Order Partial Derivatives Research Articles

On numerical calculation of probabilities according to Dirichlet distribution

The main difficulty in numerical solution of probabilistic constrained stochastic programming problems is the calculation of the probability values according to the underlying multivariate probability distribution. In addition, when we are using a nonlinear programming algorithm for the solution of the problem, the calculation of the first and second order partial derivatives may also be necessary. In this paper we give solutions to the above problems in the case of Dirichlet distribution. For the calculation of the cumulative distribution function values, the Lauricella function series expansions will be applied up to 7 dimensions. For higher dimensions we propose the hypermultitree bound calculations and a variance reduction simulation procedure based on these bounds. There will be given formulae for the calculation of the first and second order partial derivatives, too. The common property of these formulae is that they involve only lower dimensional cumulative distribution function calculations. Numerical test results will also be presented.

Optimal bivariate [formula omitted] cubic quasi-interpolation on a type-2 triangulation

In [A. Guessab, O. Nouisser, G. Schmeisser, Multivariate approximation by a combination of modified Taylor polynomials, J. Comput. Appl. Math. 196 (2006) 162–179], a general method is proposed to increase the approximation order of approximation operators. In this work, by using these enhancement techniques, we introduce and study new schemes based on a C 1 -spline quasi-interpolant on a type-2 triangulation. They are designed for approximating real-valued functions defined on R 2 . The proposed method is based on the following idea: from a discrete quasi-interpolant defined by the quadratic box-spline exact on P 2 and by judiciously choosing the first-order Taylor coefficients, we derive a cubic differential quasi-interpolant yielding optimal approximation order. In addition, when the derivatives are not available or are extremely expensive to compute, we approximate them by finite difference approximations having the desired accuracy to derive a new class of discrete quasi-interpolants. As an essential difference to some of the existing methods, we only use the given data values and, then, we do not modify the original triangulation. Finally, we present some numerical tests which confirm the efficiency of the newly quasi-interpolant and demonstrate good visual quality. In particular, we compare it with a differential quasi-interpolant done by Lai [M.-J. Lai, Approximation order from bivariate C 1 -cubics on a four-directional mesh is full, Comput. Aided Geom. Design 11 (2) (1994) 215–223] which is also exact on P 3 but uses third order partial derivatives.

A numerical differentiation library exploiting parallel architectures

We present a software library for numerically estimating first and second order partial derivatives of a function by finite differencing. Various truncation schemes are offered resulting in corresponding formulas that are accurate to order O ( h ) , O ( h 2 ) , and O ( h 4 ) , h being the differencing step. The derivatives are calculated via forward, backward and central differences. Care has been taken that only feasible points are used in the case where bound constraints are imposed on the variables. The Hessian may be approximated either from function or from gradient values. There are three versions of the software: a sequential version, an OpenMP version for shared memory architectures and an MPI version for distributed systems (clusters). The parallel versions exploit the multiprocessing capability offered by computer clusters, as well as modern multi-core systems and due to the independent character of the derivative computation, the speedup scales almost linearly with the number of available processors/cores. Program summary Program title: NDL (Numerical Differentiation Library) Catalogue identifier: AEDG_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEDG_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 73 030 No. of bytes in distributed program, including test data, etc.: 630 876 Distribution format: tar.gz Programming language: ANSI FORTRAN-77, ANSI C, MPI, OPENMP Computer: Distributed systems (clusters), shared memory systems Operating system: Linux, Solaris Has the code been vectorised or parallelized?: Yes RAM: The library uses O ( N ) internal storage, N being the dimension of the problem Classification: 4.9, 4.14, 6.5 Nature of problem: The numerical estimation of derivatives at several accuracy levels is a common requirement in many computational tasks, such as optimization, solution of nonlinear systems, etc. The parallel implementation that exploits systems with multiple CPUs is very important for large scale and computationally expensive problems. Solution method: Finite differencing is used with carefully chosen step that minimizes the sum of the truncation and round-off errors. The parallel versions employ both OpenMP and MPI libraries. Restrictions: The library uses only double precision arithmetic. Unusual features: The software takes into account bound constraints, in the sense that only feasible points are used to evaluate the derivatives, and given the level of the desired accuracy, the proper formula is automatically employed. Running time: Running time depends on the function's complexity. The test run took 15 ms for the serial distribution, 0.6 s for the OpenMP and 4.2 s for the MPI parallel distribution on 2 processors.

Towards Validation of Satellite Gradiometric Data Using Modified Version of 2nd Order Partial Derivatives of Extended Stokes' Formula

Towards Validation of Satellite Gradiometric Data Using Modified Version of 2nd Order Partial Derivatives of Extended Stokes' FormulaThe satellite gradiometric data should be validated prior to being used. One way of such a validation process is to use some integral estimators which are the second-order partial derivatives of the extended Stokes formula to regenerate the data from the gravity anomaly at the topographic surface. In this paper, we present how least-squares modification methods are used to modify such integral estimators. Our concentration will be on validation of the vertical-horizontal and horizontal-horizontal elements of the gravitational tensor at satellite level. The paper will formulate the elements of the system of equations from which the modification parameters are derived based on all types of least-squares modification. The truncation and Paul's coefficients will also be modelled.

A cubic Hermite finite element-continuation method for numerical solutions of the von Kármán equations

We study a cubic Hermite finite element method for numerical solutions of the von Kármán equations defined in a rectangular domain with totally clamped boundary conditions. A novel iterative method combined with a predictor–corrector continuation algorithm is exploited to trace solution curves of the von Kármán equations. The fourth order finite element approximations compute accurate numerical solutions for the deformation and the Airy stress function as well as their first order partial derivatives and the mixed second order partial derivatives. In this regard, the classical predictor–corrector continuation method is interpreted in a different way. Our numerical results show that the bifurcation scenario of the von Kármán equations with totally clamped boundary conditions is different from those with simply supported and partially clamped boundary conditions.

Computer system for kinematic and dynamic analysis synthesis of rigid and flexible multibody systems

AbstractIn the paper an overview of a general numerical algorithm and program system library for deriving the kinematic constraint equations and dynamic equations of motion, as well as, computation of their first and second order partial derivatives with respect to kinematic parameters of motion, design parameters and mass and inertia characteristics for rigid and flexible multibody systems is presented. These are the main basic computational modules for implementation of kinematic and dynamic synthesis, optimization and design. The main theoretical basis consists in matrix methods for deriving the kinematic constraints and dynamic equations, as well as, the generalized Newton – Euler dynamic equations for rigid and flexible bodies, and finite element discretization in relative coordinates. Block–scheme of the computational procedures and problem oriented program compilation is presented. An example of kinematic synthesis of six–link path generating mechanism with singular points is presented. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A probability measure which has Markov property

We define a probability measure which has Markov property on the unit interval, compute a higher order partial derivative of its distribution function about the parameter. As application, we obtain explicit formulas of a digital sum of the block of the binary number.

O relacionamento entre os derivados parciais das transformadas de Legendre e suas aplicações termodinâmicas

Legendre transforms and their applications in thermodynamics are revisited by the use of Jacobian representation in order to express second order partial derivatives of a Legendre transform in terms of another Legendre transform. Some examples are shown to demonstrate the simplicity and conciseness power of the method

A bivariate C1 cubic super spline space on Powell–Sabin triangulation

A bivariate C 1 cubic super spline is constructed on Powell–Sabin type-1 split with the additional smoothness at vertices in the original triangulation being C 2 , which permits the Hermite interpolation up to the second order partial derivatives exactly on all the vertices in the original triangulation. The locally supported dual basis and computational details using derivatives around each vertex are given.

Calibration and uncertainty analysis of a simple model of silica-limited diatom growth in the Elbe River

A simple Lagrangian water quality model was designed to investigate the hypothesis of sporadic silica limitations of diatom growth in the lower Elbe River in Germany. For each fluid parcel a limited reservoir of silica was specified to be consumed by diatoms. The model's simplicity notwithstanding, a set of six selected model parameters could not be fully identified from existing observations at one station. After the introduction of prior knowledge of the ranges of meaningful parameter values, calibration of the over-parameterised model manifested itself primarily in the generation of posterior parameter covariances. Estimations of the covariance matrix based on (a) second order partial derivatives of a quadratic cost function at its optimum and (b) Monte Carlo simulations exploring the whole space of parameter values gave consistent results. Diagonalisation of the covariance matrix yielded two linear parameter combinations that were most effectively controlled by data from periods with and without lack of silica, respectively. The two parameter combinations were identified as the essential inputs that govern the successful simulation of intermittently decreasing chlorophyll a concentrations in summer. A satisfactory simulation of the pronounced chlorophyll a minimum in spring, by contrast, was found to be beyond the means of the simple model.

Numerical method of slope failure probability based on Bishop model

Based on Bishop’s model and by applying the first and second order mean deviations method, an approximative solution method for the first and second order partial derivatives of functional function was deduced according to numerical analysis theory. After complicated multi-independent variables implicit functional function was simplified to be a single independent variable implicit function and rule of calculating derivative for composite function was combined with principle of the mean deviations method, an approximative solution format of implicit functional function was established through Taylor expansion series and iterative solution approach of reliability degree index was given synchronously. An engineering example was analyzed by the method. The result shows its absolute error is only 0.78% as compared with accurate solution.

Local identification of scalar hybrid models with tree structure

Standard modelling approaches, e.g. in chemical engineering, suffer from two principal difficulties: the curse of dimension and a lack of extrapolability. We propose an approach via structured hybrid models (SHMs) to resolve both issues. For simplicity, we consider reactor models which can be written as a tree-like composition of scalar input-output (i/o) functions u j . The vertices j of the finite tree structure represent known or unknown subprocesses of the overall process. Known processes are modelled by white-box functions u j ; unknown processes are represented by black boxes Uj. Oriented edges of the tree indicate composition of the i/o relations u j in a feedforward structure. The tree structure of a mixture of black and white boxes constitutes what we call an SHM. Under certain assumptions on differentiability, genericity and monotonicity, we provide an inductive algorithm which uniquely identifies all black boxes in the SHM up to a trivial scaling calibration between adjacent black boxes. Our result does not require any extra measurements interior to the SHM. Instead, we only require global, overall i/o data clustered along a d-dimensional database of inputs. More precisely, information on partial derivatives of order up to 5 is required, in all directions, but only at base points within the d-dimensional database. The dimension d need not exceed the maximal input dimension of any individual black box in the SHM. Compared to the total input dimension n of the reactor, which may be much larger than d, this dimension reduction effectively avoids the curse of dimension: the complexity of our approach is polynomial in n and exponential in d, only, rather than exponential in n. Moreover, our unique identification of all black boxes accommodates a reliable global extrapolation, far beyond the original database, to input regions of full dimension. We illustrate our results with a model of an industrial continuous polymerization plant.

A new method for calculating vorticity

It is well known that it often leads to large error to calculate the first order partial derivatives of physical quantities by the difference method from observation data with error. Especially in meteorology, observational wind has large error, and calculating vorticity from observational wind is very inaccurate. Therefore, one-dimensional numerical differentiation algorithm is applied to calculate vorticity from observational wind, and the result is compared with that of the central difference method. It turns out that the one-dimensional numerical differentiation algorithm is stable and feasible. Its accuracy is superior to the central difference method, and it has stronger recognizing ability for the smaller-scale weather systems.

Combined ℓ2 data and gradient fitting in conjunction with ℓ1 regularization

We are interested in minimizing functionals with l2 data and gradient fitting term and l1 regularization term with higher order derivatives in a discrete setting. We examine the structure of the solution in 1D by reformulating the original problem into a contact problem which can be solved by dual optimization techniques. The solution turns out to be a ’smooth’ discrete polynomial spline whose knots coincide with the contact points while its counterpart in the contact problem is a discrete version of a spline with higher defect and contact points as knots. In 2D we modify Chambolle’s algorithm to solve the minimization problem with the l1 norm of interacting second order partial derivatives as regularization term. We show that the algorithm can be implemented efficiently by applying the fast cosine transform. We demonstrate by numerical denoising examples that the l2 gradient fitting term can be used to avoid both edge blurring and staircasing effects.

Multivariate stochastic Korovkin theory given quantitatively

We introduce and study very general multivariate stochastic positive linear operators induced by general multivariate positive linear operators that are acting on multivariate continuous functions. These are acting on the space of real differentiable multivariate time stochastic processes. Under some very mild, general and natural assumptions on the stochastic processes we produce related multidimensional stochastic Shisha–Mond type inequalities of L q -type 1 ≤ q < ∞ and corresponding multidimensional stochastic Korovkin type theorems. These are regarding the stochastic q -mean convergence of a sequence of multivariate stochastic positive linear operators to the stochastic unit operator for various cases. All convergences are produced with rates and are given via the stochastic inequalities involving the maximum of the multivariate stochastic moduli of continuity of the n th order partial derivatives of the engaged stochastic process, n ≥ 0 . The astonishing fact here is that basic real Korovkin test functions assumptions are enough for the conclusions of our multidimensional stochastic Korovkin theory. We give an application.

Relationship of d-dimensional continuous multi-scale wavelet shrinkage with integro-differential equations

The goal of this paper is to extend the results of Didas and Weickert [Didas, S, Weickert, J. Integrodifferential equations for continuous multi-scale wavelet shrinkage. Inverse Prob Imag 2007;1:47–62.] to d-dimensional ( d ⩾ 1) case. Firstly, we relate a d-dimensional continuous mother wavelet ψ( x) with a fast decay and n vanishing moments to the sum of the order partial derivative of a group of functions θ k ( x)(∣ k∣ = n) with fast decay, which also makes wavelet transform equal to a sum of smoothed partial derivative operators. Moreover, d-dimensional continuous wavelet transform can be explained as a weighted average of pseudo-differential equations, too. For d = 1, our results are completely same as Didas and Weickert (2007), but for d > 1, it is different from the type of one variable. Finally, we exploit the reason with an example of 2-dimensional and 3-dimensional Mexican hat wavelet.

Adaptive dynamics for physiologically structured population models

We develop a systematic toolbox for analyzing the adaptive dynamics of multidimensional traits in physiologically structured population models with point equilibria (sensu Dieckmann et al. in Theor. Popul. Biol. 63:309-338, 2003). Firstly, we show how the canonical equation of adaptive dynamics (Dieckmann and Law in J. Math. Biol. 34:579-612, 1996), an approximation for the rate of evolutionary change in characters under directional selection, can be extended so as to apply to general physiologically structured population models with multiple birth states. Secondly, we show that the invasion fitness function (up to and including second order terms, in the distances of the trait vectors to the singularity) for a community of N coexisting types near an evolutionarily singular point has a rational form, which is model-independent in the following sense: the form depends on the strategies of the residents and the invader, and on the second order partial derivatives of the one-resident fitness function at the singular point. This normal form holds for Lotka-Volterra models as well as for physiologically structured population models with multiple birth states, in discrete as well as continuous time and can thus be considered universal for the evolutionary dynamics in the neighbourhood of singular points. Only in the case of one-dimensional trait spaces or when N = 1 can the normal form be reduced to a Taylor polynomial. Lastly we show, in the form of a stylized recipe, how these results can be combined into a systematic approach for the analysis of the (large) class of evolutionary models that satisfy the above restrictions.

On determination of jumps in terms of Abel–Poisson mean of Fourier series

In this paper, we generalize some well-known results (Theorems A, C, and D) by establishing two general results (Theorems 1 and 3). As special applications, we find that the (generalized) jumps of f can be determined by the higher order partial derivatives of its Abel–Poisson means. This is different from the determination of jumps by higher order derivatives of the partial sums. We also give some estimates of the higher order partial derivatives of the Abel–Poisson mean of an integrable function F at those points at which F is smooth.

On the complexity of the multivariate Sturm–Liouville eigenvalue problem

We study the complexity of approximating the smallest eigenvalue of - Δ + q with Dirichlet boundary conditions on the d-dimensional unit cube. Here Δ is the Laplacian, and the function q is non-negative and has continuous first order partial derivatives. We consider deterministic and randomized classical algorithms, as well as quantum algorithms using quantum queries of two types: bit queries and power queries. We seek algorithms that solve the problem with accuracy ɛ . We exhibit lower and upper bounds for the problem complexity. The upper bounds follow from the cost of particular algorithms. The classical deterministic algorithm is optimal. Optimality is understood modulo constant factors that depend on d. The randomized algorithm uses an optimal number of function evaluations of q when d ≤ 2 . The classical algorithms have cost exponential in d since they need to solve an eigenvalue problem involving a matrix with size exponential in d. We show that the cost of quantum algorithms is not exponential in d, regardless of the type of queries they use. Power queries enjoy a clear advantage over bit queries and lead to an optimal complexity algorithm.

Modified nodal cubic spline collocation for biharmonic equations

We formulate a modified nodal cubic spline collocation scheme for the solution of the biharmonic Dirichlet problem on the unit square. We prove existence and uniqueness of a solution of the scheme and show how the scheme can be solved on an N × N uniform partition of the square at a cost O(N2 log2 N + mN2) using fast Fourier transforms and m iterations of the preconditioned conjugate gradient method. We demonstrate numerically that m proportional to log2N guarantees the desired convergence rates. Numerical results indicate the fourth order accuracy of the approximations in the global maximum norm and the fourth order accuracy of the approximations to the first order partial derivatives at the partition nodes.