Estimation Of Partial Derivatives Research Articles

Estimates of Partial Derivatives for Harmonic Functions on the Unit Disc

Estimates of Partial Derivatives for Harmonic Functions on the Unit Disc

Linear operators with vector masks in digital image processing problems

The paper shows that it is expedient to use vector masks for solving some types of digital image processing problems. The main advantage of vector masks compared to matrix masks is that they reduce the computational complexity of algorithms while maintaining, and in some problems even improving, quality indicators. The article demonstrates examples of the use of vector masks in the problem of estimating the level of discrete white noise in an image, forming a basis for constructing a correctly working sigma filter, which are used for obtaining smoothed partial derivative estimates in the problem of edge detection and detecting straight lines in a contour image. The work uses results obtained by the authors in their earlier publications.

Event‐triggered model‐free adaptive fractional order sliding mode I/O constrained control for a class of nonlinear systems

AbstractTo tackle the trajectory tracking issue of nonlinear systems with unknown dynamics and input/output (I/O) constraints, a novel model‐free adaptive constrained control (MFACC) scheme is first proposed with improvement of tracking precision and reduction of computation cost. The system model is processed via the compact form dynamic linearization approach. Then, an event‐triggered observer‐based pseudo partial derivative (PPD) estimation algorithm is proposed to implement the data‐driven modeling information, which relieves the computation burden of modeling. Relying on the reconfigured data model, an improved prescribed performance control (PPC) approach with asymmetric predefined boundaries and the fractional order sliding mode control strategy are both exploited and integrated with MFACC for high‐precision and robustness assurance. Moreover, a novel prescribed‐boundary‐based event‐triggered mechanism is proposed to implement MFACC scheme so that the remarkable balance between tracking precision and computation cost can be maintained, and the I/O constraints can be handled via the proposed PPD‐based anti‐windup compensator and PPC technique. Finally, the stability analysis and contrast simulation are pursued which verifies the validity of proposed scheme.

A Physiological Control for Artificial Heart Pump based on Adaptive Regulation of Heart Rate and the improved BAS Algorithm Optimization

In order to solve the problem that the traditional artificial heart pump has poor dynamic bionic performance and cannot respond to blood perfusion in real time according to the patient’s state, this paper proposes a new physiological control method based on adaptive adjustment of heart rate. Firstly, a coupled model of the cardiovascular circulatory system and artificial heart pump was established, and the correctness of the model is verified by hemodynamic simulation. Secondly, according to the model-free adaptive control theory, the artificial heart pump physiological control strategy for adaptive regulation of heart rate is designed, including pseudo partial derivative estimation algorithm and adaptive control law. Then, the parameters of the adaptive controller are optimized by the improved beetle antennae search algorithm. Finally, the auxiliary effect and dynamic performance of the adaptive control strategy are studied by numerical methods. The results show that the physiological control method based on adaptive regulation of heart rate realizes the adaptive modulation of heart rate and blood flow, which can meet the needs of blood perfusion in different states and help to promote the recovery of damaged heart.

Uniform almost sure convergence and asymptotic distribution of the wavelet-based estimators of partial derivatives of multivariate density function under weak dependence

This paper is devoted to the estimation of partial derivatives of multivariate density functions. In this regard, nonparametric linear wavelet-based estimators are introduced, showing their attractive properties from the theoretical point of view. In particular, we prove the strong uniform consistency properties of these estimators, over compact subsets of , with the determination of the corresponding convergence rates. Then, we establish the asymptotic normality of these estimators. As a main contribution, we relax some standard dependence conditions; our results hold under a weak dependence condition allowing the consideration of mixing, association, Gaussian sequences and Bernoulli shifts.

Comparing Old and New Partial Derivative Estimates from Nonlinear Nonparametric Regressions

Partial derivatives have a special place in economics since the marginal revolution of the 1850s. We present results from multivariate partial derivative estimates using nonlinear non-parametric regressions in a finite difference method, accessible via the R-package NNS. Numerical partial derivatives are notoriously unstable, but NNS always correctly estimates their sign and comes closest to the correct magnitude compared to the coefficients in multiple linear regressions, and compared to the gradients from the popular np package for non-parametric kernel regressions.

Comparing Old and New Partial Derivative Estimates from Nonlinear Nonparametric Regressions: Supplemental Materials

This paper contains the R-code and output for all experiments of partial derivative estimation using NNS, np, and OLS techniques.

Partial Derivative Estimation for Underlying Functional-Valued Process in a Unified Framework

We consider functional data analysis when the observations at each location are functional rather than scalar. When the dynamic of underlying functional-valued process at each location is of interest, it is desirable to recover partial derivatives of a sample function, especially from sparse and noise-contaminated measures. We propose a novel approach based on estimating derivatives of eigenfunctions of marginal kernels to obtain a representation for functional-valued process and its partial derivatives in a unified framework in which the number of locations and number of observations at each location for each individual can be any rate relative to the sample size. We derive almost sure rates of convergence for the procedures and further establish consistency results for recovered partial derivatives.

Adaptive estimation‐based TILC for the finite‐time consensus control of non‐linear discrete‐time MASs under directed graph

This work explores the consensus problems under the directed graph, variable learning gains, fast convergence and data-driven control framework comprehensively and proposes an adaptive estimation-based terminal iterative learning control for a nonlinear discrete-time multi-agent system (MAS) with a constant control input. A linear iteration-incremental model is built by using an iterative dynamic linearisation where the unknown partial derivatives are estimated iteratively using I/O data. The learning control law is designed with both a constant learning gain and an iteration-time-varying learning gain. The constant one can be selected properly according to the estimation of partial derivatives and the varying one can be estimated from iteratively utilising I/O data. The result has also been extended to the nonlinear MAS with time-varying control input and an extended adaptive estimation-based TILC is developed by using time-varying control input to enhance the control performance. A fast convergence of both the proposed methods is achieved by removing the unnecessary error constraints at other time instants than the endpoint. Both the proposed methods is apparently data-driven since no model information is involved. The proposed finite time consensus control methods are confirmed to be effective under the directed graph through mathematic proof and extensive simulations.

The effects of negative interest rates on the estimation of option sensitivities: The impact of switching from a log-normal to a normal model

The estimation of partial derivatives of the price in respect to the main financial variables, called Greeks, is an essential task for a trader in order to understand the sensitivity of a derivative to the input of pricing model. The study of the level of reactivity of the mark to market is an essential task to manage properly the market risk of a portfolio. Due to the negative interest rates in Euro Area, the pricing model of interest-rates options (cap, floor and swaption) has been changed from a log-normal to a normal framework. The aim of this paper is to investigate the effects of this model change on the calculation of option sensitivities.

An Improved Multi-Level Adaptive Control Algorithm Based on Parameters Self-Tuning for a Class of SISO Nonlinear Systems

In this paper, a multi-level adaptive control algorithm using parameters self-tuning is proposed for improving the tracking performance of a class of SISO nonlinear systems. This scheme is comprised of the pseudo partial derivative estimation law and the adaptive control law. The adaptive control law is derived from a novel higher order weighted one-step-ahead criterion function which exploits more previous control information. In order to identify the laws’ penalty factors which partly imply the dynamics of the system, an approach of parameters self-tuning is proposed with on-line recursive gradient algorithm. This design is model free and depends directly on the pseudo partial derivative updating on-line with the input and output data Simulation examples are provided to illustrate the effectiveness of the proposed method.

A fast and unconditionally positive finite-difference discretization of a multidimensional equation in nonlinear population dynamics

Departing from a nonlinear diffusion–reaction equation which describes the growth of biological films, we derive a finite-difference discretization which preserves unconditionally the positivity and the boundedness of approximations. The design of this method follows a non-traditional approach in the estimation of first-order partial derivatives, and the technique is a variable step-size and exact methodology for which the properties of existence and uniqueness of non-negative and bounded solutions hold for any initial profile which is likewise non-negative and bounded. As a consequence of the exactness, the computer implementation requires less resources and yields faster results than some linear schemes available in the literature. Qualitative comparisons against known linear and nonlinear techniques show that our method produces similar computer results in the two-dimensional case. Moreover, our scheme is also able to simulate the development of microbial films in the three-dimensional scenario. This is a feature that is not inherent to the linear methodologies considered, in view of the large amount of computational resources that such approaches require.

NDL-v2.0: A new version of the numerical differentiation library for parallel architectures

We present a new version of the numerical differentiation library (NDL) used for the numerical estimation of first and second order partial derivatives of a function by finite differencing. In this version we have restructured the serial implementation of the code so as to achieve optimal task-based parallelization. The pure shared-memory parallelization of the library has been based on the lightweight OpenMP tasking model allowing for the full extraction of the available parallelism and efficient scheduling of multiple concurrent library calls. On multicore clusters, parallelism is exploited by means of TORC, an MPI-based multi-threaded tasking library. The new MPI implementation of NDL provides optimal performance in terms of function calls and, furthermore, supports asynchronous execution of multiple library calls within legacy MPI programs. In addition, a Python interface has been implemented for all cases, exporting the functionality of our library to sequential Python codes. New version program summaryProgram title: NDL-v2.0Catalog identifier: AEDG_v2_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEDG_v2_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 63036No. of bytes in distributed program, including test data, etc.: 801872Distribution format: tar.gzProgramming language: ANSI Fortran-77, ANSI C, Python.Computer: Distributed systems (clusters), shared memory systems.Operating system: Linux, Unix.Has the code been vectorized or parallelized?: Yes.RAM: The library uses O(N) internal storage, N being the dimension of the problem. It can use up to O(N2) internal storage for Hessian calculations, if a task throttling factor has not been set by the user.Classification: 4.9, 4.14, 6.5.Catalog identifier of previous version: AEDG_v1_0Journal reference of previous version: Comput. Phys. Comm. 180(2009)1404Does the new version supersede the previous version?: YesNature 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, and sensitivity analysis. For a large number of scientific and engineering applications, the underlying functions correspond to simulation codes for which analytical estimation of derivatives is difficult or almost impossible. A 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 a carefully chosen step that minimizes the sum of the truncation and round-off errors. The parallel versions employ both OpenMP and MPI libraries.Reasons for new version:The updated version was motivated by our endeavors to extend a parallel Bayesian uncertainty quantification framework [1], by incorporating higher order derivative information as in most state-of-the-art stochastic simulation methods such as Stochastic Newton MCMC [2] and Riemannian Manifold Hamiltonian MC [3]. The function evaluations are simulations with significant time-to-solution, which also varies with the input parameters such as in [1, 4]. The runtime of the N-body-type of problem changes considerably with the introduction of a longer cut-off between the bodies. In the first version of the library, the OpenMP-parallel subroutines spawn a new team of threads and distribute the function evaluations with a PARALLEL DO directive. This limits the functionality of the library as multiple concurrent calls require nested parallelism support from the OpenMP environment. Therefore, either their function evaluations will be serialized or processor oversubscription is likely to occur due to the increased number of OpenMP threads. In addition, the Hessian calculations include two explicit parallel regions that compute first the diagonal and then the off-diagonal elements of the array. Due to the barrier between the two regions, the parallelism of the calculations is not fully exploited. These issues have been addressed in the new version by first restructuring the serial code and then running the function evaluations in parallel using OpenMP tasks. Although the MPI-parallel implementation of the first version is capable of fully exploiting the task parallelism of the PNDL routines, it does not utilize the caching mechanism of the serial code and, therefore, performs some redundant function evaluations in the Hessian and Jacobian calculations. This can lead to: (a) higher execution times if the number of available processors is lower than the total number of tasks, and (b) significant energy consumption due to wasted processor cycles. Overcoming these drawbacks, which become critical as the time of a single function evaluation increases, was the primary goal of this new version. Due to the code restructure, the MPI-parallel implementation (and the OpenMP-parallel in accordance) avoids redundant calls, providing optimal performance in terms of the number of function evaluations. Another limitation of the library was that the library subroutines were collective and synchronous calls. In the new version, each MPI process can issue any number of subroutines for asynchronous execution. We introduce two library calls that provide global and local task synchronizations, similarly to the BARRIER and TASKWAIT directives of OpenMP. The new MPI-implementation is based on TORC, a new tasking library for multicore clusters [5–7]. TORC improves the portability of the software, as it relies exclusively on the POSIX-Threads and MPI programming interfaces. It allows MPI processes to utilize multiple worker threads, offering a hybrid programming and execution environment similar to MPI+OpenMP, in a completely transparent way. Finally, to further improve the usability of our software, a Python interface has been implemented on top of both the OpenMP and MPI versions of the library. This allows sequential Python codes to exploit shared and distributed memory systems.Summary of revisions:The revised code improves the performance of both parallel (OpenMP and MPI) implementations. The functionality and the user-interface of the MPI-parallel version have been extended to support the asynchronous execution of multiple PNDL calls, issued by one or multiple MPI processes. A new underlying tasking library increases portability and allows MPI processes to have multiple worker threads. For both implementations, an interface to the Python programming language has been added.Restrictions:The library uses only double precision arithmetic. The MPI implementation assumes the homogeneity of the execution environment provided by the operating system. Specifically, the processes of a single MPI application must have identical address space and a user function resides at the same virtual address. In addition, address space layout randomization should not be used for the application.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 23 ms for the serial distribution, 25 ms for the OpenMP with 2 threads, 53 ms and 1.01 s for the MPI parallel distribution using 2 threads and 2 processes respectively and yield-time for idle workers equal to 10 ms.

Schwarz–Pick type estimates of pluriharmonic mappings in the unit polydisk

In this paper, we will give Schwarz-Pick type estimates of arbitrary order partial derivatives for bounded pluriharmonic mappings defined in the unit polydisk. Our main results are generalizations of results of Colonna for planar harmonic mappings in (Indiana Univ. Math. J. 38: 829-840, 1989).

Numerical tools to estimate the flux of a gas across the air–water interface and assess the heterogeneity of its forcing functions

Abstract. A numerical tool was developed for the estimation of gas fluxes across the air–water interface. The primary objective is to use it to estimate CO2 fluxes. Nevertheless application to other gases is easily accomplished by changing the values of the parameters related to the physical properties of the gases. A user-friendly software was developed allowing to build upon a standard kernel a custom-made gas flux model with the preferred parameterizations. These include single or double layer models; several numerical schemes for the effects of wind in the air-side and water-side transfer velocities; the effects of atmospheric stability, surface roughness and turbulence from current drag with the bottom; and the effects on solubility of water temperature, salinity, air temperature and pressure. An analysis was also developed which decomposes the difference between the fluxes in a reference situation and in alternative situations into its several forcing functions. This analysis relies on the Taylor expansion of the gas flux model, requiring the numerical estimation of partial derivatives by a multivariate version of the collocation polynomial. Both the flux model and the difference decomposition analysis were tested with data taken from surveys done in the lagoon system of Ria Formosa, south Portugal, in which the CO2 fluxes were estimated using the infrared gas analyzer (IRGA) and floating chamber method, whereas the CO2 concentrations were estimated using the IRGA and degasification chamber. Observations and estimations show a remarkable fit.

Schwarz-Pick Estimates for Holomorphic Mappings with Values in Homogeneous Ball

LetBXbe the unit ball in a complex Banach spaceX. AssumeBXis homogeneous. The generalization of the Schwarz-Pick estimates of partial derivatives of arbitrary order is established for holomorphic mappings from the unit ballBntoBXassociated with the Carathéodory metric, which extend the corresponding Chen and Liu, Dai et al. results.

Higher-order Schwarz-Pick estimates for holomorphic self-mappings on classical domains

In this paper, Schwarz-Pick estimates for high order Frechet derivatives of holomorphic self-mappings on classical domains are presented. Moreover, the obtained result can deduce the early work on Schwarz-Pick estimates of higher-order partial derivatives for bounded holomorphic functions on classical domains.

The Generalized Schwarz–Pick Estimates of Arbitrary Order on the Unit Polydisk

Let Ω be a homogeneous circular convex domain (a bounded symmetric domain) in \({\mathbb{C}^N}\) containing the origin. The generalization of the Schwarz–Pick estimates of partial derivatives of arbitrary order are established for holomorphic mappings from the unit polydisk D n to Ω associated with the Caratheodory metric. The metric enables us to obtain some known results from a unified perspective and also leads some new results. In particular, our results are valid for the Cartan domains. When Ω = B N , the Schwarz–Pick estimates of high order reduces to that of Liu and Chen.

Multivariate numerical differentiation

We present an innovative method for multivariate numerical differentiation i.e. the estimation of partial derivatives of multidimensional noisy signals. Starting from a local model of the signal consisting of a truncated Taylor expansion, we express, through adequate differential algebraic manipulations, the desired partial derivative as a function of iterated integrals of the noisy signal. Iterated integrals provide noise filtering. The presented method leads to a family of estimators for each partial derivative of any order. We present a detailed study of some structural properties given in terms of recurrence relations between elements of a same family. These properties are next used to study the performance of the estimators. We show that some differential algebraic manipulations corresponding to a particular family of estimators lead implicitly to an orthogonal projection of the desired derivative in a Jacobi polynomial basis functions, yielding an interpretation in terms of the popular least squares. This interpretation allows one to (1) explain the presence of a spatial delay inherent to the estimators and (2) derive an explicit formula for the delay. We also show how one can devise, by a proper combination of different elementary estimators of a given order derivative, an estimator giving a delay of any prescribed value. The simulation results show that delay-free estimators are sensitive to noise. Robustness with respect to noise can be highly increased by utilizing voluntary-delayed estimators. A numerical implementation scheme is given in the form of finite impulse response digital filters. The effectiveness of our derivative estimators is attested by several numerical simulations.

Schwarz-pick estimates for bounded holomorphic functions on classical domains

In this paper, Schwarz-Pick estimates for high order Fréchet derivatives of bounded holomorphic functions on three kinds of classical domains are presented. We generalize the early work on Schwarz-Pick estimates of higher order partial derivatives for bounded holomorphic functions on the disk and unit ball.