Jacobian Entries Research Articles

Automatic Differentiation Accelerated Shape Optimization Approaches to Photonic Inverse Design in FDFD/FDTD

AbstractShape optimization approaches to inverse design offer low‐dimensional, physically‐guided parameterizations of structures by representing them as combinations of primitives. However, on fixed grids, computing the gradient of a user objective via the adjoint variables method requires a product of forward/adjoint field solutions and the Jacobian of the simulation material distribution with respect to the structural shape parameters. Shape parameters often perturb global parts of the simulation grid resulting in many non‐zero Jacobian entries. These are often computed by finite‐difference (FD) in practice, and hence can be non‐trivial. In this work, the gradient calculation is accelerated by invoking automatic differentiation (AD) in instantiations of structural material distributions, enabled by the development of extensible differentiable feature‐mappings from parameters to primitives and differentiable effective logic operations (denoted AutoDiffGeo or ADG). ADG can also be used to accelerate FD‐based shape optimization by efficient boundary selection. AD‐enhanced shape optimization is demonstrated using three integrated photonic examples: a blazed grating coupler, a waveguide transition taper, and a polarization‐splitting grating coupler. The accelerations of the gradient calculation by AD relative to FD with boundary selection exceed 10, resulting in total optimization wall time accelerations of – on the same hardware with no compromise to device figure‐of‐merit.

Joint-Space Kinematic Control of a Bionic Continuum Manipulator in Real-Time by Using Hybrid Approach

Continuum manipulators are a type of robot used for delicate applications, including safe human-robot interactions. Controlling these manipulators for an accurate trajectory, especially in the case of pneumatic actuation, is extremely challenging. Thus, this article proposes a real-time kinematic trajectory control of a pneumatically actuated multi-segment bionic continuum manipulator with a mobile base by combining a neural network and analytical model with a cascaded controller to overcome this challenge. The inverse kinematics solution of the multi-segment manipulator is developed by using a neural network and an inverse piecewise constant curvature approach. The neural network is trained by using a separate learning algorithm. Although hybrid inverse modeling gives better solutions than existing techniques, significant residual positional error of the manipulator tip remains due to inherent material hysteresis. Thus, a cascaded PI-controller is utilized to compensate for the residual positional error. The controller gains are updated in each step by predictions of the actuator length, where the Jacobian entries are computed from the neural network model. The proposed procedure is validated on Festo Didactics’ elephant trunk-like two-segment continuum manipulator, Robotino-XT. Three different cases are considered for real-time trajectory tracking, where the OptiTrack vision system is used for validation by tracking the manipulator tip pose. For the trajectory points outside the manipulator workspace, simultaneous trunk and base movements are used. In experimental validation, the proposed scheme is shown to give much reduced manipulator tip trajectory error as compared to the existing methods.

Generalized ROW-type methods for solving semi-explicit DAEs of index-1

A new type of Rosenbrock–Wanner (ROW) methods for solving semi-explicit DAEs of index-1 is introduced. The scheme considers arbitrary approximations to Jacobian entries resulting for the differential part and thus corresponds to a first attempt of applying W methods to DAEs. Besides, it is a generalized class covering many ROW-type methods known from literature. Order conditions are derived by a consistent approach that combines theories of ROW methods with exact Jacobian for DAEs (Roche, 1988) and W methods with arbitrary Jacobian for ODEs (Steihaug and Wolfbrandt, 1979). In this context, rooted trees based on Butcher’s theory that include a new type of vertices are used to describe non-exact differentials of the numerical solution. Resulting conditions up to order four are given explicitly, including new conditions for realizing schemes of higher order. Numerical tests emphasize the relevance of satisfying these conditions when solving DAEs together with approximations to Jacobian entries of the differential part.

Efficient 2D inversion of long ERT sections

Abstract In this work a new algorithm for the efficient and fast two dimensional (2D) inversion of long electrical resistivity tomography (ERT) sections is introduced. The algorithm is based on 2.5D finite element method (FEM) scheme to solve Poisson's equation that describes the current flow into the earth's subsurface. The adjoint equation technique was incorporated into the FEM framework to estimate the sensitivity values. The reconstructed 2D resistivity models are recovered through an iterative, non-linear smoothness constrained least-squares approach. The algorithm further incorporates an experimental procedure to avoid the calculation and storage of the entire Jacobian matrix. The basic concept of this new algorithm relies on the fact that for every measurement there is a number of model parameters which are located in parts of the 2-D model at distant locations from potential and current electrodes. The corresponding absolute Jacobian matrix values in such cases are very small (almost zero) and can be omitted by the Jacobian calculation. Around every measurement a fixed rectangular threshold region is defined a-priori based on geometrical criteria. The algorithm calculates only Jacobian matrix values for the model parameters that are included in this threshold area omitting the calculation of the Jacobian entries related to model parameters outside this region. This approach speeds up the Jacobian matrix calculations while the efficient storage of the sparse Jacobian and Smoothness matrices and the inversion using an iterative routine like LSQR method increase significantly the inversion speed and reduce the memory requirements. The new algorithm is almost more than one order of magnitude (~ 30 times) faster and consumes one order of magnitude (~ 90%) less storage memory than the original one based on full Jacobian calculations for typical applications. The application of the new algorithm to synthetic and real data sets shows that the reconstructed models exhibit comparable accuracy to the standard inversion approach.

A fully implicit Newton–Krylov–Schwarz method for tokamak magnetohydrodynamics: Jacobian construction and preconditioner formulation

Single-fluid resistive magnetohydrodynamics (MHD) is a fluid description of fusion plasmas which is often used to investigate macroscopic instabilities in tokamaks. In MHD modeling of tokamaks, it is often desirable to compute MHD phenomena to resistive time scales or a combination of resistive-Alfven time scales, which can render explicit time stepping schemes computationally expensive. We present recent advancements in the development of preconditioners for fully nonlinearly implicit simulations of single- fluid resistive tokamak MHD. Our work focuses on simulations using a structured mesh mapped into a toroidal geometry with a shaped poloidal cross-section, and a finite-volume spatial discretization of the partial differential equation model. We discretize the temporal dimension using a fully implicit or the backwards differentiation formula method, and solve the resulting nonlinear algebraic system using a standard inexact Newton-Krylov approach, provided by the sundials library. The focus of this paper is on the construction and performance of various preconditioning approaches for accelerating the convergence of the iterative solver algorithms. Effective preconditioners require information about the Jacobian entries; however, analytical formulae for these Jacobian entries may be prohibitive to derive/implement without error. We therefore compute these entries using automatic differentiation with OpenAD. We then investigate a variety of preconditioning formulations inspired by standard solution approaches in modern MHD codes, in order to investigate their utility in a preconditioning context. We first describe the code modifications necessary for the use of the OpenAD tool and sundials solver library. We conclude with numerical results for each of our preconditioning approaches in the context of pellet-injection fueling of tokamak plasmas. Of these, our optimal approach results in a speedup of a factor of 3 compared with non-preconditioned implicit tests, with that performance gap rapidly widening with increasing mesh refinement.

Dexterous Workspace Optimization of a Tricept Parallel Manipulator

The growing interest in the use of parallel manipulators in machining applications requires clear determination of the workspace and dexterity. In this paper, the workspace optimization of a Tricept parallel manipulator under joint constraints is performed. This parallel manipulator has complex degrees of freedom and, therefore, leads to dimensionally inhomogeneous Jacobian matrices. Here, we divide the Jacobian entries by units of length, thereby producing a new Jacobian that is dimensionally homogeneous. By multiplying the associated entries of the twist array to the same length, we made this array homogeneous as well. The workspace of the manipulator is parameterized using several design parameters and is optimized using a genetic algorithm. For the workspace of the manipulator, local conditioning indices and minimum singular values are calculated. For the optimal design, it is shown that by introducing the local conditioning indices and minimum singular values, the quality of the parallel manipulator is improved at the cost of workspace reduction.

High-order discontinuous Galerkin methods using an hp-multigrid approach

The goal of this paper is to investigate and develop a fast and robust algorithm for the solution of high-order accurate discontinuous Galerkin discretizations of non-linear systems of conservation laws on unstructured grids. Herein we present the development of a spectral hp-multigrid method, where the coarse “grid” levels are constructed by reducing the order ( p) of approximation of the discretization using hierarchical basis functions ( p-multigrid), together with the traditional ( h-multigrid) approach of constructing coarser grids with fewer elements. On each level we employ variants of the element-Jacobi scheme, where the Jacobian entries associated with each element are treated implicitly (i.e., inverted directly) and all other entries are treated explicitly. The methodology is developed for the two-dimensional non-linear Euler equations on unstructured grids, using both non-linear (FAS) and linear (CGC) multigrid schemes. Results are presented for the channel flow over a bump and a uniform flow over a four element airfoil. Current results demonstrate convergence rates which are independent of both order of accuracy ( p) of the discretization and level of mesh resolution ( h).

Simplified method for the computation of parameters of power-law rate equations from time-series

Modeling biological processes from time-series data is a resourceful procedure which has received much attention in the literature. For models established in the context of non-linear differential equations, parameter-dependent phenomenological tentative response functions are tested by comparing would-be solutions of those models to the experimental time-series. Those values of the parameters for which a tested solution is a best fit are then retained. It is done with the help of some appropriate optimization algorithm which simplifies the searching procedure within the range of variability of the parameters that are to be estimated. The procedure works well in problems with a small number of adjustable parameters or/and with narrow searching ranges. However, it may start to be problematic for models with a large number of problem parameters inasmuch as convergence to the best fit is not necessarily ensured. In this case, a reduction in size of the parameter estimation problem must be undertaken. We presently address this issue by proposing a systematic procedure that does so in problems in which the system's response to a sufficiently small pulse perturbation of steady-state can be obtained. The response is then assumed to be a solution of the linearized equations, the Jacobian of which can be retrieved by a simple multilinear regression. The calculated n 2 Jacobian entries provide as many relationships among problem parameters, thus cutting substantially the size of the starting problem. After this preliminary treatment is applied, only ( κ− n 2) of the initial κ adjustable parameters are left for evaluation by means of a non-linear optimization procedure. The benefits of the present variant are both in economy of computation and in accuracy in determining the parameter values. The performance of the method is established under different circumstances. It is illustrated in the context of power-law rates, although this does not preclude its applicability to more general functional responses.

Comparison of numerical and analytical Jacobians

The results are presented of comparison between numerical and analytical approaches for forming the Jacobian matrices that arise from implicit discretizations of the governing equations The derivation of the exact analytical Jacobians using the MACSYMA symbolic manipulation expert system is discussed in detail. The numerical differentiation procedure used to develop the numerical Jacobians is described. The Jacobian entries obtained from the preceding approaches were tested by comparing the density residuals obtained for selected test cases. The test cases include supersonic flat plate and compression corner geometries. The results indicate that the two approaches are virtually identical, although the numerical Jacobian approach may have some implementation advantages over the exact analytical approach.