. This condition is closely related to the one used in Ref. 4 for unrestricted implicit Lagrangian. Some new sufficient conditions are also given.

A New Merit Function For Nonlinear Complementarity Problems And A Related Algorithm

We investigate the properties of a new merit function which allows us to reduce a nonlinear complementarity problem to an unconstrained global minimization one. Assuming that the complementarity problem is defined by a $P_0$-function, we prove that every stationary point of the unconstrained problem is a global solution; furthermore, if the complementarity problem is defined by a uniform $P$-function, the level sets of the merit function are bounded. The properties of the new merit function are compared with those of Mangasarian--Solodov's implicit Lagrangian and Fukushima's regularized gap function. We also introduce a new simple active-set local method for the solution of complementarity problems and show how this local algorithm can be made globally convergent by using the new merit function.

An Implicit Lagrangian Code for Spherically Symmetric General Relativistic Hydrodynamics with an Approximate Riemann Solver

An implicit Lagrangian hydrodynamics code for general relativistic spherical collapse is presented. This scheme is based on an approximate linearized Riemann solver (Roe-type scheme) and needs no artificial viscosity. This code is aimed especially at the calculation of the late phase of collapse-driven supernovae and the nascent neutron star, where there is a remarkable contrast between the dynamical timescale of the proto-neutron star and the diffusion timescale of neutrinos, without such severe limitation of the Courant condition at the center of the neutron star. Several standard test calculations have been done, and their results show the following: (1) This code captures the shock wave accurately, although some erroneous jumps of specific internal energy are found at the contact discontinuity in the shock tube problems. (2) The scheme shows no instability even if we choose the Courant number larger than 1. (3) However, the Courant number should be kept below ~0.2 at the shock position, so that the shock can be resolved with a few meshes. (4) The scheme reproduces the well-known analytic solutions to the point blast explosion, the gravitational collapse of the uniform gas with γ = 4/3, and the general relativistic collapse of uniform dust. In addition, what is more important, calculations of hydrostatic configurations and the onset of radial instability, a preliminary neutrino transport in a proto-neutron star, and adiabatic collapses of a stellar core have also been done in order to show the performance of the code in the context of the collapse-driven supernovae. It is found that the time step can be extended far beyond the Courant limitation at the center of the neutron star, which fact is crucially significant for the purpose of this project. The details of the scheme and the results of these test calculations are discussed.