An Introduction to Numerical Analysis
Chapters (6)
Assessing the accuracy of the results of calculations is a paramount goal in numerical analysis. One distinguishes several kinds of errors which may limit this accuracy:
(1)
errors in the input data,
(2)
roundoff errors,
(3)
approximation errors.
Consider a family of functions of a single variable x, $$
\Phi \left( {x;{a_o}, \cdots ,{a_n}} \right),
$$ having n + 1 parameters αo, ..., αn whose values characterize the individual functions in this family. The interpolation problem for Φ consists of determining these parameters ai so that for n + 1 given real or complex pairs of numbers (xi, fi), i=0, ..., n, with xi ≠ xk for i ≠ k, $$
\Phi \left( {{x_i};{a_o}, \cdots ,{a_n}} \right) = {f_i},i = 0, \ldots ,n,
$$ holds. We will call the pairs (x
i, f
i) support points, the locations x
isupport abscissas, and the values f
isupport ordinates. Occasionally, the values of derivatives of Φ are also prescribed.
Calculating the definite integral of a given real function f (x), $$\int_a^b {f(x)dx,} $$ is a classic problem. For some simple integrands f (x), the indefinite integral $$\int_a^x {f\left( x \right)} dx = F\left( x \right),F'\left( x \right) = f\left( x \right),$$ can be obtained in closed form as an algebraic expression in x and wellknown transcendental functions of x. Then $$\int_a^b {f(x)dx = F(b) - F(a).} $$ See Gröbner and Hofreiter (1961) for a comprehensive collection of formulas describing such indefinite integrals and many important definite integrals.
In this chapter direct methods for solving systems of linear equations $$ Ax = b,A = \left[ \begin{array}{l}
{a_{11}}...{a_{1n}}\\
\vdots \quad \quad \vdots \\
{a_{n1}}...
\end{array} \right],b = \left[ \begin{array}{l}
{b_1}\\
\vdots \\
{b_n}
\end{array} \right] $$ will be presented. Here A is a given n × n matrix, and b is a given vector. We assume in addition that A and b are real, although this restriction is inessential in most of the methods. In contrast to the iterative methods (Chapter 8), the direct methods discussed here produce the solution in finitely many steps, assuming computations without roundoff errors.
Many practical problems in engineering and physics lead to eigenvalue problems. Typically, in all these problems, an overdetermined system of equations is given, say n + 1 equations for n unknowns ξ
1, ...,ξ
n
of the form $$f\left( {X;\lambda } \right): \equiv \left[ {\begin{array}{*{20}{c}} {{{f}_{1}}\left( {{{\xi }_{1}}, \ldots ,{{\xi }_{n}};\lambda } \right)} \\ \vdots \\ {{{f}_{n}} + \left( {{{\xi }_{1}}, \ldots ,{{\xi }_{n}};\lambda } \right)} \\ \end{array} } \right] = 0$$ (6.0.1) in which the functions f
i
also depend on an additional parameter λ. Usually, (6.0.1) has a solution x = [ξ1,...,ξn
]T
only for specific values λ = λi
, i = 1, 2,..., of this parameter. These values λi
are called eigenvalues of the eigenvalue problem (6.0.1) and a corresponding solution x = x(λi
) of (6.0.1), eigensolution belonging to the eigenvalue λi
.
Many problems in practice require the solution of very large systems of linear equations Ax = b in which the matrix A, fortunately, is sparse, i.e., has relatively few nonvanishing elements. Systems of this type arise, e.g., in the application of difference methods or finite-element methods to the approximate solution of boundary-value problems in partial differential equations. The usual elimination methods (see Chapter 4) cannot normally be applied here, since without special precautions they tend to lead to the formation of more or less dense intermediate matrices, making the number of arithmetic operations necessary for the solution much too large, even for present-day computers, not to speak of the fact that the intermediate matrices no longer fit into the usually available computer memory.
- ... , x i ], i = 0, 1, . . . , n of Newton's representation ( [18], p. 43) ...... The value of the interpolating polynomial p(x, x, y) defined by (1.8) at a given point x can also be obtained by Neville's algorithm ( [18], p. 40), which computes the matrix (1.1) for α i,j = (x − x i ) and β i,j = (x − x i−j−1 ). In this case, we have p(x, x, y) = M n n . ...... an expansion in even powers of h when f is sufficiently smooth [18], p. 136. ...In this note we extend the analysis of [5] on the numerical stability of classical New-ton's divided differences to a broader class of divided differences algorithms that includes Neville's algorithm for Lagrange interpolation and some of its particular instances, such as Richardson extrap-olation and Romberg quadrature. We show that these algorithms are backward stable and we bound the overall numerical error in their computation in finite precision. In spite of the historical connections between Neville's algorithm and Richardson extrapolation, the current literature suggests the use of the first barycentric formula for extrapolation. Our analysis shows that Neville's algorithm is as stable as the first barycentric formula for extrapolation in the real line and this consolidates a solid background for the usual representation of Richardson extrapolation and Romberg quadrature as divided differences schemes.
- ... HE implicit trapezoidal method (ITM) is widely used in power system time-domain simulations [1]. In most cases, the ITM is implemented with a fixed step, but for long-term simulations, to enhance the computation efficiency, an ITM with variable steps is preferable [2]. ...... In the PCM, we assume two numerical methods, P and Q, are of accuracy of order p and q (p<q), respectively, so the local truncation errors of P and Q are and , respectively. When P is applied at the step n+1, the truncation error can be presented as [1]: ...This letter proposes a predictor-corrector method to strike a balance between simulation accuracy and efficiency by appropriately tuning the numerical integration step length of a power system time-domain simulation. Numerical tests indicate that, by estimating the truncation error for step length tuning based on the 2-Step Adams-Moulton method and the implicit Trapezoidal method, the proposed method can provide much more precise results at little cost of efficiency compared to a conventional variable step method based on Newton's method.
- ... donde la función f satisface las hipótesis del teorema de existencia y unicidad de los PVF (Berezin & Zhidkov, 1965;Stoer & Bulirsch, 1993) α y β son constantes. ...... Sea problema valor de frontera no lineal de la forma y ´ ´ = f ( x , y , y ´ ) , a ≤ x ≤ b ; (20) y (a )=α , y (b )= β , donde la función f satisface las hipótesis del teorema de existencia y unicidad de los PVF (Stoer & Bulirsch, 1993). El método de diferencias finitas para este problema es similar al presentado en la sección anterior; sin embargo, el sistema de ecuaciones resultantes es no lineal y debe ser empleado un método iterativo para su solución. ...Many physical problems are modeled using differential equations, to which are imposed diverse conditions of initial value or frontier conditions. The presented problem of this investigation consists on the search and synthesis of a simplified and accessible numeric treatment in the solution of boundary value problem (BVP) with general conditions of frontier associated to ordinary differential equations of second order. The objective of this work is to analyze the numeric methods employees to solve the BVP that are modeled using a differential equation of second order. Numeric methods are necessary due to the fact that the analytic techniques cannot be used for the situations or cases in which differential equation cannot be solved in an exact way. Emphasis is made in the development and application of the shot methods and finite differences for lineal and not lineal problems. This methods are analyzed with frontier conditions of type Dirichlet, Neumann, and mixed; both last with more depth in the lineal cases. These methods are compared according to the convergence speed. The same ones are applied in real problems of the physics as the transfer of heat in a system with cylindrical symmetry. The effect of the rounding errors is among the particularities that are approached, which allows to toast alternative that improve the efficiency in the implementation of the same ones. The used algorithms were implemented using the software Matlab and proved through real cases of study.
- ... The following is a classical fact from elementary functional analysis. A proof can be found in, for example, [13]. φ(x) = a 1 f 1 (x) + ... + a n−1 f n−1 (x). ...... The following is a well-known lemma from numerical analysis. A proof can be found in, for example, [13]. ii) For any non-negative integer n and for i = 0, 1, ..., 2n − 1, we have ...Zernike polynomials are a basis of orthogonal polynomials on the unit disk that are a natural basis for representing smooth functions. They arise in a number of applications including optics and atmospheric sciences. In this paper, we provide a self-contained reference on Zernike polynomials, algorithms for evaluating them, and what appear to be new numerical schemes for quadrature and interpolation. We also introduce new properties of Zernike polynomials in higher dimensions. The quadrature rule and interpolation scheme use a tensor product of equispaced nodes in the angular direction and roots of certain Jacobi polynomials in the radial direction. An algorithm for finding the roots of these Jacobi polynomials is also described. The performance of the interpolation and quadrature schemes is illustrated through numerical experiments. Discussions of higher dimensional Zernike polynomials are included in appendices.
- ... ) denote the minimum time-to-climb problem with the additional artificial parameter ε > 0. From the control theory, for a fixed value of ε > 0, the candidates as minimizers are selected among a set of BC-extremals, solution of a Hamiltonian system given by the Pontryagin Maximum Principle (PMP), see Ref. [35]. The application of the PMP leads to define a Boundary Value Problem denoted (BVP ε ), in terms of state and adjoint variables, which can be solved using indirect multiple shooting methods [11]. It is well known that multiple shooting increases numerical stability and a good alternative would be to use direct multiple shooting [5] to solve the optimal control problem. ...... We present in this section the indirect multiple shooting method [11] that we use to solve the necessary conditions of optimality given by the maximum principle presented in Sect. II C. We describe the method on only one example and we refer to Ref. [23] for more details. ...In this article, we are interested in optimal aircraft trajectories in climbing phase. We consider the cost index criterion which is a convex combination of the time-to-climb and the fuel consumption. We assume that the thrust is constant and we control the air slope of the aircraft. This optimization problem is modeled as a Mayer optimal control problem with a single-input affine dynamics in the control and with two pure state constraints, limiting the Calibrated AirSpeed (CAS) and the Mach speed. The candidates as minimizers are selected among a set of extremals given by the maximum principle. We first analyze the minimum time-to-climb problem with respect to the bounds of the state constraints, combining small time analysis, indirect multiple shooting and homotopy methods. This investigation emphasizes two strategies: the common CAS/Mach procedure in aeronautics and the classical Bang-Singular-Bang policy in control theory. We then compare these two procedures for the cost index criterion.
- ... donde la función f satisface las hipótesis del teorema de existencia y unicidad de los PVF (Berezin & Zhidkov, 1965;Stoer & Bulirsch, 1993) α y β son constantes. ...... Sea problema valor de frontera no lineal de la forma y ´ ´ = f ( x , y , y ´ ) , a ≤ x ≤ b ; (20) y (a )=α , y (b )= β , donde la función f satisface las hipótesis del teorema de existencia y unicidad de los PVF (Stoer & Bulirsch, 1993). El método de diferencias finitas para este problema es similar al presentado en la sección anterior; sin embargo, el sistema de ecuaciones resultantes es no lineal y debe ser empleado un método iterativo para su solución. ...ABSTRACT Many physical problems are modeled using differential equations, to which are imposed diverse conditions of initial value or frontier conditions. The presented problem of this investigation consists on the search and synthesis of a simplified and accessible numeric treatment in the solution of boundary value problem (BVP) with general conditions of frontier associated to ordinary differential equations of second order. The objective of this work is to analyze the numeric methods employees to solve the BVP that are modeled using a differential equation of second order. Numeric methods are necessary due to the fact that the analytic techniques cannot be used for the situations or cases in which differential equation cannot be solved in an exact way. Emphasis is made in the development and application of the shot methods and finite differences for lineal and not lineal problems. This methods are analyzed with frontier conditions of type Dirichlet, Neumann, and mixed; both last with more depth in the lineal cases. These methods are compared according to the convergence speed. The same ones are applied in real problems of the physics as the transfer of heat in a system with cylindrical symmetry. The effect of the rounding errors is among the particularities that are approached, which allows to toast alternative that improve the efficiency in the implementation of the same ones. The used algorithms were implemented using the software Matlab and proved through real cases of study. KEY WORDS: boundary value problem, frontier conditions of type Dirichlet and Neumann, shot method, method of finite differences.
- ... donde la función f satisface las hipótesis del teorema de existencia y unicidad de los PVF (Berezin & Zhidkov, 1965;Stoer & Bulirsch, 1993) α y β son constantes. ...... Sea problema valor de frontera no lineal de la forma y ´ ´ = f ( x , y , y ´ ) , a ≤ x ≤ b ; (20) y (a )=α , y (b )= β , donde la función f satisface las hipótesis del teorema de existencia y unicidad de los PVF (Stoer & Bulirsch, 1993). El método de diferencias finitas para este problema es similar al presentado en la sección anterior; sin embargo, el sistema de ecuaciones resultantes es no lineal y debe ser empleado un método iterativo para su solución. ...ABSTRACT Many physical problems are modeled using differential equations, to which are imposed diverse conditions of initial value or frontier conditions. The presented problem of this investigation consists on the search and synthesis of a simplified and accessible numeric treatment in the solution of boundary value problem (BVP) with general conditions of frontier associated to ordinary differential equations of second order. The objective of this work is to analyze the numeric methods employees to solve the BVP that are modeled using a differential equation of second order. Numeric methods are necessary due to the fact that the analytic techniques cannot be used for the situations or cases in which differential equation cannot be solved in an exact way. Emphasis is made in the development and application of the shot methods and finite differences for lineal and not lineal problems. This methods are analyzed with frontier conditions of type Dirichlet, Neumann, and mixed; both last with more depth in the lineal cases. These methods are compared according to the convergence speed. The same ones are applied in real problems of the physics as the transfer of heat in a system with cylindrical symmetry. The effect of the rounding errors is among the particularities that are approached, which allows to toast alternative that improve the efficiency in the implementation of the same ones. The used algorithms were implemented using the software Matlab and proved through real cases of study. KEY WORDS: boundary value problem, frontier conditions of type Dirichlet and Neumann, shot method, method of finite differences.
- ... The piecewise cubic spline function is substituted for the linear prediction [22]. However, predicted values from large time-step networks have to be used for small time-step networks [23], [24]. ...... A straightforward method to achieve the synchronization is to introduce a delay of ∆T for the interface variables, v 1 and i 2 , which means that v 1 (t + ∆T ) = v 2 (t) and i 2 (t + ∆T ) = i 1 (t) [24]. However, this method will cause large calculation errors and simulation instability. ...With wider applications of power electronic devices in modern power systems, simulation using traditional electro-mechanical and electromagnetic simulation tools suffer from low speed and imprecision. Multi-rate technologies can greatly improve simulation efficiency by avoiding simulating the entire system using a small time-step. However, the drawbacks of the current synchronization mechanisms is that they introduce numerical errors and numerical instabilities in multi-rate parallel simulations. An improved multi-rate parallel technology, node splitting interface (NSI), is proposed to reduce errors and enhance simulation stability. A new synchronization mechanism is used to avoid prediction and signal delays. Theoretical analyses are carried out to prove the convergence and absolute stability of the proposed NSI algorithm. This algorithm is particularly suitable for simultaneously investigating long term dynamics of DC grids and fast transients of power electronic converters. Index Terms-DC grids, electromagnetic transient analysis, multi-rate interface, parallel algorithm, power system simulation.
- ... The quadrature rule is exact for a given function f (x) when the remainder E n+1 is exactly zero. For example, the standard (n + 1)-point Gauss-Legendre (GL or Gauss) quadrature is exact for the linear space of polynomials of degree at most 2n + 1 (see, for example, [12,57]). ...This chapter studies the effect of the quadrature on the isogeometric analysis of the wave propagation and structural vibration problems. The dispersion error of the isogeometric elements is minimized by optimally blending two standard Gauss-type quadrature rules. These blending rules approximate the inner products and increase the convergence rate by two extra orders when compared to those with fully-integrated inner products. To quantify the approximation errors, we generalize the Pythagorean eigenvalue error theorem of Strang and Fix. To reduce the computational cost, we further propose a two-point rule for C¹ quadratic isogeometric elements which produces equivalent inner products on uniform meshes and yet requires fewer quadrature points than the optimally-blended rules.
- ... Note that other interpolation techniques with a different number of sensors may also be employed for interpolating the discrete POD modes. In addition to the use of third-order polynomial interpolation, rational function interpolation [30,31], and trigonometric/Fourier interpolations [32] were tested on simulated data sets as well as on NDP data sets. Results showed that, for the sensor arrangement of the NDP data sets, the rational function interpolation often resulted in artificial poles while the trigonometric function interpolation led to sharp wiggles sometimes. ...To gain insight into riser motions and associated fatigue damage due to vortex-induced vibration (VIV), data loggers such as strain sensors and/or accelerometers are sometimes deployed on risers to monitor their motion in different current velocity conditions. Accurate reconstruction of the riser response and empirical estimation of fatigue damage rates over the entire riser length using measurements from a limited number of sensors can help in efficient utilization of the costly measurements recorded. Several different empirical procedures are described here for analysis of the VIV response of a long flexible cylinder subjected to uniform and sheared current profiles. The methods include weighted waveform analysis (WWA), proper orthogonal decomposition (POD), modal phase reconstruction (MPR), a modified WWA procedure, and a hybrid method which combines MPR and the modified WWA method. Fatigue damage rates estimated using these different empirical methods are compared and cross-validated against measurements. Detailed formulations for each method are presented and discussed with examples. Results suggest that all the empirical methods, despite different underlying assumptions in each of them, can be employed to estimate fatigue damage rates quite well from limited strain measurements.
- ... . It is well known that the nodes k t are eigenvalues of the following symmetric tridiagonal matrix n J of degree n known as Jacobi matrix [37] whose diagonal elements are ...In this paper, we present two methods: Modified Clenshaw-Curtis and the Gauss-Jacobi methods. These methods are commonly used in the evaluation of the finite Fourier transforms of integrands with endpoint singularities. In the first method, the integrand is truncated by the Chebyshev series, term by term, and then its singularity types are evaluated using recurrence relations. This method is more efficient for low-frequency values. On the other hand, the Gauss Jacobi method is found to be accurate in the evaluation of integrals with fairly high-frequency values; such as 1000. MATHEMATICA codes, for both methods, are provided for the purpose of testing the efficiency of automatic computation. Lastly, the illustrative examples are considered with regards to reliability, accuracy, and comparison of the methods outlined.
- ... where i = 1, 2, · · · , n−1. According to [38], these parameters satisfy the following equation ...The multivariate empirical mode decomposition (MEMD) has been pioneered recently for adaptively processing of multichannel data. Despite its high efficiency on timefrequency analysis of nonlinear and nonstationary signals, high computational load and over-decomposition have restricted wider applications of MEMD. To address these challenges, a fast MEMD (FMEMD) algorithm is proposed and featured by the following contributions: (i) A novel concept, pseudo directionindependent Multivariate Intrinsic Mode Function (IMIMF) which allows the interchange of sifting and projection operations, is defined for the purpose of developing FMEMD, (ii) FMEMD is computationally efficient. Compared with MEMD, the number of time-consuming sifting operations reduces from K·p to K for each iteration, where K and p denote the number of projection directions and signal dimension, respectively, (iii) FMEMD is consistent with EMD in terms of the dyadic filter bank property and (iv) FMEMD is more effective in working at low sampling rate. Validity of the raised approach is demonstrated on a wide variety of real world applications.
- ... where ν is a suitable adjustment factor ν ≈ 0.9 [13] . Here, p is the order of the lower order method, and 0 < ν < 1 is a safety factor whose purpose is to avoid failed steps. ...In this work, we develop a nonlinear explicit method suitable for both autonomous and non-autonomous type of initial value problems in Ordinary Differential Equations (ODEs). The method is found to be third order accurate having L -stability. It is shown that if a variable step-size strategy is employed then the performance of the proposed method is further improved in comparison with other methods of same nature and order. The method is shown to be working well for initial value problems having singular solutions, singularly perturbed and stiff problems, and blow-up ODE problems, which is illustrated using a few numerical experiments.
- ... Note that other interpolation techniques with a different number of sensors may also be employed for interpolating the discrete POD modes. In addition to the use of third-order polynomial interpolation, rational function interpolation [30,31], and trigonometric/Fourier interpolations [32] were tested on simulated data sets as well as on NDP data sets. Results showed that, for the sensor arrangement of the NDP data sets, the rational function interpolation often resulted in artificial poles while the trigonometric function interpolation led to sharp wiggles sometimes. ...To gain insight into riser motions and associated fatigue damage due to vortex-induced vibration (VIV), data loggers such as strain sensors and/or accelerometers are sometimes deployed on risers to monitor their motion in different current velocity conditions. Accurate reconstruction of the riser response and empirical estimation of fatigue damage rates over the entire riser length using measurements from a limited number of sensors can help in efficient utilization of the costly measurements recorded. Several different empirical procedures are described here for analysis of the VIV response of a long flexible cylinder subjected to uniform and sheared current profiles. The methods include weighted waveform analysis (WWA), proper orthogonal decomposition (POD), modal phase reconstruction (MPR), a modified WWA procedure, and a hybrid method which combines MPR and the modified WWA method. Fatigue damage rates estimated using these different empirical methods are compared and cross-validated against measurements. Detailed formulations for each method are presented and discussed with examples. Results suggest that all the empirical methods, despite different underlying assumptions in each of them, can be employed to estimate fatigue damage rates quite well from limited strain measurements.
- ... Reminding that the sum of Gaussian weight coefficients w r g is equal to two (cf. Stoer [35]), we have ...The so-called distance-minimizing data-driven computing method is extended to deal with boundary-value problems of continuum mechanics within the finite strain theory. In the merit of a data-driven model the solution process is carried out by using directly the experimental data instead of the conventional constitutive laws. Thus it bypasses the uncertainties in fabricating the stress-strain functional relationships from material data. Consequently, the mathematical formulation involves an optimization problem with equality constraints consisting of the equilibrium equations in continuum mechanics and the compatibility conditions on the displacement field. In the framework of finite element formulation the element tangent stiffness, the generalized internal force and the generalized external force can be computed, which renders it amenable to the implementation of finite element procedures. The proposed scheme is validated through the applications to continuum elements and convergence studies of the data-driven solution in regard to the interpolation order, the mesh size as well as the data size. The variational structure allows to recognize the overall pattern of the system of equations to be solved. This includes the structural tangent stiffness and the generalized force vectors.
- ... Finally, let s 0 = λ + jω be an eigenvalue of Σ associated with a right-eigenvector v. It is well known, see Strang (2016) and Bulirsch and Stoer (1993), that left and right eigenvalues are equal. Hence, s 0 is also an eigenvalue of Σ associated with a left-eigenvector x L , with dimension 1 × n. ...This work deals with stability and robust stabilisation of retarded time-delay systems by applying a new method for obtaining an envelope that bounds all the system poles. Through LMIs we are able to determine envelopes that can be applied to verify the stability of the system and can also be utilised to design robust state-feedback controllers which cope with design requirements regarding α - stability.
- ... An iterative differential correction scheme, is used to solve the nonlinear boundary value problem (49), (52)-(54) numeri- cally.. The steps of the numerical integration scheme is repre- sented by the flow chart in Fig. 6, and for more details the reader is referred to [22,23]. In this scheme, assume k 1 ; k 2 ; ^ b, and P are known, an arbitrary set of values for Q 2 0 ð Þ; Q 4 0 ð Þ; Q 5 0 ð Þ are assumed. ...In this paper, shapes of nonlinear blood vessels, surrounded by nonlinear soft tissues, and buckled due to radial pressure are solved for analytically and numerically. The blood flow rates through the bucked shapes are then computed numerically. A Fung-type isotropic hyperelastic stress-strain constitutive equation is used to establish a nonlinear mathematical model for radial buckling of blood vessels. The surrounding tissues are modeled as non-linear springs. Novel formulas for critical buckling pressures are derived analytically from the bifurcation analysis. This analysis shows that the nonlinearity of vessel's wall increases the critical buckling pressure. A numerical differential correction scheme is introduced to solve for post-buckling shapes. And the corresponding blood flow rates are provided before touching of the collapsed walls. The blood flow rate through a one-point wall-touching case is also provided. Numerical results show that both vessel's wall and soft tissues nonlinearities increase, locally, the flow rate through the buckled blood vessels. More importantly, a nonlinear relation between blood flow rate and the soft tissue spring constants is found.
- ... When f is of class C 2n+2 [a, b], the integration error can be expanded by Euler-Maclaurin's formula [11] T ( ...Romberg integrals are built in order to accelerate the convergence of sequences of trapezoidal rules for approximating the definite integral of a continuous function f. While every sequence of trapezoidal rules with decreasing step length converges whenever f is continuous, this does not always hold for Romberg integrals. In this note we present a concrete example for which the sequence formed by the diagonal elements of the Romberg table diverges when the number of points used to compute the trapezoidal rules grows too slowly.
- ... with ρ l,k := ρ k,l for l > k. Using Gerschgorin's Theorem (see for instance [79]), the eigenvalues of˜Pof˜ of˜P are contained in the union of the balls with center x r = S k=1,kr ρ r,k and radius x r = S k=1,kr |−ρ r,k |. These balls are all contained in the larger ball with center 0 and radius 2·max r x r . ...Signals and images with discontinuities appear in many problems in such diverse areas as biology, medicine, mechanics, and electrical engineering. The concrete data are often discrete, indirect and noisy measurements of some quantities describing the signal under consideration. A frequent task is to find the segments of the signal or image which corresponds to finding the discontinuities or jumps in the data. Methods based on minimizing the piecewise constant Mumford-Shah functional -- whose discretized version is known as Potts functional -- are advantageous in this scenario, in particular, in connection with segmentation. However, due to their non-convexity, minimization of such functionals is challenging. In this paper we propose a new iterative minimization strategy for the multivariate Potts functional dealing with indirect, noisy measurements. We provide a convergence analysis and underpin our findings with numerical experiments.
- ... We believe that one of the advantages of using PMT for finding fixed points of a given function is that only the signs of the components of it have to be controlled on some suitable sets, which is straightforward in the case that either the equations are polynomial or the problem can be polynomialized (see for instance the proof of Theorem 6 in Section 5). Recall that the use of Sturm sequences for polynomials in Q [x] allows to control their signs on intervals with rational endpoints ( [32]). ...We present a systematic methodology to determine and locate analytically isolated periodic points of discrete and continuous dynamical systems with algebraic nature. We apply this method to a wide range of examples, including a one-parameter family of counterexamples to the discrete Markus-Yamabe conjecture (La Salle conjecture); the study of the low periods of a Lotka-Volterra-type map; the existence of three limit cycles for a piece-wise linear planar vector field; a new counterexample of Kouchnirenko's conjecture; and an alternative proof of the existence of a class of symmetric central configuration of the $(1+4)$-body problem.
- ... During our computation we found that c = 0 satisfies these conditions for both problems. The same concept is applied while solving a boundary value problem by the shooting technique numerically [33]. ...This work proposes two analytical techniques to obtain approximate analytical solutions for nonlinear problems containing two-point Neuman boundary conditions. We use the rational homotopy perturbation method (RHPM) and the homotopy analysis method (HAM) to obtain the solutions. Found that both methods can lead to good representations of the considered nonlinear problems. Both techniques are tested solving a couple of nonlinear problems with Neumann boundary conditions. Additionally, a novel technique for the distribution of the Neumann boundary conditions among the different iterations of RHPM to generate suitable adjustment parameters to obtain accurate compact and simple computable analytical solutions is proposed. Also, we demonstrate how HAM coupled with the shooting method provides an acceptable approximate solution to the considered problems. Finally, obtained solutions are compared to the exact solutions.
- ... In approximation theory, exponential splines are modelling data that ex- hibit sudden growth or decay and for which polynomials are ill-suited be- cause of their oscillatory behavior [34]. Analogously to polynomial B-splines, exponential B-splines can be defined as finite convolution products of the exponential functions e a j (·) | [0,1] , a j = 0. (Cf. ...
- ... In approximation theory, exponential splines are modelling data that ex- hibit sudden growth or decay and for which polynomials are ill-suited be- cause of their oscillatory behavior [34]. Analogously to polynomial B-splines, exponential B-splines can be defined as finite convolution products of the exponential functions e a j (·) | [0,1] , a j = 0. (Cf. ...Several classes of classical cardinal B-splines can be obtained as solutions of operator equations of the form $Ly = 0$ where $L$ is a linear differential operator of integral order. (Cf., for instance, \cite{akhiezer,Golomb,Krein,micchelli,schoenberg}.) In this article, we consider classes of generalized B-splines consisting of cardinal polynomial B-splines of complex and hypercomplex orders and cardinal exponential B-splines of complex order, and derive the fractional linear differential operators that are naturally associated with them. For this purpose, we also present the spaces of distributions onto which these fractional differential operators act.
- ... The ge- ometric positions of the major planets and the Moon are pro- vided by INPOP planetary theory ( Fienga et al., 2014). Those of small SSOs (asteroids, comets, Centaurs, trans-neptunian objects) are calculated by numerical integration of the N- body perturbed problem (Gragg-Bulirsch-Stoer algorithm, see Bulirsch and Stoer, 1966;Stoer and Bulirsch, 1980), using their latest published osculating elements, from the astorb ( Bowell et al., 1993) andcometpro (Rocher andCavelier, 1996) databases. The overall accuracy of asteroid and comet ephemerides provided by ViSiON is at the level of tens of milli- arcseconds, mainly depending on the accuracy of their osculat- ing elements. ...7 pages, 3 figures, 4 tables, accepted for publication in P&SS
- ... This is a well-investigated problem, for which a great variety of numerical solvers are available. In this work, we employ a standard Runge-Kutta integration scheme (see, for example, Stoer and Bulirsch, 2002). ...In recent years, the use of radiographic inspection with cosmic-ray muons has spread into multiple research and industrial fields. This technique is based on the high-penetration power of cosmogenic muons. Specifically, it allows the resolution of internal density structures of large-scale geological objects through precise measurements of the muon absorption rate. So far, in many previous works, this muon absorption rate has been considered to depend solely on the density of traversed material (under the assumption of a standard rock) but the variation in chemical composition has not been taken seriously into account. However, from our experience with muon tomography in Alpine environments, we find that this assumption causes a substantial bias in the muon flux calculation, particularly where the target consists of high {Z²∕A} rocks (like basalts and limestones) and where the material thickness exceeds 300m. In this paper, we derive an energy loss equation for different minerals and we additionally derive a related equation for mineral assemblages that can be used for any rock type on which mineralogical data are available. Thus, for muon tomography experiments in which high {Z²∕A} rock thicknesses can be expected, it is advisable to plan an accompanying geological field campaign to determine a realistic rock model.
- ... These methods use the Jacobian (or an approximation) of the function G in order to find a descent direction. Augmenting the number of subintervals improves the stability, as the determinant of this Jacobian, seen as a function of t i+1 − t i , is an exponentially growing function (see [100]). ...This thesis focuses on the optimal control of Linear Complementarity Systems (LCS). LCS are dynamical systems defined through Differential Algebraic Equations (DAE), where one of the variable is defined by a Linear Complementarity Problem.These systems can be found in the modeling of various phenomena, as Nash equilibria, hybrid dynamical systems or modeling of electrical circuits. Properties of the solution to these DAE essentially depend on properties that the matrix D in the complementarity must meet. These complementarity constraints induce two different challenges. First, the analysis of these dynamical systems often use state of the art tools, and their study still has some unansweredquestions. Second, the optimal control of these systems causes troubles due to on one hand the presence of the state in the constraints, on the other hand the violation of Constraint Qualifications, that are a recurring hypothesis for optimisation problems.The research presented in this manuscript focuses on the optimal control of these systems. We mainly focus on the quadratic optimal control problem (minimisation of a quadratic functional involving the state and the control), and the minimal time control. The results present two different aspects: first, we start with an analytical approach in order to find necessary conditions of optimality (if possible, these conditions are proved to be sufficient); secondly, a numerical approach is tackled, with the aim of getting precise results with a reduced computational time.
- ... , n. A Lagrange polynomial based on these points can be defined as follows (Stoer and Bulirsch, 2002: p. 39, equations (2.1.1.2) and (2.1.1.3)) ...
- ... • Step 1: Extract the upper envelope e max (t) and lower envelope e min (t) of signal s(t) by connecting all local maxima and local minima with cubic spline functions (Stoer & Bulirsch, 2013), respectively. ...Parkinson's disease (PD) is a common neurodegenerative disorder that affects human's quality of life, especially leading to locomotor deficits such as postural instability and gait disturbances. Gait signal is one of the best features to characterize and detect movement disorders caused by a malfunction in parts of the brain and nervous system of the patients with PD. Various classification approaches using spatiotemporal gait variables have been presented earlier to classify Parkinson's gait. In this study we propose a novel method for gait pattern classification between patients with PD and healthy controls, based upon phase space reconstruction (PSR), empirical mode decomposition (EMD) and neural networks. First, vertical ground reaction forces (GRFs) at specific positions of human feet are captured and then phase space is reconstructed. The properties associated with the gait system dynamics are preserved in the reconstructed phase space. Three-dimensional (3D) PSR together with Euclidean distance (ED) has been used. These measured parameters demonstrate significant difference in gait dynamics between the two groups and have been utilized to form a reference variable set. Second, reference variables are decomposed into Intrinsic Mode Functions (IMFs) using EMD, and the third IMFs are extracted and served as gait features. Third, neural networks are then used as the classifier to distinguish between patients with PD and healthy controls based on the difference of gait dynamics preserved in the gait features between the two groups. Finally, experiments are carried out on 93 PD patients and 73 healthy subjects to assess the effectiveness of the proposed method. By using 2-fold, 10-fold and leave-one-out cross-validation styles, the correct classification rates are reported to be 91.46%, 96.99% and 98.80%, respectively. Compared with other state-of-the-art methods, the results demonstrate superior performance and the proposed method can serve as a potential candidate for the automatic and non-invasive classification between patients with PD and healthy subjects.
- ... Growth is treated continuously and deterministically, while mutations are introduced stochasti- cally (probabilistically). At each simulation step, first the growth portion of the system is simu- lated deterministically by a fourth order Runge-Kutta method, [59] then the cells that are divided are calculated and the mutants are computed analogous to tau leaping methods, [60] [61] by a Poisson random number with the rate of mutation. The time step is taken to be 60 minutes. ...Antisense molecules used as antibiotics offer the potential to keep up with acquired resistance, by redesigning the sequence of an antisense. Once bacteria acquire resistance by mutating the targeted sequence, new antisense can readily be designed by using sequence information of a target gene. However, antisense molecules require additional delivery vehicles to get into bacteria and be protected from degradation. Based on progress in the last few years it appears that, while redesigning or finding new delivery vehicle will be more difficult than redesigning the antisense cargo, it will perhaps be less difficult than finding new conventional small molecule antibiotics. In this study we propose a protocol that maximizes the combined advantages of engineered delivery vehicle and antisense cargo by decreasing the immediate growth advantage to the pathogen of mutating the entry mechanisms and increasing the advantage to the pathogen of antisense target mutations. Using this protocol, we show by computer simulation an appropriately designed antisense therapy can potentially be effective many times longer than conventional antibiotics before succumbing to resistance. While the simulations describe an in-vitro situation, based on comparison with other in-vitro studies on acquired resistance we believe the advantages of the combination antisense strategy have the potential to provide much more sustainability in vivo than conventional antibiotic therapy.
- The local convergence of a family of iterative methods based on a parameter θ for multiple roots of nonlinear equations is established. It is established under the assumption that the derivative g(m+ 1) of the function g satisfies the Hölder continuity condition. The existence–uniqueness theorem along with a posteriori error bounds are provided. The R-order convergence is shown to be equal to (2 + p), where p ∈ (0, 1]. The well-known methods of Chebyshev (θ = 0) and of Osada (θ = 1) belong to the family. A number of numerical examples are worked out. The radii of convergence balls for different values of the parameter θ are compared and tabulated. To the best of our knowledge, our work is the first attempt to establish the local convergence of a family of iterative methods based on a parameter θ for computing multiple roots of nonlinear equations. Moreover, its dynamical study is investigated and the parameter plane is constructed to find the best value of the parameter θ. The values of θ are investigated according to the character of the strange fixed point and the multiplicity of the root.
- The accurate description of the dissociative chemisorption of a molecule on a metal surface requires a chemically accurate description of the molecule-surface interaction. Previously, it was shown that the specific reaction parameter approach to density functional theory (SRP-DFT) enables accurate descriptions of the reaction of dihydrogen with metal surfaces in, for instance H2+Pt(111), H2+Cu(111) and H2+Cu(100). SRP-DFT likewise allowed a chemically accurate description of dissociation of methane on Ni(111) and Pt(111), and the SRP functional for CH4+Ni(111) was transferable to CH4+Pt(111), where Ni and Pt belong to the same group. Here we investigate whether the SRP density functional derived for H2+Cu(111) also gives chemically accurate results for H2+Ag(111), where Ag belong to the same group as Cu. To do this, we have performed quasi-classical trajectory calculations using the six-dimensional PES of H2+Ag(111) within the Born-Oppenheimer static surface approximation. The computed reaction probabilities are compared with both state-resolved associative desorption and molecular beam sticking experiments. Our results do not yet show transferability, as the computed sticking probabilities and initial-state selected reaction probabilities are shifted relative to experiment to higher energies by about 2-3 kcal/mol. The lack of transferability may be due to the different character of the SRP functionals for H2+Cu and CH4+group 10 metals, the latter containing a van der Waals correlation functional and the former not.
- In this paper, we consider inverse problems of the heat conduction process in one and two-dimensional homogeneous bodies of finite size, subject to the given initial and boundary conditions. The considered inverse problems are severely ill-posed since their solutions; if they exist, they do not depend continuously on the input data. We obtain stable solutions for several inverse problems by proposing a meshless regularization technique based on the combination of the method of fundamental solutions and the Tikhonov’s regularization method. In particular, we use the given information at the terminal state to estimate the space-dependent heat source in the one-dimensional case and the space- and time-dependent heat source in the two-dimensional case. Numerical results demonstrate high accuracy and low computational cost.
- We consider discrete-modulation protocols for continuous-variable quantum key distribution (CV-QKD) that employ a modulation constellation consisting of a finite number of coherent states and that use a homodyne-detection receiver. We establish a security proof for arbitrary collective attacks in the asymptotic regime, and we provide a formula for an achievable secret-key rate. Previous works established security proofs for Gaussian-modulation CV-QKD protocols or for discrete-modulation protocols with two or three coherent states. The main constituents of our approach include approximating a complex, isotropic Gaussian probability distribution by a finite-size Gauss-Hermite constellation, applying entropic continuity bounds, and leveraging previous security proofs for Gaussian-modulation protocols. As an application of our method, we calculate secret-key rates achievable over a pure-loss bosonic channel. Our results indicate that in the high-loss regime the achievable key rates scale optimally, i.e., proportional to the channel's transmissivity, and approach that achieved by a Gaussian-modulation protocol as the constellation size is increased.
- Rapid increase in number of cellular users and high demand for data has lead to the formation of multi-tier networks. Non-orthogonal multiple access (NOMA) has proved to be an efficient method to cater to the paradigm shift from 4G to 5G. This paper employs NOMA in an heterogeneous cellular network consisting of a macro base station (MBS) tier underlaid with femto base station (FBS) tier and device-to-device (D2D) tier, where NOMA is employed in FBS and D2D tier only. The congestion at the MBS tier is relieved by offloading macro users (MU) to the FBS tier. The offloaded MU are further supported by the D2D tier when the FBS tier fails to find a corresponding pairing user for the incoming offloaded MU. Since, absence of pairing user means outage for offloaded MU, D2D cooperation is employed, which decreases the rate outage probability by $86.87\%$ for the MU offloaded as cell edge user (CEU) in comparison to no cooperation. Also, a three times increase in ergodic rate and four times increase in sum ergodic rate for MU offloaded as CEU is achieved using cooperation from D2D tier. Verification of the results is done using Monte Carlo simulations.
- Complex engineering and economic systems are often dynamic in nature and its time-varying variables are subject to algebraic restrictions that are cast in the form of equalities and inequalities; furthermore, these variables may be related to each other via some logical conditions. For such constrained dynamical systems, the classical approach based on ordinary differential equations (ODEs) alone is inadequate to fully capture the intricate details of the system evolution. Combining an ODE with a variational condition on an auxiliary algebraic variable that represents either a constrained optimization problem in finite dimensions or its generalization of a variational inequality, the mathematical paradigm of differential variational inequalities (DVIs) was born. On one hand, a DVI addresses the need to extend the century-old ODE paradigm to incorporate such elements as algebraic inequalities, mode changes, and logical relations, and on the other hand, introduces a temporal element into a static optimization or equilibrium problems. The resulting DVI paradigm transforms the basic science of dynamical systems by leveraging the modeling power and computational advances of constrained optimization and its extension of a variational inequality, bringing the former (dynamical systems) to new heights that necessitate renewed analysis and novel solution methods and endowing the latter (optimization) with a novel perspective that is much needed for its sustained development. Introducing the subject of the DVI and giving a survey of its recent developments, this report is an expanded version of five lectures that the author gave at the CIME Summer School on Centralized and Distributed Multi-agent Optimization Models and Algorithms held in Cetraro, Italy, June 23–27, 2014.
- This study investigates the use of trigonometry to solve optimal control problems with mixed state and control constraints. Regularization of bang-bang and singular control problems by use of the Epsilon-Trig regularization method and the Trigonometrization technique for optimal control problems with constraints on control, which are based on trigonometry, form the motivation for this study. The proposed methodology converts the control into a trigonometric form and places implicit bounds on it. These bounds on the control are variable and are dependent on the states. The optimal control problems with mixed constraints on control and state traditionally require the solution to a multi-point boundary value problem, which is very difficult and complicated. The Rayleigh mixed constraint problem is chosen as a benchmark to showcase this issue and its effective resolution through the Trigonometrization approach. A space shuttle re-entry problem with a very complicated mixed state control heat constraint is solved as an aerospace application for this study. The great simplicity and power of the Trigonometrization technique has been highlighted in this study.
- Nowadays, the Web has evolved from a static Web where users were only able to consume information, to a Web where users are also able to produce information. This evolution is commonly known as Social Web or Web 2.0. Social platforms and networks are certainly the most adopted technologies in this new era. These platforms are commonly used as a means to interact with peers, exchange messages, share resources, etc. Thus, these collaborative tasks that make users more active in generating content are one of the most important factors for the increasingly growing quantity of available data. From the research perspective, this brings important and interesting challenges for many research fields. In such a context, a mostly crucial problem is to enable users to find relevant information with respect to their interests and needs. This task is commonly referred to as Information Retrieval (IR). IR is performed every day in an obvious way over the Web, typically under a search engine. However, classic models of IR don’t consider the social dimension of the Web. They model web pages as a mixture of a static homogeneous terms generated by the same creators. Then, ranking algorithms are often based on: (i) a query and document text similarity and (ii) the existing hypertext links that connect these web pages, e.g. PageRank. Therefore, classic models of IR and even the IR paradigm should be adapted to the socialization of the Web, in order to fully leverage the social context that surround web pages and users. This thesis presents many approaches that go in this direction. In particular, three methods are introduced in this thesis: (i) a Personalized Social Query Expansion (PSQE) framework, which achieves social and personalized expansions of a query with respect to each user, i.e. for the same query, different users will obtain different expanded queries. (ii) a Personalized Social Document Representation (PSDR) framework that uses social information to enhance, improve and provide a personalized social representation of documents to each user. (iii) a Social Personalized Ranking function called SoPRa, which takes into account social features that are related to users and documents. All these approaches have the particularity of being scalable to large-scale datasets, flexible and adaptable according to the high dynamicity of social data, and efficient since they have been intensively evaluated and compared to the closest works. From a practical point of view, this thesis led to the development of an experimental social Web search engine called LAICOS that includes all the algorithms developed throughout this thesis.
- Despite a Network Anomaly Detection System (NADS) being capable of detecting existing and zero-day attacks, it is still not universally implemented in industry and real applications, with current systems producing high False Positive Rates (FPRs) and low Detection Rates (DRs). The challenges involved in designing a NADS architecture are 1) the methodology adopted for creating as comprehensive a profile as possible from diverse normal patterns and 2) in establishing an adaptive and lightweight Decision Engine (DE) which efficiently distinguishes between legitimate and anomalous activities at high speeds in large network environments. The need for such a method to be trained and validated on a decent dataset with the characteristics of current network environments is a significant challenge. This thesis provides substantial contributions to research on the building of a scalable, adaptive and lightweight NADS framework. It considers three aspects: a data source, relevant features and observations, and new DE approaches for achieving a reliable NADS architecture. The first key contribution is the creation of a new dataset called UNSWNB15 that has a hybrid of realistic modern legitimate and synthetic malicious activities, with statistical analyses and evaluations of it fully explained. Also, its complexity is assessed using existing techniques to demonstrate the extent of current sophisticated types of anomalous events. The second core contribution is the development of a new theory for selecting important features and observations from network packets without redundancy to construct a legitimate profile from which any deviation is considered an attack, that is, establish an efficient NADS from analyses of the protocols and services of the OSI model. The third major contribution is the development of two scalable frameworks with two new DE techniques for successfully detecting malicious activities in less processing times than current methods. These techniques, called the Geometric Area Analysis-ADS (GAA-ADS) and Dirichlet Mixture Model-ADS (DMM-ADS), are based on mixture models for modelling all possible normal patterns and detecting abnormal events that deviate from them using new outlier approaches.
- Prolate spheroidal wave functions provide a natural and effective tool for computing with bandlimited functions defined on an interval. As demonstrated by Slepian et al., the so called generalized prolate spheroidal functions (GPSFs) extend this apparatus to higher dimensions. While the analytical and numerical apparatus in one dimension is fairly complete, the situation in higher dimensions is less satisfactory. This report attempts to improve the situation by providing analytical and numerical tools for GPSFs, including the efficient evaluation of eigenvalues, the construction of quadratures, interpolation formulae, etc. Our results are illustrated with several numerical examples.
- Matrices are divided into different classes, depending on the form and specific properties of the matrix. Matrix factorizations depend on the properties of certain class of matrices, hence matrix factorization are of great importance in the matrix theory, in the analysis of numerical algorithms and even in numerical linear algebra. A factorization of the matrix A is a representation of A as a product of several "simpler" matrices, which makes the problem at hand easier to solve. Factorizations of matrices into some special sorts of matrices with similarity are of fundamental importance in matrix theory, like Schur decomposition, spectral decomposition and the singular value decomposition. Furthermore, the basic tool for solving systems of linear equations, as one of the basic problems of numerical linear algebra, is the LU factorization. Also, it is important to mention QR factorization and its calculation through rotation and reflectors.
- Random walk is a popular and efficient algorithm for image segmentation, especially for extracting regions of interest (ROI). One difficulty with the random walk algorithm is the requirement for solving a huge sparse linear system when applied to large images. Another limitation is its sensitivity to seeds distribution, i.e. the segmentation result depends on the number of seeds as well as their placement, which puts a burden on users. In this paper, we first propose a continuous random walk model with explicit coherence regularization (CRWCR) for the extracted ROI, which helps to reduce the seeds sensitivity, so as to reduce the user interactions. Then, a very efficient algorithm to solve the CRWCR model will be developed, which helps to remove the difficulty of solving huge linear systems. Our algorithm consists of two stages: initialization by performing one dimensional random walk sweeping based on user-provided seeds, followed by the alternating direction scheme, i.e. Peaceman-Rachford scheme for further correction. The first stage aims to provide a good initial guess for the ROI, and it is very fast since we just solve a limited number of one dimensional random walk problems. Then this initial guess is evolved to the ideal solution by applying the second stage, which should be also very efficient since it fits well for GPU computing and 10 iterations are usually sufficient for convergence. Numerical experiments are provided to validate the proposed model as well as the efficiency of the two-stage algorithm.
- In this paper, aliasing effects are investigated for random fields defined on the d-dimensional sphere, and reconstructed from discrete samples. First, we introduce the concept of an aliasing function on the sphere. The aliasing function allows to identify explicitly the aliases of a given harmonic coefficient in the Fourier decomposition. Then, we exploit this tool to establish the aliases of the harmonic coefficients approximated by means of the quadrature procedure named spherical uniform sampling. Subsequently, we study the consequences of the aliasing errors in the approximation of the angular power spectrum of an isotropic random field, the harmonic decomposition of its covariance function. Finally, we show that band-limited random fields are aliases-free, under the assumption of a sufficiently large amount of nodes in the quadrature rule.
- Accurately describing surface temperature effects for the dissociation of H2 on Cu(111) remains challenging. While Ab initio Molecular Dynamics (AIMD), the current state-of-the-art method for modelling such systems, can produce accurate results, it is computationally very expensive to use for extensive testing of, for example, density functionals. A chemically accurate static corrugation model for H2 and D2 on Cu(111) dissociation was made by introducing effective three-body interactions as well as an H2-bond dependence and fitting the model to density functional theory energies for 15 113 different configurations. Reaction probabilities and rovibrational (in)elastic scattering probabilities were computed and compared to experiments and other calculations. Theoretical and experimental results are in good agreement, except for the reaction of (v = 0, J = 0) H2 where both AIMD and the newly developed static corrugation model, both based on the same underlying density functional, predict a similar deviation from the experiment.
- Based on the fact that an ensemble of moving Rydberg atoms in two counterpropagating laser beams in the limit of complete dipole blocking is isomorphic to a Jaynes–Cummings model, a scheme for robust and efficient excitation of atomic Rydberg states is proposed. It is shown that the Doppler frequency shifts play an important role in atomic population transfer processes. The suggested method can be employed to detect the symmetric entangled states and paves the way to preparing entangled states with a single excited atom in a Rydberg state. It is shown that this process is robust with respect to parameter fluctuations, such as the laser pulse area, the relative spatial offset (the delay) of the laser beams and the number of atoms. Graphical abstract Open image in new window
- Functional data are observations that are either themselves functions or are naturally representable as functions. When these functions can be considered smooth, it is natural to use their derivatives in exploring their variation. Principal differential analysis (PDA) identifies a linear differential operator L = w0I+w1D + . . . + wm‐1Dm‐1 + Dm that comes as close as possible to annihilating a sample of functions. Convenient procedures for estimating the m weighting functions wj are developed. The estimated differential operator L is analogous to the projection operator used as the data annihilator in principal components analysis and thus can be viewed as a type of data reduction or exploration tool. The corresponding linear differential equation may also have a useful substantive interpretation. Modelling and regularization features can also be incorporated into PDA.
- In this article, the minimum time optimal control problem of an aircraft in its climbing phase is studied. First, a reduction of the initial dynamics into a three dimensional single-input system with a linear dependence with respect to the control is performed. This reduced system is then studied using geometric control techniques. In particular, the maximum principle leads to describe a multi-point boundary value problem which is solved by indirect methods. These methods are the implementation of the maximum principle and are initialized by direct methods. We check that the extremal solution of the boundary value problem satisfies necessary and sufficient conditions of optimality. From this reference case and considering small-time optimal trajectories, we give a local classification with respect to the initial mass and final velocity of BC-extremals for the climbing phase.
- In this paper, we have investigated numerically the unsteady separated stagnation-point flow of an incompressible viscous fluid over a porous flat plate subjected to continuous suction or blowing. The analysis covers the complete range of the suction/blowing parameter d, including \(d = 0\) and \(d \rightarrow \pm \infty \), in conjunction with the flow strength parameter \(a (> 0)\) and the unsteadiness parameter \(\beta \). Two types of solutions, namely attached flow solution (AFS) and reverse flow solution (RFS), have been found for a negative value of \(\beta \). A novel result that emerges from our analysis is the characteristic features of the boundary layer flows which firmly depend on the sum values of \((a + \beta )\) in case of massive blowing \((d \ll 0)\) given at the wall. In fact, a negative value of \((a + \beta )\) opposes the flow like an adverse pressure gradient and the solution domain ends off with a RFS. On the other hand, a positive value of \((a + \beta )\) assists the flow like a favourable pressure gradient for which the solution continues with an AFS. An interesting result of this analysis is that after a certain negative value of \((a + \beta )\), dependent on a and \(\beta \), the solution of this flow problem does not exist for blowing, and this trend persists even for suction \((d>0)\). On the other hand, for large positive values of \((a + \beta )\), the attached flow solution exists in case of both strong suction and massive blowing, whereas for small positive values of \((a + \beta )\) and especially in case of hard blowing, the solution of the governing boundary layer equation does not appear to have the boundary layer character. Furthermore, when \((a + \beta ) = 0\), the velocity gradient at the wall is zero for both AFS and RFS flows, and ultimately, the governing boundary layer equation produces a trivial solution which does not satisfy the outer boundary condition.
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