Analytical Derivatives Research Articles

Low-Complexity Tracking Control of Strict-Feedback Systems With Unknown Control Directions

This paper focuses on the problem of output tracking with prescribed transient and steady-state performance for strict-feedback systems with unknown nonlinear functions and unmatched disturbances. In lieu of Nussbaum gain techniques, parameter estimation algorithms and switching control strategies, a continuous static low-complexity control solution is provided by means of a novel combination of smooth orientation functions and error transformation functions. The proposed method possesses inherent robustness against model uncertainties, disturbances, and virtual control signal derivatives, thus eliminating the needs to introduce extra robust control schemes and compute analytic derivatives. Comparative simulation results further illustrate the above theoretical findings.

Accelerated complex-step finite difference for expedient deformable simulation

In deformable simulation, an important computing task is to calculate the gradient and derivative of the strain energy function in order to infer the corresponding internal force and tangent stiffness matrix. The standard numerical routine is the finite difference method, which evaluates the target function multiple times under a small real-valued perturbation. Unfortunately, the subtractive cancellation prevents us from setting this perturbation sufficiently small, and the regular finite difference is doomed for computing problems requiring a high-accuracy derivative evaluation. In this paper, we graft a new finite difference scheme, namely the complex-step finite difference (CSFD), with physics-based animation. CSFD is based on the complex Taylor series expansion, which avoids subtractions in first-order derivative approximation. As a result, one can use a very small perturbation to calculate the numerical derivative that is as accurate as its analytic counterpart. We accelerate the original CSFD method so that it is also as efficient as the analytic derivative. This is achieved by discarding high-order error terms, decoupling real and imaginary calculations, replacing costly functions based on the theory of equivalent infinitesimal, and isolating the propagation of the perturbation in composite/nesting functions. CSFD can be further augmented with multicomplex Taylor expansion and Cauchy-Riemann formula to handle higher-order derivatives and tensor-valued functions. We demonstrate the accuracy, convenience, and efficiency of this new numerical routine in the context of deformable simulation - one can easily deploy a robust simulator for general hyperelastic materials, including user-crafted ones to cater specific needs in different applications. Higher-order derivatives of the energy can be readily computed to construct modal derivative bases for reduced real-time simulation. Inverse simulation problems can also be conveniently solved using gradient/Hessian-based optimization procedures.

X-CAD

We propose a novel generic shape optimization method for CAD models based on the eXtended Finite Element Method (XFEM). Our method works directly on the intersection between the model and a regular simulation grid, without the need to mesh or remesh, thus removing a bottleneck of classical shape optimization strategies. This is made possible by a novel hierarchical integration scheme that accurately integrates finite element quantities with sub-element precision. For optimization, we efficiently compute analytical shape derivatives of the entire framework, from model intersection to integration rule generation and XFEM simulation. Moreover, we describe a differentiable projection of shape parameters onto a constraint manifold spanned by user-specified shape preservation, consistency, and manufacturability constraints. We demonstrate the utility of our approach by optimizing mass distribution, strength-to-weight ratio, and inverse elastic shape design objectives directly on parameterized 3D CAD models.

Application of Artificial Neural Network accelerating a porous media FE2 homogenization scheme

AbstractMultiscale techniques, which include information of discrete lower level substructures of real material, are state of the art methods of current researches. This technology has the advantage of achieving more accurate results, by imaging the real geometry information from the microscopic level. In addition, it provides the opportunity to design a certain microstructure which fulfills the specific requirements at a macroscopic level. The drawback lies on the increasing computational effort. Simulation of a 3‐dimensional, nonlinear, time‐dependent, coupled, two‐scale problem with industrial relevance, could cause unacceptable runtimes. There are several strategies to overcome this disadvantage, such as parallelization, analytical derivatives and various surrogate models. This contribution shows the feasibility of storing microstructural information in an Artificial Neural Network, in order to reduce computational runtime.

Two-order approximate and large stepsize numerical direction based on the quadratic hypothesis and fitting method

Many effective optimization algorithms require partial derivatives of objective functions, while some optimization problems ʼ objective functions have no derivatives. According to former research studies, some search directions are obtained using the quadratic hypothesis of objective functions. Based on derivatives, quadratic function assumptions, and directional derivatives, the computational formulas of numerical first-order partial derivatives, second-order partial derivatives, and numerical second-order mixed partial derivatives were constructed. Based on the coordinate transformation relation, a set of orthogonal vectors in the fixed coordinate system was established according to the optimization direction. A numerical algorithm was proposed, taking the second order approximation direction as an example. A large stepsize numerical algorithm based on coordinate transformation was proposed. Several algorithms were validated by an unconstrained optimization of the two-dimensional Rosenbrock objective function. The numerical second order approximation direction with the numerical mixed partial derivatives showed good results. Its calculated amount is 0.2843 % of that of without second-order mixed partial derivative. In the process of rotating the local coordinate system 360 ° , because the objective function is more complex than the quadratic function, if the numerical direction derivative is used instead of the analytic partial derivative, the optimization direction varies with a range of 103.05 ° . Because theoretical error is in the numerical negative gradient direction, the calculation with the coordinate transformation is 94.71 % less than the calculation without coordinate transformation. If there is no theoretical error in the numerical negative gradient direction or in the large-stepsize numerical optimization algorithm based on the coordinate transformation, the sawtooth phenomenon occurs. When each numerical mixed partial derivative takes more than one point, the optimization results cannot be improved. The numerical direction based on the quadratic hypothesis only requires the objective function to be obtained, but does not require derivability and does not take into account truncation error and rounding error. Thus, the application scopes of many optimization methods are extended.

Subsurface Structural Mapping over Koton Karifi and Adjoining Areas, Southern Bida Basin, Nigeria, using High-Resolution Aeromagnetic Data

In this current work, we aim to delineate the subsurface structural trends, determine the depth to basement surface, and provide an illustrative 3D model for its subsurface structure. Four digitized aeromagnetic maps were acquired from the Nigerian Geological Survey Agency, Abuja. The total field aeromagnetic anomalies over Koton Karifi and adjoining areas have been evaluated. In order to map the subsurface structures and estimate the depth to basement surface the spectral analysis method was applied. To achieve such goals, a detailed analysis of the aeromagnetic data for the study area was performed. 2D interpretation was carried out for the aeromagnetic data. The processes used include contouring of the Total Magnetic Intensity (TMI) data, separation of regional and residual anomalies, structural detection methods such as analytic signal, vertical derivatives, and magnetic lineament mapping were used to map the contacts and faults within the study area. The first vertical derivative and the magnetic lineament maps show major geologic lineaments trending in East-West with minor ones trending Northeast-Southwest. In the south-eastern part of the study area, there is a dome-shaped linear feature. The result obtained using the spectral analysis method reveals two source depth models. The depths to deeper magnetic sources range from 2.81 km to 3.24 km with an average depth of 2.90 km. The deeper magnetic source bodies are identified with the magnetic basement. The shallower magnetic sources which range from 0.45 km to 1.81 km with an average depth of 1.13 km could be attributed to near surface magnetic sources which are magnetic rocks that intruded into the sedimentary formations or magnetised bodies within the sedimentary cover. Based on the sedimentary thickness range of 0.45 to 3.24 km, there is an indication that the possibility of hydrocarbon generation in the study area is feasible.

Analytic first-order derivatives of partially contracted n-electron valence state second-order perturbation theory (PC-NEVPT2)

A balanced treatment of dynamic and static electron correlation is important in computational chemistry, and multireference perturbation theory (MRPT) is able to do this at a reasonable computational cost. In this paper, analytic first-order derivatives, specifically gradients and dipole moments, are developed for a particular MRPT method, state-specific partially contracted n-electron valence state second-order perturbation theory (PC-NEVPT2). Only one linear equation needs to be solved for the derivative calculation if the Z-vector method is employed, which facilitates the practical application of this approach. A comparison of the calculated results with experimental geometrical parameters of O3 indicates excellent agreement although the calculated results for O3 - are slightly outside the experimental error bars. The 0-0 transition energies of various methylpyrimidines and trans-polyacetylene are calculated by performing geometry optimizations and seminumerical second-order geometrical derivative calculations. In particular, the deviations of 0-0 transition energies of trans-polyacetylene from experimental values are consistently less than 0.1 eV with PC-NEVPT2, indicating the reliability of the method. These results demonstrate the importance of adding dynamic electron correlation on top of methods dominated by static electron correlation and of developing analytic derivatives for highly accurate methods.

A spherical harmonic transform approach to the indexing of electron back-scattered diffraction patterns

A new approach is proposed for the indexing of electron back-scattered diffraction (EBSD) patterns. The algorithm employs a spherical master EBSD pattern and computes its cross-correlation with a back-projected experimental pattern using the spherical harmonic transform (SHT). This approach is significantly faster than the recent dictionary indexing algorithm, but shares the latter’s robustness against noise. The underlying theory is presented, followed by example applications, one on a series of Ni EBSD data sets recorded with decreasing signal-to-noise ratio, the other on a large shot-peened Al data set. The dependence of indexing speed and memory usage on the SHT bandwidth is explored. The speed gains of the new algorithm are achieved by executing real-valued Fast Fourier Transforms, explicitly incorporating crystallographic symmetry in the cross-correlation computation, and using efficient loop ordering to improve the caching behavior. The algorithm produces a cross-correlation array in the zyz Euler space; an orientation refinement procedure is proposed based on analytical derivatives of the Wigner d functions. The new approach can be applied to any diffraction modality for which the scattered intensity can be represented on a spherical surface.

Irradiance in Mixed Coherent/Incoherent Structures: An Analytical Approach

We propose a new method for a light energy flux density (or irradiance) calculationin an arbitrary multilayer stack containing coherent and incoherent layers. Although the wellknown General Transfer-Matrix Method (GTMM) can be successfully used for the overall reflectanceand transmittance calculation, it does not allow us to obtain the corresponding irradiance depthprofile straightforwardly. We show in this paper that subsequent phase-shift integrations over theincoherent layers result in the reflectance and transmittance expressions identical to those of theGTMMformulation. However, the alternative mathematical approach allows us to derive an analyticalexpression for irradiance at an arbitrary depth of the multilayer stack, thus making it possible tocalculate the absorptance depth profile. In fact, the GTMM expressions for the overall reflectanceand transmittance turn out to be special cases of the irradiance calculation at the incident andemergent surface of the multilayer stack. Consequently, the proposed Phase-shift Integration Method(PIM) represents a continuous irradiance calculation model without any energy imbalances onlayer interfaces. In addition, since we are able to obtain analytical layer thickness derivatives,the PIM is suitable for use with gradient optimization methods. We verify the method on threecases of an encapsulated bifacial heterojunction silicon (HJ Si) solar cell, a perovskite solar cell,and a perovskite/silicon tandem solar cell, which all consist of thin and thick layers.

Automatic targetless camera–LIDAR calibration by aligning edge with Gaussian mixture model

AbstractThis paper presents a calibration algorithm that does not require an artificial target object to precisely estimate a rigid‐body transformation between a camera and a light detection and ranging (LIDAR) sensor. The proposed algorithm estimates calibration parameters by minimizing a cost function that evaluates the edge alignment between two sensor measurements. In particular, the proposed cost function is constructed using a projection model‐based many‐to‐many correspondence of the edges to fully exploit measurements with different densities (dense photometry and sparse geometry). The alignment of the many‐to‐many correspondence is represented using the Gaussian mixture model (GMM) framework. Here, each component of the GMM, including weight, displacement, and standard deviation, is derived to suitably capture the intensity, location, and influential range of the edge measurements, respectively. The derived cost function is optimized by the gradient descent method with an analytical derivative. A coarse‐to‐fine scheme is also applied by gradually decreasing the standard deviation of the GMM to enhance the robustness of the algorithm. Extensive indoor and outdoor experiments validate the claim that the proposed GMM strategy improves the performance of the proposed algorithm. The experimental results also show that the proposed algorithm outperforms previous methods in terms of precision and accuracy by providing calibration parameters of standard deviations less than 0.6° and 2.1 cm with a reprojection error of 1.78 for a 2.1‐megapixel image (2,048 × 1,024) in the best case.

PyCycle: A Tool for Efficient Optimization of Gas Turbine Engine Cycles

Aviation researchers are increasingly focusing on unconventional vehicle designs with tightly integrated propulsion systems to improve overall aircraft performance and reduce environmental impact. Properly analyzing these types of vehicle and propulsion systems requires multidisciplinary models that include many design variables and physics-based analysis tools. This need poses a challenge from a propulsion modeling standpoint because current state-of-the-art thermodynamic cycle analysis tools are not well suited to integration into vehicles level models or to the application of efficient gradient-based optimization techniques that help to counteract the increased computational costs. Therefore, the objective of this research effort was to investigate the development a new thermodynamic cycle analysis code, called pyCycle, to address this limitation and enable design optimization of these new vehicle concepts. This paper documents the development, verification, and application of this code to the design optimization of an advanced turbofan engine. The results of this study show that pyCycle models compute thermodynamic cycle data within 0.03% of an identical Numerical Propulsion System Simulation (NPSS) model. pyCycle also provides more accurate gradient information in three orders of magnitude less computational time by using analytic derivatives. The ability of pyCycle to accurately and efficiently provide this derivative information for gradient-based optimization was found to have a significant benefit on the overall optimization process with wall times at least seven times faster than using finite difference methods around existing tools. The results of this study demonstrate the value of using analytic derivatives for optimization of cycle models, and provide a strong justification for integrating derivatives into other important engineering analyses.

Analytically Deriving Risk-Neutral Densities from Volatility Smiles

This paper develops a method to analytically derive RNDs (risk-neutral distributions) from volatility smiles. The strike price measure in the volatility smile does not have to be the strike price itself. It can be moneyness, the delta, or other popular strike price measures. The method utilizes analytical derivatives of the volatility smile that come with parametric interpolations of observed data on options. This makes numerical second derivatives of the call price function unnecessary. A worked-out example shows that the analytically derived RNDs are free of distortions associated with numerical second derivatives. The proposed method should be useful to practitioners.

Analytic QM/MM atomic charge derivatives avoiding the scaling of coupled perturbed equations with the MM subsystem size.

In electrostatic embedding mixed quantum and molecular mechanics (QM/MM) approaches, the QM charge distribution is polarized by the electrostatic interaction with the MM environment. Analytic derivatives of expectation values of operators are required to extract properties such as vibrational spectra. These derivatives usually require solving a set of coupled perturbed equations for each nucleus/atom in the system, thus becoming prohibitive when the MM subsystem contains thousands of atoms. In the context of Electrostatic Potential Fitting (ESPF) QM/MM, we can easily overcome this bottleneck by defining a set of auxiliary coupled perturbed equations called the Q-vector equations. The Q-vector method scales only with the size of the QM subsystem, producing an effective charge tensor that leads to the atomic charge derivative after contraction with the MM electrostatic potential gradient. As an example, we use the charge derivatives as an analysis tool to identify the most important chromophore-polarizing amino-acids in plant cryptochrome. This finding opens up the route of defining polarizable force fields and simulating vibrational spectroscopy using ESPF QM/MM electrostatic embedding at an affordable computational cost.

Surrogate models for parameterized representation of wake‐induced loads in wind farms

AbstractA procedure for mapping wake‐induced load predictions computed with the dynamic wake meandering model to a computationally efficient surrogate model approximation is defined and demonstrated. Using the mapping function, the load variation can be efficiently estimated for a wind farm with arbitrary layout. The resulting load assessment procedure provides continuous, differentiable output with known analytical derivatives and can be used for applications such as wind turbine layout optimization, estimation of turbine lifetime, and uncertainty analysis.

Efficient Derivative Evaluation for Rigid-Body Dynamics Based on Recursive Algorithms Subject to Kinematic and Loop Constraints

Simulation and control of robotic and bio-mechanical systems depend on a mathematical model description, typically a rigid-body system connected by joints, for which efficient algorithms to compute the forward or inverse dynamics exist. Gradient-based optimization and control methods require derivatives of the dynamics, often approximated by numerical differentiation (FD). However, they benefit from accurate gradients, which promote faster convergence, less iterations, and improved handling of nonlinearities or ill-conditioning of the problem formulations, which are particularly observed when kinematic constraints are involved. In this letter, we apply algorithmic differentiation (AD) to propagate sensitivities through dynamics algorithms. To this end, we augment the computational graph of these algorithms with derivative information. We provide analytic derivatives for elementary operations, in particular matrix factorizations of the descriptor form of the equation of motions, which yields a very efficient derivative evaluation for constrained dynamics. The proposed approach is implemented within the free software package rigid body dynamics library (RBDL), which heavily employs so-called spatial transformations in its implementation of the dynamics algorithms. Thus, manipulations of spatial transformations are treated as elementary operations. The efficiency is improved further by sparsity exploitation. We validate and benchmark the implementation against its FD counterpart for a lifting motion of a human model.

Analytical shape functions and derivatives approximation formulas in local radial point interpolation methods with applications to financial option pricing problems

The radial point interpolation method is increasingly being applied for the numerical solution of partial differential equations in different fields. Most implementations in the literature for obtaining the matrix coefficients make use of numerical approximations of the shape functions and their derivatives. To avoid the solution of linear systems required for computation of derivative approximations, this work derives analytical shape functions for three and five-node support domains in a local radial point interpolation method (LRPIM) with multiquadrics as basis functions. A weak form algorithm for the Black–Scholes equation using three-node analytical shape functions is developed and its unconditional stability and convergence are theoretically established. LRPIM finite-difference (FD) formulas are derived and applied to the solution of one and two-asset financial options. A five-node LRPIM-FD method in one-dimension is shown to yield fourth-order accuracy and applications to two-asset problems also yield accurate prices for options on a minimum of two risky assets and exchange options.

The effectiveness of reference-free modified embedded atom method potentials demonstrated for NiTi and NbMoTaW

One of the effective potentials that has proven to be very versatile and useful for describing metals is the modified embedded atom method (MEAM) potential. The reference-free version of the MEAM (RF-MEAM) potential provides more flexibility for fitting than the 2NN-MEAM because it also describes the pair potential as an explicit function. In this work, we present a methodology to fit RF-MEAM potentials to DFT data. We then evaluate the performance of the fitted potential by comparing MD simulations with experimental and DFT data. As an example, the methodology is applied to a binary and a quaternary alloy, namely NiTi and NbMoTaW. In the case of the equi-atomic NiTi shape memory alloy, our attention focuses on designing a potential that properly captures its mechanical behavior, given that the existing potentials fail to predict elastic constants in agreement with experiments. To reach our aim, we included the stress tensors of different high temperature NiTi configurations in the fitting database. The obtained RF-MEAM potential outperforms existing EAM and MEAM potentials in predicting the lattice and elastic constants of austenitic and martensitic phases as well as the corresponding transformation temperatures. To demonstrate the suitability of this methodology also for more complex systems, a RF-MEAM potential is fitted to model the multi-component NbMoTaW high-entropy alloy. Validation is achieved through comparison between observables obtained through the MD output and ab initio data. The article also reports key improvements to the optimization code MEAMfit v2 and the freely-available LAMMPS implementation of the RF-MEAM formalism. Most notably, resorting to analytic derivatives of the objective function with respect to the potential parameters rather than derivatives through finite differences, the time necessary for fitting has decreased by an order of magnitude.

Reduced-Order Modeling with Artificial Neurons for Gravitational-Wave Inference.

Gravitational-wave data analysis is rapidly absorbing techniques from deep learning, with a focus on convolutional networks and related methods that treat noisy time series as images. We pursue an alternative approach, in which waveforms are first represented as weighted sums over reduced bases (reduced-order modeling); we then train artificial neural networks to map gravitational-wave source parameters into basis coefficients. Statistical inference proceeds directly in coefficient space, where it is theoretically straightforward and computationally efficient. The neural networks also provide analytic waveform derivatives, which are useful for gradient-based sampling schemes. We demonstrate fast and accurate coefficient interpolation for the case of a four-dimensional binary-inspiral waveform family and discuss promising applications of our framework in parameter estimation.

A generally applicable atomic-charge dependent London dispersion correction.

The so-called D4 model is presented for the accurate computation of London dispersion interactions in density functional theory approximations (DFT-D4) and generally for atomistic modeling methods. In this successor to the DFT-D3 model, the atomic coordination-dependent dipole polarizabilities are scaled based on atomic partial charges which can be taken from various sources. For this purpose, a new charge-dependent parameter-economic scaling function is designed. Classical charges are obtained from an atomic electronegativity equilibration procedure for which efficient analytical derivatives with respect to nuclear positions are developed. A numerical Casimir-Polder integration of the atom-in-molecule dynamic polarizabilities then yields charge- and geometry-dependent dipole-dipole dispersion coefficients. Similar to the D3 model, the dynamic polarizabilities are precomputed by time-dependent DFT and all elements up to radon (Z = 86) are covered. The two-body dispersion energy expression has the usual sum-over-atom-pairs form and includes dipole-dipole as well as dipole-quadrupole interactions. For a benchmark set of 1225 molecular dipole-dipole dispersion coefficients, the D4 model achieves an unprecedented accuracy with a mean relative deviation of 3.8% compared to 4.7% for D3. In addition to the two-body part, three-body effects are described by an Axilrod-Teller-Muto term. A common many-body dispersion expansion was extensively tested, and an energy correction based on D4 polarizabilities is found to be advantageous for larger systems. Becke-Johnson-type damping parameters for DFT-D4 are determined for more than 60 common density functionals. For various standard energy benchmark sets, DFT-D4 slightly but consistently outperforms DFT-D3. Especially for metal containing systems, the introduced charge dependence of the dispersion coefficients improves thermochemical properties. We suggest (DFT-)D4 as a physically improved and more sophisticated dispersion model in place of DFT-D3 for DFT calculations as well as other low-cost approaches like semi-empirical models.

Robust calculation of slopes in detrended fluctuation analysis and its application to envelopes of human alpha rhythms

Detrended fluctuation analysis (DFA) is a popular method to analyze long-range temporal correlations in time series of many different research areas but in particular also for electrophysiological recordings. Using the classical DFA method, the cumulative sum of data are divided into segments, and the variance of these sums is studied as a function of segment length after linearly detrending them in each segment. The starting point of the proposed new method is the observation that the classical method is inherently non-stationary without justification by a corresponding non-stationarity of the data. This leads to unstable estimates of fluctuations to the extent that it is impossible to estimate slopes of the fluctuations other than by fitting a line over a wide range of temporal scales. We here use a modification of the classical method by formulating the detrending as a strictly stationary operation. With this modification the detrended fluctuations can be expressed as a weighted average across the power spectrum of a signal. Most importantly, we can also express the slopes, calculated as analytic derivatives of the fluctuations with respect to the scales, as statistically robust weighted averages across the power spectra. The method is applied to amplitudes of brain oscillations measured with magnetoencephalography in resting state condition. We found for envelopes of the the alpha rhythm that fluctuations as a function of time scales in a double-logarithmic plot differ substantially from a linear relation for time scales below 10 seconds. In particular we will show that model selections fail to determine accurate scaling laws, and that standard parameter settings are likely to yield results depending on signal to noise ratios than on true long range temporal correlations.