Error estimates for approximate approximations on compact intervals

The aim of this paper is the investigation of the error which results from the method of approximate approximations applied to functions defined on compact in- tervals, only. This method, which is based on an approximate partition of unity, was introduced by V. Mazya in 1991 and has mainly been used for functions defied on the whole space up to now. For the treatment of differential equations and boundary integral equations, however, an efficient approximation procedure on compact intervals is needed. In the present paper we apply the method of approximate approximations to functions which are defined on compact intervals. In contrast to the whole space case here a truncation error has to be controlled in addition. For the resulting total error pointwise estimates and L1-estimates are given, where all the constants are determined explicitly.

Numerical Methods for Non-Stationary Stokes Flow

We consider a first order implicit time stepping procedure (Euler scheme) for the non-stationary Stokes equations in smoothly bounded domains of R3. Using energy estimates we can prove optimal convergence properties in the Sobolev spaces Hm(G) (m = 0;1;2) uniformly in time, provided that the solution of the Stokes equations has a certain degree of regularity. For the solution of the resulting Stokes resolvent boundary value problems we use a representation in form of hydrodynamical volume and boundary layer potentials, where the unknown source densities of the latter can be determined from uniquely solvable boundary integral equations’ systems. For the numerical computation of the potentials and the solution of the boundary integral equations a boundary element method of collocation type is used. Some simulations of a model problem are carried out and illustrate the efficiency of the method.

A multiwave range test for obstacle reconstructions with unknown physical properties

We develop a new multiwave version of the range test for shape reconstruction in inverse scattering theory. The range test [R. Potthast, et al., A ‘range test’ for determining scatterers with unknown physical properties, Inverse Problems 19(3) (2003) 533–547] has originally been proposed to obtain knowledge about an unknown scatterer when the far field pattern for only one plane wave is given. Here, we extend the method to the case of multiple waves and show that the full shape of the unknown scatterer can be reconstructed. We further will clarify the relation between the range test methods, the potential method [A. Kirsch, R. Kress, On an integral equation of the first kind in inverse acoustic scattering, in: Inverse Problems (Oberwolfach, 1986), Internationale Schriftenreihe zur Numerischen Mathematik, vol. 77, Birkhäuser, Basel, 1986, pp. 93–102] and the singular sources method [R. Potthast, Point sources and multipoles in inverse scattering theory, Habilitation Thesis, Göttingen, 1999]. In particular, we propose a new version of the Kirsch–Kress method using the range test and a new approach to the singular sources method based on the range test and potential method. Numerical examples of reconstructions for all four methods are provided.

Complexity of Monte Carlo algorithms for a class of integral equations

In this work we study the computational complexity of a class of grid Monte Carlo algorithms for integral equations. The idea of the algorithms consists in an approximation of the integral equation by a system of algebraic equations. Then the Markov chain iterative Monte Carlo is used to solve the system. The assumption here is that the corresponding Neumann series for the iterative matrix does not necessarily converge or converges slowly. We use a special technique to accelerate the convergence. An estimate of the computational complexity of Monte Carlo algorithm using the considered approach is obtained. The estimate of the complexity is compared with the corresponding quantity for the complexity of the grid-free Monte Carlo algorithm. The conditions under which the class of grid Monte Carlo algorithms is more efficient are given.

Generalised Dirichelt-to-Neumann map in time dependent domains

We study the heat, linear Schrodinger and linear KdV equations in the domain l(t) < x < ∞, 0 < t < T, with prescribed initial and boundary conditions and with l(t) a given differentiable function. For the first two equations, we show that the unknown Neumann or Dirichlet boundary value can be computed as the solution of a linear Volterra integral equation with an explicit weakly singular kernel. This integral equation can be derived from the formal Fourier integral representation of the solution. For the linear KdV equation we show that the two unknown boundary values can be computed as the solution of a system of linear Volterra integral equations with explicit weakly singular kernels. The derivation in this case makes crucial use of analyticity and certain invariance properties in the complex spectral plane. The above Volterra equations are shown to admit a unique solution.

Stability results for the time-harmonic Maxwell equations with impedance boundary conditions

We consider the time-harmonic Maxwell equations with constant coefficients in a bounded, uniformly star-shaped polyhedron. We prove wavenumber-explicit norm bounds for weak solutions. This result is pivotal for convergence proofs in numerical analysis and may be a tool in the analysis of electromagnetic boundary integral operators.

Scattering by infinite one-dimensional rough surfaces

We consider the Dirichlet boundary-value problem for the Helmholtz equation in a non-locally perturbed half-plane. This problem models time-harmonic electromagnetic scattering by a one-dimensional, infinite, rough, perfectly conducting surface; the same problem arises in acoustic scattering by a sound-soft surface. ChandlerWilde & Zhang have suggested a radiation condition for this problem, a generalization of the Rayleigh expansion condition for diffraction gratings, and uniqueness of solution has been established. Recently, an integral equation formulation of the problem has also been proposed and, in the special case when the whole boundary is both Lyapunov and a small perturbation of a flat boundary, the unique solvability of this integral equation has been shown by Chandler-Wilde & Ross by operator perturbation arguments. In this paper we study the general case, with no limit on surface amplitudes or slopes, and show that the same integral equation has exactly one solution in the space of bounded and continuous functions for all wavenumbers. As an important corollary we prove that, for a variety of incident fields including the incident plane wave, the Dirichlet boundary-value problem for the scattered field has a unique solution.

On the solvability of a class of second kind integral equations on unbounded domains

We consider integral equations of the form ψ(x) = φ(x) + ∫Ωk(x, y)z(y)ψ(y) dy(in operator form ψ = φ + Kzψ), where Ω is some subset ofRn(n ≥ 1). The functionsk,z, and φ are assumed known, withz ∈ L∞(Ω) and φ ∈ Y, the space of bounded continuous functions on Ω. The function ψ ∈ Yis to be determined. The class of domains Ω and kernelskconsidered includes the case Ω = Rnandk(x, y) = κ(x − y) with κ ∈ L1(Rn), in which case, ifzis the characteristic function of some setG, the integral equation is one of Wiener–Hopf type. The main theorems, proved using arguments derived from collectively compact operator theory, are conditions on a setW ⊂ L∞(Ω) which ensure that ifI − Kzis injective for allz ∈ WthenI − Kzis also surjective and, moreover, the inverse operators (I − Kz)−1onYare bounded uniformly inz. These general theorems are used to recover classical results on Wiener–Hopf integral operators of21and19, and generalisations of these results, and are applied to analyse the Lippmann–Schwinger integral equation.

Sparse polynomial approximation in positive order Sobolev spaces with bounded mixed derivatives and applications to elliptic problems with random loading

In the present paper we study the approximation of functions with bounded mixed derivatives by sparse tensor product polynomials in positive order tensor product Sobolev spaces. We introduce a new sparse polynomial approximation operator which exhibits optimal convergence properties in L2 and tensorized View the MathML source simultaneously on a standard k-dimensional cube. In the special case k=2 the suggested approximation operator is also optimal in L2 and tensorized H1 (without essential boundary conditions). This allows to construct an optimal sparse p-version FEM with sparse piecewise continuous polynomial splines, reducing the number of unknowns from O(p2), needed for the full tensor product computation, to View the MathML source, required for the suggested sparse technique, preserving the same optimal convergence rate in terms of p. We apply this result to an elliptic differential equation and an elliptic integral equation with random loading and compute the covariances of the solutions with View the MathML source unknowns. Several numerical examples support the theoretical estimates.

Design of boundary-scan testing in CMOS image sensors for industrial applications: internship repor

Tests on printed circuit boards and integrated circuits are widely used in industry,resulting in reduced design time and cost of a project. The functional and connectivity tests in this type of circuits soon began to be a concern for the manufacturers, leading to research for solutions that would allow a reliable, quick, cheap and universal solution. Initially, using test schemes were based on a set of needles that was connected to inputs and outputs of the integrated circuit board (bed-of-nails), to which signals were applied, in order to verify whether the circuit was according to the specifications and could be assembled in the production line. With the development of projects, circuit miniaturization, improvement of the production processes, improvement of the materials used, as well as the increase in the number of circuits, it was necessary to search for another solution. Thus Boundary-Scan Testing was developed which operates on the border of integrated circuits and allows testing the connectivity of the input and the output ports of a circuit. The Boundary-Scan Testing method was converted into a standard, in 1990, by the IEEE organization, being known as the IEEE 1149.1 Standard. Since then a large number of manufacturers have adopted this standard in their products. This master thesis has, as main objective: the design of Boundary-Scan Testing in an image sensor in CMOS technology, analyzing the standard requirements, the process used in the prototype production, developing the design and layout of Boundary-Scan and analyzing obtained results after production. Chapter 1 presents briefly the evolution of testing procedures used in industry, developments and applications of image sensors and the motivation for the use of architecture Boundary-Scan Testing. Chapter 2 explores the fundamentals of Boundary-Scan Testing and image sensors, starting with the Boundary-Scan architecture defined in the Standard, where functional blocks are analyzed. This understanding is necessary to implement the design on an image sensor. It also explains the architecture of image sensors currently used, focusing on sensors with a large number of inputs and outputs.Chapter 3 describes the design of the Boundary-Scan implemented and starts to analyse the design and functions of the prototype, the used software, the designs and simulations of the functional blocks of the Boundary-Scan implemented. Chapter 4 presents the layout process used based on the design developed on chapter 3, describing the software used for this purpose, the planning of the layout location (floorplan) and its dimensions, the layout of individual blocks, checks in terms of layout rules, the comparison with the final design and finally the simulation. Chapter 5 describes how the functional tests were performed to verify the design compliancy with the specifications of Standard IEEE 1149.1. These tests were focused on the application of signals to input and output ports of the produced prototype. Chapter 6 presents the conclusions that were taken throughout the execution of the work.

Thermodynamics of a Peyrard-Bishop One-Dimensional Lattice with On-site Hump Potential

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

On dynamical gluon mass generation

The effective gluon propagator constructed with the pinch technique is governed by a Schwinger-Dyson equation with special structure and gauge properties, that can be deduced from the correspondence with the background field method. Most importantly the non-perturbative gluon self-energy is transverse order-by-order in the dressed loop expansion, and separately for gluonic and ghost contributions, a property which allows for a meanigfull truncation. A linearized version of the truncated Schwinger-Dyson equation is derived, using a vertex that satisfies the required Ward identity and contains massless poles. The resulting integral equation, subject to a properly regularized constraint, is solved numerically, and the main features of the solutions are briefly discussed.

Antiparticle Contribution in the Cross Ladder Diagram for Bethe-Salpeter Equation in the Light-Front

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

ITERATIVE NUMERICAL-SOLUTION OF SCATTERING PROBLEMS

An iterative Neumann series method, employing a real auxiliary scattering integral equation, is used to calculate scattering lengths and phase shifts for the atomic Yukawa and exponential potentials. For these potentials the original Neumann series diverges. The present iterative method yields results that are far better, in convergence, stability and precision, than other momentum space methods. Accurate result is obtained in both cases with an estimated error of about 1 in 10(10) after some 8-10 iterations.

Mechanism of phase formation in Pb(ZrxTi1-x)O3 synthesized by a partial oxalate method

The phase formation mechanism, as well as the morphotropic phase boundary, of lead zirconate titanate (PZT) processed by a partial oxalate method was investigated by simultaneous thermal analysis (TG-DTA) and by qualitative and quantitative X-ray diffraction (XRD). The results show that the ZrxTi1-xO2 (ZT) phase reacts with PbO forming the PZT phase without intermediate phases. XRD analysis showed the coexistence of rhombohedral and tetragonal phases for 0.47 ≤ x ≤ 0.55 with the phase boundary composition for x = 0.51. For low calcination temperatures, preferential formation of the PZT rhombohedral phase was observed. A model for phase formation of PZT by the partial oxalate method is proposed based on the existence of two interfaces of reaction (PbO-PZT and PZT-ZT) and diffusion of cations.