A split least-squares characteristic mixed element method for nonlinear nonstationary convection–diffusion problem

In this paper, a split least-squares characteristic mixed finite element method is proposed for solving nonlinear nonstationary convection–diffusion problem. By selecting the least-squares functional property, the resulting least-squares procedure can be split into two independent symmetric positive definite sub-schemes. The first sub-scheme is for the unknown variable u, which is the same as the standard characteristic Galerkin finite element approximation. The second sub-scheme is for the unknown flux σ. Theoretical analysis shows that the method yields the approximate solutions with optimal accuracy in L ²(Ω) norm for the primal unknown and in H(div; Ω) norm for the unknown flux, respectively. Some numerical examples are given to confirm our theory results.     

New explicit filters and smoothers for diffusions with nonlinear drift and measurements

The optimal least-squares filtering of a diffusion x(t) from its noisy measurements {y(τ); 0 τ t} is given by the conditional mean E[x(t)|y(τ); 0 τ t]. When x(t) satisfies the stochastic diffusion equation dx(t) = f(x(t)) dt + dw(t) and y(t) = ∫₀^tx(s) ds + b(t), where f(·) is a global solution of the Riccati equation /xf(x) + f(x)² = f(x)² = αx² + βx + γ, for some , and w(·), b(·) are independent Brownian motions, Benes gave an explicit formula for computing the conditional mean. This paper extends Benes results to measurements y(t) = ∫₀^tx(s) ds + ∫₀^t dx(s) + b(t) (and its multidimensional version) without imposing additional conditions on f(·). Analogous results are also derived for the optimal least-squares smoothed estimate E[x(s)|y(τ); 0 τ t], s < t. The methodology relies on Girsanov's measure transformations, gauge transformations, function space integrations, Lie algebras, and the Duncan-Mortensen-Zakai equation.     

A New Nonsymmetric Discontinuous Galerkin Method for Time Dependent Convection Diffusion Equations

We propose a discontinuous Galerkin finite element method for convection diffusion equations that involves a new methodology handling the diffusion term. Test function derivative numerical flux term is introduced in the scheme formulation to balance the solution derivative numerical flux term. The scheme has a nonsymmetric structure. For general nonlinear diffusion equations, nonlinear stability of the numerical solution is obtained. Optimal kth order error estimate under energy norm is proved for linear diffusion problems with piecewise P ^k polynomial approximations. Numerical examples under one-dimensional and two-dimensional settings are carried out. Optimal (k+1)th order of accuracy with P ^k polynomial approximations is obtained on uniform and nonuniform meshes. Compared to the Baumann-Oden method and the NIPG method, the optimal convergence is recovered for even order P ^k polynomial approximations.     

Superconvergence of Discontinuous Finite Element Solutions for Transient Convection–diffusion Problems

We present a study of the local discontinuous Galerkin method for transient convection–diffusion problems in one dimension. We show that p-degree piecewise polynomial discontinuous finite element solutions of convection-dominated problems are O(Δx ^p+2) superconvergent at Radau points. For diffusion- dominated problems, the solution’s derivative is O(Δx ^p+2) superconvergent at the roots of the derivative of Radau polynomial of degree p+1. Using these results, we construct several asymptotically exact a posteriori finite element error estimates. Computational results reveal that the error estimates are asymptotically exact.This revised version was published online in July 2005 with corrected volume and issue numbers.     

Compact difference method for solving the fractional reaction–subdiffusion equation with Neumann boundary value condition

In this paper, we derive a high-order compact finite difference scheme for solving the reaction–subdiffusion equation with Neumann boundary value condition. The L1 method is used to approximate the temporal Caputo derivative, and the compact difference operator is applied for spatial discretization. We prove that the compact finite difference method is unconditionally stable and convergent with order O(τ^2?α+h⁴) in L₂ norm, where τ, α, and h are the temporal step size, the order of time fractional derivative and the spatial step size, respectively. Finally, some numerical experiments are carried out to show the effectiveness of the proposed difference scheme.     

On the Sample Complexity of Weak Learning

In this paper, we study the sample complexity of weak learning. That is, we ask how many data must be collected from an unknown distribution in order to extract a small but significant advantage in prediction. We show that it is important to distinguish between those learning algorithms that output deterministic hypotheses and those that output randomized hypotheses. We prove that in the weak learning model, any algorithm using deterministic hypotheses to weakly learn a class of Vapnik-Chervonenkis dimension d(n) requires Ω ([formula]) examples. In contrast, when randomized hypotheses are allowed, we show that Θ (1) examples suffice in some cases. We then show that there exists an efficient algorithm using deterministic hypotheses that weakly learns against any distribution on a set of size d(n) with only O(d(n)^2/3) examples. Thus for the class of symmetric Boolean functions over n variables, where the strong learning sample complexity is Θ (n), the sample complexity for weak learning using deterministic hypotheses is Ω ([formula]) and O(n^2/3), and the sample complexity for weak learning using randomized hypotheses is Θ (1). Next we prove the existence of classes for which the distribution-free sample size required to obtain a slight advantage in prediction over random guessing is essentially equal to that required to obtain arbitrary accuracy. Finally, for a class of small circuits, namely all parity functions of subsets of n Boolean variables, we prove a weak learning sample complexity of Θ(n). This bound holds even if the weak learning algorithm is allowed to replace random sampling with membership queries, and the target distribution is uniform on {0, 1}ⁿ.     

Empirical Bayes estimation of the scale parameter in a Pareto distribution

Let f(xθ) = αθ^αx^−(α+1)I(x>θ) be the pdf of a Pareto distribution with known shape parameter α>0, and unknown scale parameter θ. Let {(X_i, θ_i)} be a sequence of independent random pairs, where X_i's are independent with pdf f(xα_i), and θ_i are iid according to an unknown distribution G in a class of distributions whose supports are included in an interval (0, m), where m is a positive finite number. Under some assumption on the class and squared error loss, at (n + 1)th stage we construct a sequence of empirical Bayes estimators of θ_n+1 based on the past n independent observations X₁,…, X_n and the present observation X_n+1. This empirical Bayes estimator is shown to be asymptotically optimal with rate of convergence O(n^−1/2). It is also exhibited that this convergence rate cannot be improved beyond n^−1/2 for the priors in class .     

A full discrete two-grid finite-volume method for a nonlinear parabolic problem

A fully discrete two-grid finite-volume method (FVM) for a nonlinear parabolic problem is studied in this paper. This method involves solving a nonlinear parabolic equation on coarse mesh space and a linearized parabolic equation on fine grid. Both L ² and H ¹ norm error estimates of the standard FVM for the nonlinear parabolic problem are derived. Compared with the standard FVM, the two-level method is of the same order as the one-level method in the H ¹-norm as long as the mesh sizes satisfy h=𝒪(H ^3/2). However, the two-level method involves much less work than the standard method. Numerical results are provided to demonstrate the effectiveness of our algorithm.     

Stability of a Class of Differential Inclusions

Stability of differential inclusions F(x(t)) is studied by using minorant and majorant mappings F ^– and F ⁺, F ^–(x)F(x)F ⁺(x). Properties of F ^–,F ⁺ are developed in terms of partial orderings, with the condition that F ^–, F ⁺ are either heterotone or pseudoconcave. The main results concern asymptotically stable absorbing sets, including the case of a single equilibrium point, and are illustrated by examples of control systems.     

Toward a theory of Pollard's rho method

Pollard's “rho” method for integer factorization iterates a simple polynomial map and produces a nontrivial divisor of n when two such iterates agree modulo this divisor. Experience and heuristic arguments suggest that a prime divisor p should be detected in steps, but this has never been proved. Indeed, nothing seems to be have been rigorously proved about the probability of success that would improve the obvious lower bound of 1/p. This paper shows that for fixed k, this probability is at least (₂^k)/p + O(p^−3/2) as p → ∞. This leads to an Ω(log²p)/p estimate of the success probability.     

Piecemeal Graph Exploration by a Mobile Robot

We study how a mobile robot can learn an unknown environment in a piecemeal manner. The robot's goal is to learn a complete map of its environment, while satisfying the constraint that it must return every so often to its starting position (for refueling, say). The environment is modeled as an arbitrary, undirected graph, which is initially unknown to the robot. We assume that the robot can distinguish vertices and edges that it has already explored. We present a surprisingly efficient algorithm for piecemeal learning an unknown undirected graph G=(V, E) in which the robot explores every vertex and edge in the graph by traversing at most O(E+V^1+o(1)) edges. This nearly linear algorithm improves on the best previous algorithm, in which the robot traverses at most O(E+V²) edges. We also give an application of piecemeal learning to the problem of searching a graph for a “treasure.”     

Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms

This paper examine the Euler-Lagrange equations for the solution of the large deformation diffeomorphic metric mapping problem studied in Dupuis et al. (1998) and Trouvé (1995) in which two images I ₀, I ₁ are given and connected via the diffeomorphic change of coordinates I ₀○ϕ⁻¹=I ₁ where ϕ=Φ₁ is the end point at t= 1 of curve Φ _t , t∈[0, 1] satisfying ^.Φ _t =v _t (Φ _t ), t∈ [0,1] with Φ₀=id. The variational problem takes the form

QFT template generation for time-delay plants based on zero-inclusion test

Plant template generation is the key step in applying quantitative feedback theory (QFT) to design robust control for uncertain systems. In this paper we propose a technique for generating plant templates for a class of linear systems with an uncertain time delay and affine parameter perturbations in coefficients. The main contribution lies in presenting a necessary and sufficient condition for the zero inclusion of the value set f(T,Q)={f(τ,q): τT[τ⁻,τ⁺], qQ∏_k=0^m−1[q_k⁻,q_k⁺]}, where f(τ,q)=g(q)+h(q)e^−jτω*, g(q) and h(q) are both complex-valued affine functions of the m-dimensional real vector q, and ω^* is a fixed frequency. Based on this condition, an efficient algorithm which involves, in the worst case, evaluation of m algebraic inequalities and solution of m2^m−1 one-variable quadratic equations, is developed for testing the zero inclusion of the value set f(T,Q). This zero-inclusion test algorithm allows one to utilize a pivoting procedure to generate the outer boundary of a plant template with a prescribed accuracy or resolution. The proposed template generation technique has a linear computational complexity in resolution and is, therefore, more efficient than the parameter gridding and interval methods. A numerical example illustrating the proposed technique and its computational superiority over the interval method is included.     

Perturbing Bézier coefficients for best constrained degree reduction in the L2-norm

This paper first shows how the Bézier coefficients of a given degree n polynomial are perturbed so that it can be reduced to a degree m (<n) polynomial with the constraint that continuity of a prescribed order is preserved at the two endpoints. The perturbation vector, which consists of the perturbation coefficients, is determined by minimizing a weighted Euclidean norm. The optimal degree n−1 approximation polynomial is explicitly given in Bézier form. Next the paper proves that the problem of finding a best L₂-approximation over the interval [0,1] for constrained degree reduction is equivalent to that of finding a minimum perturbation vector in a certain weighted Euclidean norm. The relevant weights are derived. This result is applied to computing the optimal constrained degree reduction of parametric Bézier curves in the L₂-norm.     

Control of singular problem via differentiation of a Min-Max

For Ω a smooth domain in Rⁿ with boundary Λ = Λ₀Λ₁, we are concerned with the wave equation y″ − Δy = S in Q_T =]0, T[ × Ω with = ∂/∂t, at source term satisfying S, S′ ε L¹(0, T L² (Ω)). A Dirichlet condition is imposed on Λ₀ and we consider an absorbing condition ∂y/∂n + uy′ = 0 in [0, T] × gL₁ where u is the control.parameter. We introduce the cost function. and using the Min-Max formulation of J we by-pasas the sensitivity analysis of u → y and obtain the gradient of J with a usual adjoint problem. We first present an abstract frame for this kind of problems. using the differentiability results of a Min-Max [1,2], which we very shortly deduce here, we show that the well posedness of the adjoint equation implies differentiability of the cost function governed by a linear well posed problem.     

Approximation by Bézier type of Meyer–König and Zeller operators

In this paper, we give direct, inverse and equivalence approximation theorems for the Bézier type of Meyer–König and Zeller operator with unified Ditzian–Totik modulus ω_φ^λ(f,t) (0≤λ≤1).     

A least distance algorithm for a smooth strictly convex norm

An algorithm for approximating a non negative solution of inconsistent systems of linear equations is presented. We define a best approximate solution of a system Ax = b x≥0 to be the vector x≥0 which minimizes the norm of the residual r(x) = b ? Ax, for a smooth and strictly convex norm. The algorithm is shown to be feasible and globally convergent. The special case of the ? ^p norm is included. In particular, the method converges for 1 < p < 2. A generalization of this algorithm is also given. Numerical results are included.     

Existence of triple positive solutions of two-point right focal boundary value problems on time scales

We consider the following boundary value problem, (−1)ⁿ⁻¹y^Δn(t)=(−1)^p+1F(t,y(σⁿ⁻¹(t))),t[a,b]∩T, y^Δn(a)=0,0≤i≤p−1, y^Δn(σ(b))=0,p≤i≤n−1,where n ≥ 2, 1 ≤ p ≤ n - 1 is fixed and T is a time scale. By applying fixed-point theorems for operators on a cone, existence criteria are developed for triple positive solutions of the boundary value problem. We also include examples to illustrate the usefulness of the results obtained.     

A Hermitian Box-scheme for One-dimensional Elliptic Equations – Application to Problems with High Contrasts in the Ellipticity

We introduce a new box-scheme, called ``hermitian box-scheme' on the model of the one-dimensional Poisson problem. The scheme combines features of the box-scheme of Keller, [20], [13], with the hermitian approximation of the gradient on a compact stencil, which is characteristic of compact schemes, [9], [21]. The resulting scheme is proved to be 4th order accurate for the primitive unknown u and its gradient p. The proved convergence rate is 1.5 for (u,p) in the discrete L ² norm. The connection with a non standard mixed finite element method is given. Finally, numerical results are displayed on pertinent 1-D elliptic problems with high contrasts in the ellipticity, showing in practice convergence rates ranging from 1 to 2.5 in the discrete H ¹ norm. This work has been performed with the support of the GDR MOMAS, (ANDRA, CEA, EDF, BRGM and CNRS): Modélisation pour le stockage des déchets radioactifs. The author thanks especially A. Bourgeat for his encouragements and his interest in this work.