The eigensystem of the Pulliam-Chaussee diagonalized form of the approximate-factorization algorithm for the three-dimensional Euler and Navier-Stokes equations is revisited to remove an apparent dimensional inconsist...
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The eigensystem of the Pulliam-Chaussee diagonalized form of the approximate-factorization algorithm for the three-dimensional Euler and Navier-Stokes equations is revisited to remove an apparent dimensional inconsistency. The original set of eigenvectors in curvilinear coordinates were derived systematically and has been widely used and referenced. Although mathematically correct, the original eigenvectors for the advected modes appear dimensionally inconsistent and yield a set of matrices with large condition numbers for some flows. A new set of eigenvectors is presented that remove the inconsistency and improves the robustness of the diagonalized scheme. (C) 2020 Elsevier Inc. All rights reserved.
3-D registration has always been performed invoking singular value decomposition (SVD) or eigenvalue decomposition (EIG) in real engineering practices. However, these numerical algorithms suffer from uncertainty of co...
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3-D registration has always been performed invoking singular value decomposition (SVD) or eigenvalue decomposition (EIG) in real engineering practices. However, these numerical algorithms suffer from uncertainty of convergence in many cases. A novel fast symbolic solution is proposed in this article by following our recent publication in this journal. The equivalence analysis shows that our previous solver can be converted to deal with the 3-D registration problem. Rather, the computation procedure is studied for further simplification of computing without complex-number support. Experimental results show that the proposed solver does not loose accuracy and robustness but improves the execution speed to a large extent by almost 50%-80%, on both a personal computer (PC) and an embedded processor. Note to Practitioners-3-D registration usually has a large computational burden in engineering tasks. The proposed symbolic solution can directly solve the eigenvalue and its associated eigenvector. A lot of computation resources can then be saved for better overall system performance. The deterministic behavior of the proposed solver also ensures long-endurance stability and can help an engineer better design thread timing logic.
This paper presents a numerical algorithm for computing six-degree-of-freedom free-final-time powered descent guidance trajectories. The trajectory generation problem is formulated using a unit dual quaternion represe...
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This paper presents a numerical algorithm for computing six-degree-of-freedom free-final-time powered descent guidance trajectories. The trajectory generation problem is formulated using a unit dual quaternion representation of the rigid body dynamics and several standard path constraints. Our formulation also includes a special line-of-sight constraint that is enforced only within a specified band of slant ranges relative to the landing site: a novel feature that is especially relevant to terrain and hazard relative navigation. We use the newly introduced state-triggered constraints to formulate these range constraints in a manner that is amenable to real-time implementations. The resulting nonconvex optimal control problem is solved iteratively as a sequence of convex second-order cone programs that locally approximate the nonconvex problem. Each second-order cone program is solved using a customizable interior point method solver. Also introduced are a scaling method and a new heuristic technique that guide the convergence process toward dynamic feasibility. To demonstrate the capabilities of our algorithm, two numerical case studies are presented. The first studies the effect of including a slant-range-triggered line-of-sight constraint on the resulting trajectories. The second study performs a Monte Carlo analysis to assess the algorithm's robustness to initial conditions and real-time performance.
Recent low-thrust space missions have highlighted the importance of designing trajectories that are robust against uncertainties. In its complete form, this process is formulated as a nonlinear constrained stochastic ...
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Recent low-thrust space missions have highlighted the importance of designing trajectories that are robust against uncertainties. In its complete form, this process is formulated as a nonlinear constrained stochastic optimal control problem. This problem is among the most complex in control theory, and no practically applicable method to low-thrust trajectory optimization problems has been proposed to date. This paper presents a new algorithm to solve stochastic optimal control problems with nonlinear systems and constraints. The proposed algorithm uses the unscented transform to convert a stochastic optimal control problem into a deterministic problem, which is then solved by trajectory optimization methods such as differential dynamic programming. Two numerical examples, one of which applies the proposed method to low-thrust trajectory design, illustrate that it automatically introduces margins that improve robustness. Finally, Monte Carlo simulations are used to evaluate the robustness and optimality of the solution.
Ewen's sampling formula is a foundational theoretical result that connects probability and number theory with molecular genetics and molecular evolution;it was the analytical result required for testing the neutra...
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Ewen's sampling formula is a foundational theoretical result that connects probability and number theory with molecular genetics and molecular evolution;it was the analytical result required for testing the neutral theory of evolution, and has since been directly or indirectly utilized in a number of population genetics statistics. Ewen's sampling formula, in turn, is deeply connected to Stirling numbers of the first kind. Here, we explore the cumulative distribution function of these Stirling numbers, which enables a single direct estimate of the sum, using representations in terms of the incomplete beta function. This estimator enables an improved method for calculating an asymptotic estimate for one useful statistic, Fu's Fs. By reducing the calculation from a sum of terms involving Stirling numbers to a single estimate, we simultaneously improve accuracy and dramatically increase speed.
An analytical solution of Euler's equation is exhibited using the Volterra series theory in the frequency domain. Based on the Volterra series, the nonlinear output frequency response functions of Euler's equa...
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An analytical solution of Euler's equation is exhibited using the Volterra series theory in the frequency domain. Based on the Volterra series, the nonlinear output frequency response functions of Euler's equation are formulated by a numerical algorithm to reveal an energy-transfer phenomenon. The output responses of Euler's equation have some higher frequency parts than those of the inputs. It provides motivation to design a finite-frequency controller for Euler's equation to accommodate the high-frequency parts of the outputs. A hybrid passive/finite gain control scheme fused with the generalized Kalman-Yakubovich-Popov lemma is used to generate a controller that is effective for stabilizing the angular velocities of Euler's equation. Additionally, quaternions are considered in the proposed hybrid finite-frequency controller to stabilize the attitude of spacecraft. Simulation results are demonstrated to validate the effectiveness of the proposed control schemes.
The standard implementation of the conjugate gradient algorithm suffers from communication bottlenecks on parallel architectures, due primarily to the two global reductions required every iteration. In this paper, we ...
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The standard implementation of the conjugate gradient algorithm suffers from communication bottlenecks on parallel architectures, due primarily to the two global reductions required every iteration. In this paper, we study conjugate gradient variants which decrease the runtime per iteration by overlapping global synchronizations, and in the case of pipelined variants, matrix-vector products. Through the use of a predict-and-recompute scheme, whereby recursively updated quantities are first used as a predictor for their true values and then recomputed exactly at a later point in the iteration, these variants are observed to have convergence behavior nearly as good as the standard conjugate gradient implementation on a variety of test problems. We provide a rounding error analysis which provides insight into this observation. It is also verified experimentally that the variants studied do indeed reduce the runtime per iteration in practice and that they scale similarly to previously studied communication-hiding variants. Finally, because these variants achieve good convergence without the use of any additional input parameters, they have the potential to be used in place of the standard conjugate gradient implementation in a range of applications.
Recently, we proposed novel temperature and pressure evaluations in molecular dynamics (MD) simulations to preserve the accuracy up to the third order of a time step, delta t [J. Jung, C. Kobayashi, and Y. Sugita, J. ...
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Recently, we proposed novel temperature and pressure evaluations in molecular dynamics (MD) simulations to preserve the accuracy up to the third order of a time step, delta t [J. Jung, C. Kobayashi, and Y. Sugita, J. Chem. Theory Comput. 15, 84-94 (2019);J. Jung, C. Kobayashi, and Y. Sugita, J. Chem. Phys. 148, 164109 (2018)]. These approaches allow us to extend delta t of MD simulations under an isothermal-isobaric condition up to 5 fs with a velocity Verlet integrator. Here, we further improve the isothermal-isobaric MD integration by introducing the group-based evaluations of system temperature and pressure to our previous approach. The group-based scheme increases the accuracy even using inaccurate temperature and pressure evaluations by neglecting the high-frequency vibrational motions of hydrogen atoms. It also improves the overall performance by avoiding iterations in thermostat and barostat updates and by allowing a multiple time step integration such as r-RESPA (reversible reference system propagation algorithm) with our proposed high-precision evaluations of temperature and pressure. Now, the improved integration scheme conserves physical properties of lipid bilayer systems up to delta t = 5 fs with velocity Verlet as well as delta t = 3.5 fs for fast motions in r-RESPA, respectively.
We describe and analyze an algorithm for computing the homology (Betti numbers and torsion coefficients) of closed semialgebraic sets given by Boolean formulas without negations over lax polynomial inequalities. The a...
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We describe and analyze an algorithm for computing the homology (Betti numbers and torsion coefficients) of closed semialgebraic sets given by Boolean formulas without negations over lax polynomial inequalities. The algorithm works in weak exponential time. This means that outside a subset of data having exponentially small measure, the cost of the algorithm is single exponential in the size of the data. All previous algorithms solving this problem have doubly exponential complexity. Our algorithm thus represents an exponential acceleration over state-of-the-art algorithms for all input data outside a set that vanishes exponentially fast.
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