Showing papers in "Applications of Mathematics in 2006"

Regularity of minima: an invitation to the Dark Side of the Calculus of Variations

Abstract: I am presenting a survey of regularity results for both minima of variational integrals, and solutions to non-linear elliptic, and sometimes parabolic, systems of partial differential equations. I will try to take the reader to the Dark Side...

From scalar to vector optimization

Abstract: Initially, second-order necessary optimality conditions and sufficient optimality conditions in terms of Hadamard type derivatives for the unconstrained scalar optimization problem ϕ(x) → min, x ∈ ℝ m , are given. These conditions work with arbitrary functions ϕ: ℝ m → ℝ, but they show inconsistency with the classical derivatives. This is a base to pose the question whether the formulated optimality conditions remain true when the “inconsistent” Hadamard derivatives are replaced with the “consistent” Dini derivatives. It is shown that the answer is affirmative if ϕ is of class $$\mathcal{C}^{1,1}$$ (i.e., differentiable with locally Lipschitz derivative). Further, considering $$\mathcal{C}^{1,1}$$ functions, the discussion is raised to unconstrained vector optimization problems. Using the so called “oriented distance” from a point to a set, we generalize to an arbitrary ordering cone some second-order necessary conditions and sufficient conditions given by Liu, Neittaanmaki, Krizek for a polyhedral cone. Furthermore, we show that the conditions obtained are sufficient not only for efficiency but also for strict efficiency.

Sparse finite element methods for operator equations with stochastic data

Abstract: Let A: V → V′ be a strongly elliptic operator on a d-dimensional manifold D (polyhedra or boundaries of polyhedra are also allowed) An operator equation Au = f with stochastic data f is considered The goal of the computation is the mean field and higher moments $$\mathcal{M}^1 u \in V,\mathcal{M}^2 u \in V \otimes V,,\mathcal{M}^k u \in V \otimes \otimes V$$ of the solution We discretize the mean field problem using a FEM with hierarchical basis and N degrees of freedom We present a Monte-Carlo algorithm and a deterministic algorithm for the approximation of the moment $$\mathcal{M}^k u$$ for k⩾1 The key tool in both algorithms is a “sparse tensor product” space for the approximation of $$\mathcal{M}^k u$$ with O(N(log N) k−1) degrees of freedom, instead of N k degrees of freedom for the full tensor product FEM space A sparse Monte-Carlo FEM with M samples (ie, deterministic solver) is proved to yield approximations to $$\mathcal{M}^k u$$ with a work of O(M N(log N) k−1) operations The solutions are shown to converge with the optimal rates with respect to the Finite Element degrees of freedom N and the number M of samples The deterministic FEM is based on deterministic equations for $$\mathcal{M}^k u$$ in D k ⊂ ℝkd Their Galerkin approximation using sparse tensor products of the FE spaces in D allows approximation of $$\mathcal{M}^k u$$ with O(N(log N) k−1) degrees of freedom converging at an optimal rate (up to logs) For nonlocal operators wavelet compression of the operators is used The linear systems are solved iteratively with multilevel preconditioning This yields an approximation for $$\mathcal{M}^k u$$ with at most O(N (log N) k+1) operations

Approximation of an Eigenvalue Problem Associated with the Stokes Problem by the Stream Function-Vorticity-Pressure Method

Abstract: By means of eigenvalue error expansion and integral expansion techniques, we propose and analyze the stream function-vorticity-pressure method for the eigenvalue problem associated with the Stokes equations on the unit square. We obtain an optimal order of convergence for eigenvalues and eigenfuctions. Furthermore, for the bilinear finite element space, we derive asymptotic expansions of the eigenvalue error, an efficient extrapolation and an a posteriori error estimate for the eigenvalue. Finally, numerical experiments are reported.

Error Estimates for Linear Finite Elements on Bakhvalov-Type Meshes

Abstract: For convection-diffusion problems with exponential layers, optimal error estimates for linear finite elements on Shishkin-type meshes are known. We present the first optimal convergence result in an energy norm for a Bakhvalov-type mesh.

On Large Eddy Simulation and Variational Multiscale Methods in the numerical simulation of turbulent incompressible flows

Abstract: Numerical simulation of turbulent flows is one of the great challenges in Computational Fluid Dynamics (CFD) In general, Direct Numerical Simulation (DNS) is not feasible due to limited computer resources (performance and memory), and the use of a turbulence model becomes necessary The paper will discuss several aspects of two approaches of turbulent modeling—Large Eddy Simulation (LES) and Variational Multiscale (VMS) models Topics which will be addressed are the detailed derivation of these models, the analysis of commutation errors in LES models as well as other results from mathematical analysis

Partition of unity method for Helmholtz equation: $q$-convergence for plane-wave and wave-band local bases

Abstract: In this paper we study the q-version of the Partition of Unity Method for the Helmholtz equation. The method is obtained by employing the standard bilinear finite element basis on a mesh of quadrilaterals discretizing the domain as the Partition of Unity used to paste together local bases of special wave-functions employed at the mesh vertices. The main topic of the paper is the comparison of the performance of the method for two choices of local basis functions, namely a) plane-waves, and b) wave-bands. We establish the q-convergence of the method for the class of analytical solutions, with q denoting the number of plane-waves or wave-bands employed at each vertex, for which we get better than exponential convergence for sufficiently small h, the mesh-size of the employed mesh. We also discuss the a-posteriori estimation for any solution quantity of interest and the problem of quadrature for all integrals employed. The goal of the paper is to stimulate theoretical development which could explain various numerical features. A main open question is the analysis of the pollution and its disappearance as function of h and q.

On two-scale convergence and related sequential compactness topics

Abstract: A general concept of two-scale convergence is introduced and two-scale compactness theorems are stated and proved for some classes of sequences of bounded functions in L 2(Ω) involving no periodicity assumptions. Further, the relation to the classical notion of compensated compactness and the recent concepts of two-scale compensated compactness and unfolding is discussed and a defect measure for two-scale convergence is introduced.

Exponential smoothing for irregular data

Abstract: Various types of exponential smoothing for data observed at irregular time intervals are surveyed. Double exponential smoothing and some modifications of Holt’s method for this type of data are suggested. A real data example compares double exponential smoothing and Wright’s modification of Holt’s method for data observed at irregular time intervals.

The density of infinitely differentiable functions in Sobolev spaces with mixed boundary conditions

Abstract: We present a detailed proof of the density of the set $$C^\infty (\bar \Omega ) \cap V$$ in the space of test functions V ⊂ H 1 (Ω) that vanish on some part of the boundary ∂Ω of a bounded domain Ω.

Approximation of the Stieltjes integral and applications in numerical integration

Abstract: Some inequalities for the Stieltjes integral and applications in numerical integration are given. The Stieltjes integral is approximated by the product of the divided difference of the integrator and the Lebesgue integral of the integrand. Bounds on the approximation error are provided. Applications to the Fourier Sine and Cosine transforms on finite intervals are mentioned as well.

Finite volume schemes for multi-dimensional hyperbolic systems based on the use of bicharacteristics

Abstract: In this paper we present recent results for the bicharacteristic based finite volume schemes, the so-called finite volume evolution Galerkin (FVEG) schemes. These methods were proposed to solve multi-dimensional hyperbolic conservation laws. They combine the usually conflicting design objectives of using the conservation form and following the characteristics, or bicharacteristics. This is realized by combining the finite volume formulation with approximate evolution operators, which use bicharacteristics of the multi-dimensional hyperbolic system. In this way all of the infinitely many directions of wave propagation are taken into account. The main goal of this paper is to present a self-contained overview on the recent results. We study the L1-stability of the finite volume schemes obtained by various approximations of the flux integrals. Several numerical experiments presented in the last section confirm robustness and correct multi-dimensional behaviour of the FVEG methods.

On a multiplicative type sum form functional equation and its role in information theory

Abstract: In this paper, we obtain all possible general solutions of the sum form functional equations $$\sum\limits_{i = 1}^k {\sum\limits_{j = 1}^l {f(p_i q_j )} } = \sum\limits_{i = 1}^k {g(p_i )} \sum\limits_{j = 1}^l {h(q_j )} $$ and $$\sum\limits_{i = 1}^k {\sum\limits_{j = 1}^l {F(p_i q_j )} } = \sum\limits_{i = 1}^k {G(p_i ) + } \sum\limits_{j = 1}^l {H(q_j ) + \lambda } \sum\limits_{i = 1}^k {G(p_i )} \sum\limits_{j = 1}^l {H(q_j )} $$ valid for all complete probability distributions (p1, ..., pk), (q1, ..., ql), k ≥ 3, l ≥ 3 fixed integers; λ ∈ ℝ, λ ≠ 0 and F, G, H, f, g, h are real valued mappings each having the domain I = [0, 1], the unit closed interval.

A mathematical model describing the thyroid-pituitary axis with time delays in hormone transportation

Abstract: In the present paper, a mathematical model, originally proposed by Danziger and Elmergreen and describing the thyroid-pituitary homeostatic mechanism, is modified and analyzed for its physiological and clinical significance. The inuence of different system parameters on the stability behavior of the system is discussed. The transportation delays of different hormones in the bloodstream, both in the discrete and distributed forms, are considered. Delayed models are analyzed regarding the stability and bifurcation behavior. Clinical treatment of periodic catatonic schizophrenia is discussed in presence of transportation delays. Numerical simulations are presented to support analytic results.

Weak solutions to a nonlinear variational wave equation and some related problems

Abstract: In this paper we present some results on the global existence of weak solutions to a nonlinear variational wave equation and some related problems. We first introduce the main tools, the L p Young measure theory and related compactness results, in the first section. Then we use the L p Young measure theory to prove the global existence of dissipative weak solutions to the asymptotic equation of the nonlinear wave equation, and comment on its relation to Camassa-Holm equations in the second section. In the third section, we prove the global existence of weak solutions to the original nonlinear wave equation under some restrictions on the wave speed. In the last section, we present global existence of renormalized solutions to two-dimensional model equations of the asymptotic equation, which is also the so-called vortex density equation arising from sup-conductivity.

A comparison of solvers for linear complementarity problems arising from large-scale masonry structures

Abstract: We compare the numerical performance of several methods for solving the discrete contact problem arising from the finite element discretisation of elastic systems with numerous contact points. The problem is formulated as a variational inequality and discretised using piecewise quadratic finite elements on a triangulation of the domain. At the discrete level, the variational inequality is reformulated as a classical linear complementarity system. We compare several state-of-art algorithms that have been advocated for such problems. Computational tests illustrate the use of these methods for a large collection of elastic bodies, such as a simplified bidimensional wall made of bricks or stone blocks, deformed under volume and surface forces.

Worst scenario method in homogenization. Linear case

Abstract: The paper deals with homogenization of a linear elliptic boundary problem with a specific class of uncertain coefficients describing composite materials with periodic structure. Instead of stochastic approach to the problem, we use the worst scenario method due to Hlavacek (method of reliable solution). A few criterion functionals are introduced. We focus on the range of the homogenized coefficients from knowledge of the ranges of individual components in the composite, on the values of generalized gradient in the places where these components change and on the average of homogenized solution in some critical subdomain.

On the search for singularities in incompressible flows

Abstract: In these notes we give some examples of the interaction of mathematics with experiments and numerical simulations on the search for singularities.

Explicit solution for Lamé and other PDE systems

Abstract: We provide a general series form solution for second-order linear PDE system with constant coefficients and prove a convergence theorem The equations of three dimensional elastic equilibrium are solved as an example Another convergence theorem is proved for this particular system We also consider a possibility to represent solutions in a finite form as partial sums of the series with terms depending on several complex variables

What is the smallest possible constant in Céa’s lemma?

Abstract: We consider finite element approximations of a second order elliptic problem on a bounded polytopic domain in ℝd with d ∈ {1, 2, 3, ...} The constant C ⩾ 1 appearing in Cea’s lemma and coming from its standard proof can be very large when the coefficients of an elliptic operator attain considerably different values. We restrict ourselves to regular families of uniform partitions and linear simplicial elements. Using a lower bound of the interpolation error and the supercloseness between the finite element solution and the Lagrange interpolant of the exact solution, we show that the ratio between discretization and interpolation errors is equal to \(1 + \mathcal{O}(h)\) as the discretization parameter h tends to zero. Numerical results in one and two-dimensional case illustrating this phenomenon are presented.

On discontinuous Galerkin method and semiregular family of triangulations

Abstract: Discretization of second order elliptic partial differential equations by discontinuous Galerkin method often results in numerical schemes with penalties. In this paper we analyze these penalized schemes in the context of quite general triangular meshes satisfying only a semiregularity assumption. A new (modified) penalty term is presented and theoretical properties are proven together with illustrative numerical results.

On valuation of derivative securities: A lie group analytical approach

Abstract: This paper proposes a Lie group analytical approach to tackle the problem of pricing derivative securities. By exploiting the infinitesimal symmetries of the Boundary Value Problem (BVP) satisfied by the price of a derivative security, our method provides an effective algorithm for obtaining its explicit solution.

Regularity and uniqueness for the stationary large eddy simulation model

Abstract: In the note we are concerned with higher regularity and uniqueness of solutions to the stationary problem arising from the large eddy simulation of turbulent ows. The system of equations contains a nonlocal nonlinear term, which prevents straightforward application of a difference quotients method. The existence of weak solutions was shown in A. Świerczewska: Large eddy simulation. Existence of stationary solutions to the dynamical model, ZAMM, Z. Angew. Math. Mech. 85 (2005), 593–604 and P. Gwiazda, A. Świerczewska: Large eddy simulation turbulence model with Young measures, Appl. Math. Lett. 18 (2005), 923–929.

On stability of the P n mod /P n element for incompressible flow problems

Abstract: It is well known that finite element spaces used for approximating the velocity and the pressure in an incompressible flow problem have to be stable in the sense of the inf-sup condition of Babuska and Brezzi if a stabilization of the incompressibility constraint is not applied. In this paper we consider a recently introduced class of triangular nonconforming finite elements of nth order accuracy in the energy norm called P n mod elements. For n ≤ 3 we show that the stability condition holds if the velocity space is constructed using the P n mod elements and the pressure space consists of continuous piecewise polynomial functions of degree n.

Some remarks to multivariate regression model

Abstract: Some remarks to problems of point and interval estimation, testing and problems of outliers are presented in the case of multivariate regression model.

The Neumann problem for some degenerate elliptic equations

Abstract: In the paper we study the equation L u = f, where L is a degenerate elliptic operator, with Neumann boundary condition in a bounded open set μ. We prove existence and uniqueness of solutions in the space H(μ) for the Neumann problem.

On an elasto-dynamic evolution equation with non dead load and friction

Abstract: In this paper, we are interested in the dynamic evolution of an elastic body, acted by resistance forces depending also on the displacements. We put the mechanical problem into an abstract functional framework, involving a second order nonlinear evolution equation with initial conditions. After specifying convenient hypotheses on the data, we prove an existence and uniqueness result. The proof is based on Faedo-Galerkin method.