Non-degeneracy Requirement Research Articles

Approximation of SDE solutions using local asymptotic expansions

We develop an asymptotic expansion for small time of the density of the solution of a non-degenerate system of stochastic differential equations with smooth coefficients, and apply this to the stepwise approximation of solutions. The asymptotic expansion, which takes the form of a multivariate Edgeworth-type expansion, is obtained from the Kolmogorov forward equation using some standard PDE results. To generate one step of the approximation to the solution, we use a Cornish–Fisher type expansion derived from the Edgeworth expansion. To interpret the approximation generated in this way as a strong approximation we use couplings between the (normal) random variables used and the Brownian path driving the SDE. These couplings are constructed using techniques from optimal transport and Vaserstein metrics. The type of approximation so obtained may be regarded as intermediate between a conventional strong approximation and a weak approximation. In principle approximations of any order can be obtained, though for higher orders the algebra becomes very heavy. In order 1/2 the method gives the usual Euler approximation; in order 1 it gives a variant of the Milstein method, but which needs only normal variables to be generated. However the method is somewhat limited by the non-degeneracy requirement.

Multisymplectic 3-forms on 7-dimensional manifolds

A 3-form ω∈Λ3R7⁎ is called multisymplectic if it satisfies some natural non-degeneracy requirement. It is well known that there are 8 orbits (or types) of multisymplectic 3-forms on R7 under the canonical action of GL(7,R) and that two types are open. This leads to 8 types of global multisymplectic 3-forms on 7-dimensional manifolds without boundary. The existence of a global multisymplectic 3-form of a fixed type is a classical problem in differential topology which is equivalent to the existence of a certain G-structure. The open types are the most interesting cases as they are equivalent to a G2 and G˜2-structure, respectively. The existence of these two structures is a well known and solved problem. In this article is solved (under some convenient assumptions) the problem of the existence of multisymplectic 3-forms of the remaining types.

NONORTHOGONAL GEOMETRIC REALIZATIONS OF COXETER GROUPS

AbstractWe define in an axiomatic fashion a Coxeter datum for an arbitrary Coxeter group $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}W$. This Coxeter datum will specify a pair of reflection representations of $W$ in two vector spaces linked only by a bilinear pairing without any integrality or nondegeneracy requirements. These representations are not required to be embeddings of $W$ in the orthogonal group of any vector space, and they give rise to a pair of inter-related root systems generalizing the classical root systems of Coxeter groups. We obtain comparison results between these nonorthogonal root systems and the classical root systems. Further, we study the equivalent of the Tits cone in these nonorthogonal representations.

Addressing Rank Degeneracy in Constraint-Reduced Interior-Point Methods for Linear Optimization

In earlier works (Tits et al. SIAM J. Optim., 17(1):119–146, 2006; Winternitz et al. Comput. Optim. Appl., 51(3):1001–1036, 2012), the present authors and their collaborators proposed primal–dual interior-point (PDIP) algorithms for linear optimization that, at each iteration, use only a subset of the (dual) inequality constraints in constructing the search direction. For problems with many more variables than constraints in primal form, this can yield a major speedup in the computation of search directions. However, in order for the Newton-like PDIP steps to be well defined, it is necessary that the gradients of the constraints included in the working set span the full dual space. In practice, in particular in the case of highly sparse problems, this often results in an undesirably large working set—or in an expensive trial-and-error process for its selection. In this paper, we present two approaches that remove this non-degeneracy requirement, while retaining the convergence results obtained in the earlier work.

Conformal Structures of Static Vacuum Data

In the Cauchy problem for asymptotically flat vacuum data the solution-jets along the cylinder at space-like infinity develop in general logarithmic singularities at the critical sets at which the cylinder touches future/past null infinity. The tendency of these singularities to spread along the null generators of null infinity obstructs the development of a smooth conformal structure at null infinity. For the solution-jets arising from time reflection symmetric data to extend smoothly to the critical sets it is necessary that the Cotton tensor of the initial three-metric h satisfies a certain conformally invariant condition (*) at space-like infinity, it is sufficient that h be asymptotically static at space-like infinity. The purpose of this article is to characterize the gap between these conditions. We show that with the class of metrics which satisfy condition (*) on the Cotton tensor and a certain non-degeneracy requirement is associated a one-form $\kappa$ with conformally invariant differential $d\kappa$. We provide two criteria: If $h$ is real analytic, $\kappa$ is closed, and one of it integrals satisfies a certain equation then h is conformal to static data near space-like infinity. If h is smooth, $\kappa$ is asymptotically closed, and one of it integrals satisfies a certain equation asymptotically then h is asymptotically conformal to static data at space-like infinity.

Nondegeneracy tests for hexahedral cells

The aims of the paper are to consider the nondegeneracy requirement for computational grids and to analyze eight tests used to check the nondegeneracy of hexahedral cells. The paper starts with consideration of nondegeneracy requirement and formulation of definitions and common theorems utilized for estimation of nondegeneracy of grids both structured and unstructured. Then hexahedral cells are introduced and sufficient nondegeneracy conditions (Ushakova, 2000) for them are given. Sufficient nondegeneracy conditions are 27 inequalities for 32 tetrahedral volumes. Besides sufficient nondegeneracy conditions other conditions are applied as nondegeneracy tests in grid generation theory and practice. Considered nondegeneracy tests are the checks for positivity of different values. Tests 1, 2, 3, 4, 5, 6 check the positivity of 8, 10, 24, 32, 58, 48 tetrahedral volumes, correspondingly. Test 7 verifies the positivity of the volume of a cell. Test 8 checks the positivity of the Jacobian of the mapping used for generation of a cell. The check is performed at the corners of a cell and hex center. Tests 1, 7, 8 are often used in commercial packages. For the most part, nondegeneracy tests are not sufficient nondegeneracy conditions, however they are used for the purpose of constructing nondegenerate grids and, some times, instead of sufficient nondegeneracy conditions. The effectiveness and reliability of such substitutions are investigated in special numerical experiments with random numbers. In the numerical experiment for each test, hexahedral cells are generated randomly. Results of such experiments are the following. Among eight tests, test 2 is considered the best since it verifies the volumes of only 10 tetrahedra for positiveness, guarantees the nondegeneracy in most of cases (68.7% randomly generated hexahedral cells satisfying test 2) and covers a wide class of cells (about 60% of nondegenerate cells). Tests 1, 3, 4, 5, 6, 7, 8 have success in 31.7%, 83.1%, 100%, 100%, 39.5%, 0.2%, 34% of cases and cover 100%, 7.9%, 7.9%, 4.2%, 59.5%, 100%, 100% of nondegenerate cells, correspondingly. Because of high rate of success, tests 3, 4, 5 also can be used for grid generation purpose. All tests are illustrated by the examples of structured grids.

Sample path Large Deviations and optimal importance sampling for stochastic volatility models

Sample path Large Deviation Principles (LDP) of the Freidlin–Wentzell type are derived for a class of diffusions, which govern the price dynamics in common stochastic volatility models from Mathematical Finance. LDP are obtained by relaxing the non-degeneracy requirement on the diffusion matrix in the standard theory of Freidlin and Wentzell. As an application, a sample path LDP is proved for the price process in the Heston stochastic volatility model. Using the sample path LDP for the Heston model, the problem is considered of selecting an importance sampling change of drift, for both the price and volatility, which minimize the variance of Monte Carlo estimators for path-dependent option prices. An asymptotically optimal change of drift is identified as a solution to a two-dimensional variational problem. The case of the arithmetic average Asian put option is solved in detail.

Whitham systems and deformations

We consider the deformations of Whitham systems including the “dispersion terms” and having the form of Dubrovin-Zhang deformations of Frobenius manifolds. The procedure is connected with the B. A. Dubrovin problem of deformations of Frobenius manifolds corresponding to the Whitham systems of integrable hierarchies. Under some nondegeneracy requirements we suggest a general scheme of the deformation of the hyperbolic Whitham systems using the initial nonlinear system. The general form of the deformed Whitham system coincides with the form of the “low-dispersion” asymptotic expansions used by B. A. Dubrovin and Y. Zhang in the theory of deformations of Frobenius manifolds.

Chaotic analyses of weakly damped parametrically excited cross waves with surface tension

The Wiggins-Holmes extension of the Generalized Melnikov Method (GMM) to higher dimensions and the extension of the Generalized Herglotz Algorithm (GHA) to non-autonomous systems are applied to weakly damped parametrically excited cross waves with surface tension in a long rectangular wave channel in order to demonstrate that cross waves are chaotic. The Luke Lagrangian density function for surface gravity waves with surface tension and dissipation is expressed in three generalized coordinates (or, equivalently, three degrees of freedom) that are the time-dependent components of three velocity potentials that represent three standing waves. The generalized momenta are computed from the Lagrangian, and the Hamiltonian is computed from a Legendre transform of the Lagrangian. This Hamiltonian contains both autonomous and non-autonomous components that must be suspended by applying an extension of the Herglotz algorithm for non-autonomous transformations in order to apply the Kolmogorov-Arnold-Moser (KAM) averaging operation and the GMM. Three canonical transformations are applied to (i) eliminate cross product terms by a rotation of axes; (ii) to transform to action-angle canonical variables and to eliminate two degrees of freedom; and (iii) to suspend the non-autonomous terms and to apply the Hamilton-Jacobi transformation. The system of nonlinear non-autonomous evolution equations determined from Hamilton’s equations of motion of the second kind must be averaged in order to obtain an autonomous system that may be analyzed by the GMM. Hyperbolic saddle points that are connected by heteroclinic separatrices are computed from the unperturbed autonomous system. The non-dissipative perturbed Hamiltonian system with surface tension satisfies the KAM non-degeneracy requirements, and the Melnikov integral is calculated to demonstrate that the motion is chaotic. For the perturbed dissipative system with surface tension, the only hyperbolic fixed point that survives the averaged equations is a fixed point of weak chaos that is not connected by a homoclinic separatrix; consequently, the Melnikov integral is identically zero. The chaotic motion for the perturbed dissipative system with surface tension is demonstrated by numerical computation of positive Liapunov characteristic exponents.

On the compatible weakly nonlocal Poisson brackets of hydrodynamic type

We consider the pairs of general weakly nonlocal Poisson brackets of hydrodynamic type (Ferapontov brackets) and the corresponding integrable hierarchies. We show that, under the requirement of the nondegeneracy of the corresponding “first” pseudo‐Riemannian metric and also some nondegeneracy requirement for the nonlocal part, it is possible to introduce a “canonical” set of “integrable hierarchies” based on the Casimirs, momentum functional and some “canonical Hamiltonian functions.” We prove also that all the “higher” “positive” Hamiltonian operators and the “negative” symplectic forms have the weakly nonlocal form in this case. The same result is also true for “negative” Hamiltonian operators and “positive” symplectic structures in the case when both pseudo‐Riemannian metrics and are nondegenerate.

Qualitative behavior of nonlinear differential equations describing adaptive filters using nonideal multipliers

We analyze the behavior of a well-known class of analog adaptive filters in which all the multipliers are replaced by nonideal multipliers for which the linearity requirement with respect to one of the inputs is relaxed. Applications of these filters have been proposed in many different engineering contexts which have in common the following idealized problem: a system has a vector input x <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</inf> and a scalar output <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z_{t} = h' x_{t}</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h</tex> is an unknown lime-invariant coefficient vector. From a knowledge of x <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</inf> and z <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</inf> it is required to estimate <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h</tex> . The estimate vector is <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f(\hat{h_{t}})</tex> where h <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</inf> is generated in the filter as the solution of a differential equation and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f(\cdot)</tex> is a nonlinear map defined by the characteristics of the nonideal multipliers. The effectiveness of the filter is determined by the convergence properties of the misalignment vector, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r=h-f(\hat{h})</tex> . With a weak nondegeneracy requirement on x <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</inf> , the "mixing condition," we prove the exponential convergence of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\parallel r_{t} \parallel</tex> . Considerable emphasis is placed on the analysis of the effects of various important departures from the idealized problem as when noise is present, the coefficient vector <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h</tex> is time varying, etc. The results show that in every important aspect the qualitative behavior is similar to that of the conventional filter in which ideal multipliers are used. However, new techniques are required for the analysis of this inherently nonlinear system.

New results on the performance of a well-known class of adaptive filters

We derive a broad range of theoretical results concerning the performance and limitations of a class of analog adaptive filters. Applications of these filters have been proposed in many different engineering contexts which have in common the following idealized identification problem: A system has a vector input x(t) and a scalar output z(t)=h'x(t), where h is an unknown time-invariant coefficient vector. From a knowledge of x(t) and z(t) it is required to estimate h. The filter considered here adjusts an estimate vector h <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">^</sup> (t) in a control loop, thus d/dt h <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">^</sup> = KF[z(t)-z <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">^</sup> (t)]x(t) where z <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">^</sup> (t) = h <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">^</sup> 'x(t), F is a suitable, in general nonlinear, function, and K is the loop gain. The effectiveness of the filter is determined by the convergence properties of the misalignment vector, r = h - h <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">^</sup> . With weak nondegeneracy requirements on x(t) we prove the exponential convergence to zero of the norm ||r(t)||. For all values of K, we give upper and lower bounds on the convergence rate which are tight in that both bounds have similar qualitative dependence on K. The dependence of these bounds on K is unexpected and important since it reveals basic limitations of the filters which are not predicted by the conventional approximate method of analysis, the "method of averaging." By analyzing the effects of an added forcing term u(t) in the control equation we obtain upper bounds to the effects on the convergence process of various important departures from the idealized model as when noise is present as an additional component of z(t), the coefficient vector h is time-varying, and the integrators in a hardware implementation have finite memory.