Interval bounds on the solutions of semi-explicit index-one DAEs. Part 2: computation

This article presents two methods for computing interval bounds on the solutions of nonlinear, semi-explicit, index-one differential-algebraic equations (DAEs). Part 1 presents theoretical developments, while Part 2 discusses implementation and numerical examples. The primary theoretical contributions are (1) an interval inclusion test for existence and uniqueness of a solution, and (2) sufficient conditions, in terms of differential inequalities, for two functions to describe componentwise upper and lower bounds on this solution, point-wise in the independent variable. The first proposed method applies these results sequentially in a two-phase algorithm analogous to validated integration methods for ordinary differential equations (ODEs). The second method unifies these steps to characterize bounds as the solutions of an auxiliary system of DAEs. Efficient implementations of both are described using interval computations and demonstrated on numerical examples.     

Singularity analysis of quasi-linear DAEs

A reduction method is introduced to explore the quasi-linear DAE. On the basis of reduction, three kinds of singularities of quasi-linear DAE are discussed.     

Ultracontractive bounds on Hamilton-Jacobi solutions

Following the equivalence between logarithmic Sobolev inequality, hypercontractivity of the heat semigroup showed by Gross and hypercontractivity of Hamilton-Jacobi equations, we prove, like the Varopoulos theorem, the equivalence between Euclidean-type Sobolev inequality and an ultracontractive control of the Hamilton-Jacobi equations. We obtain also ultracontractive estimations under general Sobolev inequality which imply in the particular case of a probability measure, transportation inequalities.     

Local error estimates for moderately smooth problems: Part I – ODEs and DAEs

The paper consists of two parts. In the first part, we propose a procedure to estimate local errors of low order methods applied to solve initial value problems in ordinary differential equations (ODEs) and index 1 differential-algebraic equations (DAEs). Based on the idea of defect correction we develop local error estimates for the case when the problem data is only moderately smooth. Numerical experiments illustrate the performance of the mesh adaptation based on the error estimation developed in this paper. In the second part of the paper, we will consider the estimation of local errors in context of stochastic differential equations with small noise. AMS subject classification (2000)  65L06, 65L80, 65L50, 65L05     

Error bounds for suboptimal solutions to kernel principal component analysis

Suboptimal solutions to kernel principal component analysis are considered. Such solutions take on the form of linear combinations of all n-tuples of kernel functions centered on the data, where n is a positive integer smaller than the cardinality m of the data sample. Their accuracy in approximating the optimal solution, obtained in general for n = m, is estimated. The analysis made in Gnecco and Sanguineti (Comput Optim Appl 42:265–287, 2009) is extended. The estimates derived therein for the approximation of the first principal axis are improved and extensions to the successive principal axes are derived.     

Bivariational bounds on solutions of two-point boundary-value problems

Bivariational principles for a linear equation in a Hilbert space are used to derive complementary upper and lower bounds on solutions of two-point boundary-value problems. The functional dependence of the bounds is exhibited, and various simplified versions of them are discussed. Illustrative examples are presented, showing encouraging accuracy with simple trial vectors.     

Computable bounds on parametric solutions of convex problems

We present a new method for computing bounds on parametric solutions of convex problems. The approach is based on a uniform quadratic underestimation of the objective function and a simple technique for the calculation of bounds on the optimal value function.Research supported by Grant ECS-8619859, National Science Foundation and Contract N00017-86-K-0052, Office of Naval Research.     

Classical solutions of two-dimensional stokes problems on non-smooth domains. Part 1: The radon integral operators

The applicability of the Neumann indirect method of potentials to the Dirichlet and Neumann problems for the two-dimensional Stokes operator on a non-smooth boundary Γ is subject to two kinds of sufficient and/or necessary conditions on Γ. The first one, occurring in electrostatic, is equivalent to the boundedness on C(Γ) of the velocity double-layer potential W as well as to the existence of jump relations of potentials. The second condition, which forces Γ to be a simple rectifiable curve and which, compared to the Laplacian, is a stronger restriction on the corners of Γ, states that the Fredholm radius of W is greater than 2. Under these conditions, the Radon boundary integral equations defined by the above-mentioned jump relations are solvable by the Fredholm theory; the double- (for Dirichlet) and the single- (for Neumann) layer potentials corresponding to their solutions are classical solutions of the Stokes problems.     

Component-wise perturbation analysis and error bounds for linear least squares solutions

Perturbation bounds for the linear least squares problem min _x Ax –b₂ corresponding tocomponent-wise perturbations in the data are derived. These bounds can be computed using a method of Hager and are often much better than the bounds derived from the standard perturbation analysis. In particular this is true for problems where the rows ofA are of widely different magnitudes. Generalizing a result by Oettli and Prager, we can use the bounds to compute a posteriori error bounds for computed least squares solutions.     

Improved bounds on equilibria solutions in the network design game

In the network design game with n players, every player chooses a path in an edge-weighted graph to connect her pair of terminals, sharing costs of the edges on her path with all other players fairly. It has been shown that the price of stability of any network design game is at most \(H_n\), the n-th harmonic number. This bound is tight for directed graphs.For undirected graphs, it has only recently been shown that the price of stability is at most \(H_n \left( 1-\frac{1}{\Theta (n^4)} \right) \), while the worst-case known example has price of stability around 2.25. We improve the upper bound considerably by showing that the price of stability is at most \(H_{n/2} + \varepsilon \) for any \(\varepsilon \) starting from some suitable \(n \ge n(\varepsilon )\).We also study quality measures of different solution concepts for the multicast network design game on a ring topology. We recall from the literature a lower bound of \(\frac{4}{3}\) and prove a matching upper bound for the price of stability. Therefore, we answer an open question posed by Fanelli et al. (Theor Comput Sci 562:90–100, 2015). We prove an upper bound of 2 for the ratio of the costs of a potential optimizer and of an optimum, provide a construction of a lower bound, and give a computer-assisted argument that it reaches 2 for any precision. We then turn our attention to players arriving one by one and playing myopically their best response. We provide matching lower and upper bounds of 2 for the myopic sequential price of anarchy (achieved for a worst-case order of the arrival of the players). We then initiate the study of myopic sequential price of stability and for the multicast game on the ring we construct a lower bound of \(\frac{4}{3}\), and provide an upper bound of \(\frac{26}{19}\). To the end, we conjecture and argue that the right answer is \(\frac{4}{3}\).     

Some remarks on blvarlational bounds and the pointwise estimation of solutions

When a given equation can be realised in some Hilbert space H as an operator equation in the from A[d] then it is known that complementary variational bounds can be obtained for the inner product [d]. Recently it has been shown that complementary bivariational bounds can be obtained for the inner product [d]associated with the equation A[d]= f and an arbitary elementy [d]. These latter bounds are a natural starting point for investigatng pointwise estimatesd of solutions provided thta certain questions relating, on the one hand, to the self-adjointness and the invertibility of associated operators and on the other to the availability of a suitable large Hilbert space can be resolved. We show that if the underlying problem can be given an operator realisation in a suitably equipped Hilbert space then pointwise estimates of solution can be obtained     

Stability and bifurcation properties of index-1 DAEs

It is well known that an equilibrium of a semi-explicit, index-1 differential-algebraic equation under a parameter variation may encounter the singularity manifold. It is a generic property of this encounter that one eigenvalue of the linear stability mapping associated with the equilibrium will pass from one half of the complex plane to the other without passing through the imaginary axis. This is known as singularity-induced bifurcation and an equivalent result is proven in this paper. While this property is generic, it is shown how more than one eigenvalue can diverge in an analogous manner, with applications in electrical power systems. This revised version was published online in June 2006 with corrections to the Cover Date.     

Suboptimal policy determination for large-scale Markov decision processes,Part 1: Description and bounds

This paper is the first of two papers that present and evaluate an approach for determining suboptimal policies for large-scale Markov decision processes (MDP). Part 1 is devoted to the determination of bounds that motivate the development and indicate the quality of the suboptimal design approach; Part 2 is concerned with the implementation and evaluation of the suboptimal design approach. The specific MDP considered is the infinite-horizon, expected total discounted cost MDP with finite state and action spaces. The approach can be described as follows. First, the original MDP is approximated by a specially structured MDP. The special structure suggests how to construct associated smaller, more computationally tractable MDP's. The suboptimal policy for the original MDP is then constructed from the solutions of these smaller MDP's. The key feature of this approach is that the state and action space cardinalities of the smaller MDP's are exponential reductions of the state and action space cardinalities of the original MDP.     

The bounds of the solutions to generalized modular equations

In this paper the authors study a monotonicity of several functions involving ma(r) and μa(r). By using these results, the authors obtain some new bounds of the solutions of Ramanujan's generalized modular equations.     

Nonlinear integrodifferential equations anda priori bounds on periodic solutions

Summary This paper studies the existence of aperiodic solution of a nonlinear integrodifferential system of the form , for each continuous periodic function p and under suitable assumptions on f, k and g. A topological transversality method is employed to obtain the existence of periodic solutions. This method relies ona priori bounds on periodic solutions. Several examples are provided where a variant of Liapunov's direct method is employed to obtaina priori bounds on periodic solutions.     

Equivariant cosymmetry and front solutions of the Dubreil–Jacotin–Long equation. Part 1: Boussinesq limit

The problem of two-dimensional internal travelling waves in a perfect fluid with smooth density being close to linear stratification is considered. Approximate front solutions connecting uniform flow with a conjugate shear flow of the first mode are constructed. It is demonstrated that the number of the front branches essentially depends on the fine-scale stratification for linear density background. To cite this article: N. Makarenko, C. R. Acad. Sci. Paris, Ser. I 337 (2003).