Relevant bibliographies by topics / Solvable approximation

Contents

  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers

Academic literature on the topic 'Solvable approximation'

Author: Grafiati
Published: 25 May 2024

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Journal articles on the topic "Solvable approximation"

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Hopkins, William E., and Wing Shing Wong. "Approximation of almost solvable bilinear systems." Systems & Control Letters 6, no. 2 (July 1985): 131–40. http://dx.doi.org/10.1016/0167-6911(85)90011-8.

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Valtancoli, P. "Exactly solvable f(R) inflation." International Journal of Modern Physics D 28, no. 07 (May 2019): 1950087. http://dx.doi.org/10.1142/s0218271819500871.

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Lemm, J. C. "Inhomogeneous Random Phase Approximation: A Solvable Model." Annals of Physics 244, no. 1 (November 1995): 201–38. http://dx.doi.org/10.1006/aphy.1995.1111.

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Shi, Ronggang, and Barak Weiss. "Invariant measures for solvable groups and Diophantine approximation." Israel Journal of Mathematics 219, no. 1 (April 2017): 479–505. http://dx.doi.org/10.1007/s11856-017-1472-y.

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Co’, Giampaolo, and Stefano De Leo. "Hartree–Fock and random phase approximation theories in a many-fermion solvable model." Modern Physics Letters A 30, no. 36 (November 3, 2015): 1550196. http://dx.doi.org/10.1142/s0217732315501965.

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We present an ideal system of interacting fermions where the solutions of the many-body Schrödinger equation can be obtained without making approximations. These exact solutions are used to test the validity of two many-body effective approaches, the Hartree–Fock and the random phase approximation theories. The description of the ground state done by the effective theories improves with increasing number of particles.
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SOLENOV, DMITRY, and VLADIMIR PRIVMAN. "EVALUATION OF DECOHERENCE FOR QUANTUM COMPUTING ARCHITECTURES: QUBIT SYSTEM SUBJECT TO TIME-DEPENDENT CONTROL." International Journal of Modern Physics B 20, no. 11n13 (May 20, 2006): 1476–95. http://dx.doi.org/10.1142/s0217979206034066.

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We present an approach that allows quantifying decoherence processes in an open quantum system subject to external time-dependent control. Interactions with the environment are modeled by a standard bosonic heat bath. We develop two unitarity-preserving approximation schemes to calculate the reduced density matrix. One of the approximations relies on a short-time factorization of the evolution operator, while the other utilizes expansion in terms of the system-bath coupling strength. Applications are reported for two illustrative systems: an exactly solvable adiabatic model, and a model of a rotating-wave quantum-computing gate function. The approximations are found to produce consistent results at short and intermediate times.
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Mota, V., and E. S. Hern�ndez. "A solvable version of the collisional random phase approximation." Zeitschrift f�r Physik A Atomic Nuclei 328, no. 2 (June 1987): 177–87. http://dx.doi.org/10.1007/bf01290660.

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Kudryashov, Vladimir V., and Yulian V. Vanne. "Explicit summation of the constituent WKB series and new approximate wave functions." Journal of Applied Mathematics 2, no. 6 (2002): 265–75. http://dx.doi.org/10.1155/s1110757x02112046.

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The independent solutions of the one-dimensional Schrödinger equation are approximated by means of the explicit summation of the leading constituent WKB series. The continuous matching of the particular solutions gives the uniformly valid analytical approximation to the wave functions. A detailed numerical verification of the proposed approximation is performed for some exactly solvable problems arising from different kinds of potentials.
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Sollich, Peter, and Anason Halees. "Learning Curves for Gaussian Process Regression: Approximations and Bounds." Neural Computation 14, no. 6 (June 1, 2002): 1393–428. http://dx.doi.org/10.1162/089976602753712990.

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We consider the problem of calculating learning curves (i.e., average generalization performance) of gaussian processes used for regression. On the basis of a simple expression for the generalization error, in terms of the eigenvalue decomposition of the covariance function, we derive a number of approximation schemes. We identify where these become exact and compare with existing bounds on learning curves; the new approximations, which can be used for any input space dimension, generally get substantially closer to the truth. We also study possible improvements to our approximations. Finally, we use a simple exactly solvable learning scenario to show that there are limits of principle on the quality of approximations and bounds expressible solely in terms of the eigenvalue spectrum of the covariance function.
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Shen, Jinrong, Wei Liu, Baiyu Wang, and Xiangyang Peng. "The Centrosymmetric Matrices of Constrained Inverse Eigenproblem and Optimal Approximation Problem." Mathematical Problems in Engineering 2020 (March 10, 2020): 1–8. http://dx.doi.org/10.1155/2020/4590354.

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In this paper, a kind of constrained inverse eigenproblem and optimal approximation problem for centrosymmetric matrices are considered. Necessary and sufficient conditions of the solvability for the constrained inverse eigenproblem of centrosymmetric matrices in real number field are derived. A general representation of the solution is presented for a solvable case. The explicit expression of the optimal approximation problem is provided. Finally, a numerical example is given to illustrate the effectiveness of the method.
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Dissertations / Theses on the topic "Solvable approximation"

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Manríquez, Peñafiel Ronald. "Local approximation by linear systems and Almost-Riemannian Structures on Lie groups and Continuation method in rolling problem with obstacles." Electronic Thesis or Diss., université Paris-Saclay, 2022. https://theses.hal.science/tel-03716186.

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L'objectif de cette thèse est d'étudier deux sujets en géométrie sub-Riemannienne. D'une part, l'approximation locale d'une structure presque riemannienne aux points singuliers, et d'autre part, le système cinématique d’une variété à 2 dimensions roulant (sans torsion ni glissement) sur le plan euclidien avec des régions interdites. Une structure presque riemannienne de dimension n peut être définie localement par n champs vectoriels satisfaisant la condition de rang de l'algèbre de Lie, jouant le rôle d'un cadre orthonormé. L'ensemble des points où ces champs vectoriels sont colinéaires est appelé l'ensemble singulier Z. Aux points de tangence, c'est-à-dire aux points où l'espace linéaire engendré par champs vectoriels est égale à l'espace tangent de Z, l'approximation nilpotente peut être remplacée par l'approximation solvable. Dans cette thèse, sous des conditions génériques, nous établissons l'ordre d'approximation de la distance originale par d ̃ (la distance induite par l'approximation solvable) et nous prouvons que d ̃ est plus proche que la distance induite par l'approximation nilpotente de la distance originale. En ce qui concerne les structures des systèmes d'approximation, l'algèbre de Lie générée par cette nouvelle famille de champs vectoriels est de dimension finie et solvable (dans le cas générique). De plus, l'approximation solvable est équivalente à un ARS linéaire sur un espace homogène ou un groupe de Lie. D'autre part, les systèmes non-holonomes ont attiré l'attention de nombreux auteurs de différentes disciplines pour leurs applications variées, principalement en robotique. Le problème du corps roulant (sans glissement ni rotation) d'une variété riemannien bidimensionnel sur une autre variété peut être écrit comme un système non-holonomique. De nombreuses méthodes, algorithmes et techniques ont été développés pour le résoudre. Une implémentation numérique de la méthode de continuation pour résoudre le problème dans lequel une surface convexe roule sur le plan euclidien avec des régions interdites (ou obstacles) sans glisser ou tourner est effectuée. Plusieurs exemples sont illustrés
The aim of this thesis is to study two topics in sub-Riemannian geometry. On the one hand, the local approximation of an almost-Riemannian structure at singular points, and on the other hand, the kinematic system of a 2-dimensional manifold rolling (without twisting or slipping) on the Euclidean plane with forbidden regions. A n-dimensional almost-Riemannian structure can be defined locally by n vector fields satisfying the Lie algebra rank condition, playing the role of an orthonormal frame. The set of points where these vector fields are colinear is called the singular set (Z). At tangency points, i.e., points where the linear span of the vector fields is equal to the tangent space of Z, the nilpotent approximation can be replaced by the solvable one. In this thesis, under generic conditions, we state the order of approximation of the original distance by d ̃ (the distance induced by the solvable approximation), and we prove that d ̃ is closer than the distance induced by the nilpotent approximation to the original distance. Regarding the structure of the approximating system, the Lie algebra generated by this new family of vector fields is finite-dimensional and solvable (in the generic case). Moreover, the solvable approximation is equivalent to a linear ARS on a homogeneous space or a Lie group. On the other hand, nonholonomic systems have attracted the attention of many authors from different disciplines for their varied applications, mainly in robotics. The rolling-body problem (without slipping or spinning) of a 2-dimensional Riemannian manifold on another one can be written as a nonholonomic system. Many methods, algorithms, and techniques have been developed to solve it. A numerical implementation of the Continuation Method to solve the problem in which a convex surface rolls on the Euclidean plane with forbidden regions (or obstacles) without slipping or spinning is performed. Several examples are illustrated
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Books on the topic "Solvable approximation"

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Research Institute for Advanced Computer Science (U.S.), ed. Explicitly solvable complex Chebyshev approximation problems related to sine polynomials. [Moffett Field, Calif.]: Research Institute for Advanced Computer Science, NASA Ames Research Center, 1989.

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Christensen, Jens Gerlach. Trends in harmonic analysis and its applications: AMS special session on harmonic analysis and its applications : March 29-30, 2014, University of Maryland, Baltimore County, Baltimore, MD. Providence, Rhode Island: American Mathematical Society, 2015.

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Combinatorics and Random Matrix Theory. American Mathematical Society, 2016.

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Book chapters on the topic "Solvable approximation"

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Lobbe, Alexander. "Deep Learning for the Benes Filter." In Mathematics of Planet Earth, 195–210. Cham: Springer International Publishing, 2022. http://dx.doi.org/10.1007/978-3-031-18988-3_12.

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AbstractThe filtering problem is concerned with the optimal estimation of a hidden state given partial and noisy observations. Filtering is extensively studied in the theoretical and applied mathematical literature. One of the central challenges in filtering today is the numerical approximation of the optimal filter. Here, accurate and fast methods are actively sought after, especially for such high-dimensional settings as numerical weather prediction, for example. In this paper we present a brief study of a new numerical method based on the mesh-free neural network representation of the density of the solution of the filtering problem achieved by deep learning. Based on the classical SPDE splitting method, our algorithm includes a recursive normalisation procedure to recover the normalised conditional distribution of the signal process. The present work uses the Benes model as a benchmark. The Benes filter is a well-known continuous-time stochastic filtering model in one dimension that has the advantage of being explicitly solvable. Within the analytically tractable setting of the Benes filter, we discuss the role of nonlinearity in the filtering model equations for the choice of the domain of the neural network. Further, we present the first study of the neural network method with an adaptive domain for the Benes model.
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Sikorski, Krzysztof A. "Fixed Points- Noncontractive Functions." In Optimal Solution of Nonlinear Equations. Oxford University Press, 2001. http://dx.doi.org/10.1093/oso/9780195106909.003.0007.

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In this chapter we consider the approximation of fixed points of noncontractive functions with respect to the absolute error criterion. In this case the functions may have multiple and/or whole manifolds of fixed points. We analyze methods based on sequential function evaluations as information. The simple iteration usually does not converge in this case, and the problem becomes much more difficult to solve. We prove that even in the two-dimensional case the problem has infinite worst case complexity. This means that no methods exist that solve the problem with arbitrarily small error tolerance for some “bad” functions. In the univariate case the problem is solvable, and a bisection envelope method is optimal. These results are in contrast with the solution under the residual error criterion. The problem then becomes solvable, although with exponential complexity, as outlined in the annotations. Therefore, simplicial and/or homotopy continuation and all methods based on function evaluations exhibit exponential worst case cost for solving the problem in the residual sense. These results indicate the need of average case analysis, since for many test functions the existing algorithms computed ε-approximations with polynomial in 1/ε cost.
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Mussardo, Giuseppe. "Approximate Solutions." In Statistical Field Theory, 106–58. Oxford University Press, 2020. http://dx.doi.org/10.1093/oso/9780198788102.003.0003.

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Chapter 3 discusses the approximation schemes used to approach lattice statistical models that are not exactly solvable. In addition to the mean field approximation, it also considers the Bethe–Peierls approach to the Ising model. Moreover, there is a thorough discussion of the Gaussian model and its spherical version, both of which are two important systems with several points of interest. A chapter appendix provides a detailed analysis of the random walk on different lattices: apart from the importance of the subject on its own, it explains how the random walk is responsible for the critical properties of the spherical model.
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Rodriguez, Ricardo, Ivo Bukovsky, and Noriyasu Homma. "Potentials of Quadratic Neural Unit for Applications." In Advances in Abstract Intelligence and Soft Computing, 343–54. IGI Global, 2013. http://dx.doi.org/10.4018/978-1-4666-2651-5.ch023.

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The paper discusses the quadratic neural unit (QNU) and highlights its attractiveness for industrial applications such as for plant modeling, control, and time series prediction. Linear systems are still often preferred in industrial control applications for their solvable and single solution nature and for the clarity to the most application engineers. Artificial neural networks are powerful cognitive nonlinear tools, but their nonlinear strength is naturally repaid with the local minima problem, overfitting, and high demands for application-correct neural architecture and optimization technique that often require skilled users. The QNU is the important midpoint between linear systems and highly nonlinear neural networks because the QNU is relatively very strong in nonlinear approximation; however, its optimization and performance have fast and convex-like nature, and its mathematical structure and the derivation of the learning rules is very comprehensible and efficient for implementation. These advantages of QNU are demonstrated by using real and theoretical examples.
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Tuck, Adrian F. "Non-Equilibrium Statistical Mechanics." In Atmospheric Turbulence. Oxford University Press, 2008. http://dx.doi.org/10.1093/oso/9780199236534.003.0010.

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The Earth’s atmosphere is far from equilibrium; it is constantly in motion from the combined effects of gravity and planetary rotation, is constantly absorbing and emitting radiation, and hosts ongoing chemical reactions which are ultimately fuelled by solar photons. It has fluxes of material and energy across its boundaries with the planetary surface, both terrestrial and marine, and also emits a continual outward flux of infrared photons to space. The gaseous atmosphere is manifestly a kinetic system, meaning that its evolution must be described by time dependent differential equations. The equations doing this under the continuum fluid approximation are the Navier–Stokes equations, which are not analytically solvable and which support many non-linear instabilities. We have also seen that the generation of turbulence is a fundamentally difficult yet central feature of air motion, originating on the molecular scale. Non-equilibrium statistical mechanics may offer insight into which steady states a system far from equilibrium as a result of fluxes and anisotropies may migrate, without the need for detailed solution of the explicit path between the states. However, it does not seem possible to demonstrate mathematically that such steady states exist for the atmosphere. A physical view of the planet’s past and probable future suggests that the past and future evolution of the sun and its outgoing fluxes of energy may mean that the air-water-earth system may never have been or will ever be in a rigorously defined steady state. Also, to the human population, the detailed, time-dependent evolution is what matters in many respects. Nevertheless, non-equilibrium statistical mechanics is a discipline which should be applicable in principle to yield information about approximate steady states. These steady states may as a practical matter be definable from the observational record, for example the ice ages and the intervening periods evident in the geological record, or between states with two differing global average abundances of a radiatively active gas such as carbon dioxide. There has been great progress recently in non-equilibrium statistical mechanics, stemming from recent work on the concept of the maximization of entropy production.
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Kalyuzhnyi, Yu V., and P. T. Cummings. "6 Equations of state from analytically solvable integral equation approximations." In Equations of State for Fluids and Fluid Mixtures, 169–254. Elsevier, 2000. http://dx.doi.org/10.1016/s1874-5644(00)80017-x.

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Juan Peña, José, Jesús Morales, and Jesús García-Ravelo. "Perspective Chapter: Relativistic Treatment of Spinless Particles Subject to a Class of Multiparameter Exponential-Type Potentials." In Schrödinger Equation - Fundamentals Aspects and Potential Applications [Working Title]. IntechOpen, 2023. http://dx.doi.org/10.5772/intechopen.112184.

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By using the exactly-solvable Schrödinger equation for a class of multi-parameter exponential-type potential, the analytical bound state solutions of the Klein-Gordon equation are presented. The proposal is based on the fact that the Klein-Gordon equation can be reduced to a Schrödinger-type equation when the Lorentz-scalar and vector potential are equal. The proposal has the advantage of avoiding the use of a specialized method to solve the Klein-Gordon equation for a specific exponential potential due that it can be derived by means of an appropriate choice of the involved parameters. For this, to show the usefulness of the method, the relativistic treatment of spinless particles subject to some already published exponential potentials are directly deduced and given as examples. So, beyond the particular cases considered in this work, this approach can be used to solve the Klein-Gordon equation for new exponential-type potentials having hypergeometric eigenfunctions. Also, it can be easily adapted to other approximations of the centrifugal term different to the Green-Aldrich used in this work.
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Conference papers on the topic "Solvable approximation"

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Todorov, Emanuel. "Eigenfunction approximation methods for linearly-solvable optimal control problems." In 2009 IEEE Symposium on Adaptive Dynamic Programming and Reinforcement Learning (ADPRL). IEEE, 2009. http://dx.doi.org/10.1109/adprl.2009.4927540.

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Pedram, Ali Reza, and Takashi Tanaka. "Linearly-Solvable Mean-Field Approximation for Multi-Team Road Traffic Games." In 2019 IEEE 58th Conference on Decision and Control (CDC). IEEE, 2019. http://dx.doi.org/10.1109/cdc40024.2019.9029579.

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Elperin, Tov, Andrew Fominykh, and Boris Krasovitov. "Modeling of Simultaneous Gas Absorption and Evaporation of Large Droplet." In ASME 2005 International Mechanical Engineering Congress and Exposition. ASMEDC, 2005. http://dx.doi.org/10.1115/imece2005-79924.

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In this study we investigated numerically simultaneous heat and mass transfer during evaporation/condensation on the surface of a stagnant droplet in the presence of inert admixtures containing non-condensable solvable gas. The performed analysis is pertinent to slow droplet evaporation/condensation when Mach number is small (M≪1). The system of transient conjugate nonlinear energy and mass conservation equations was solved using anelastic approximation. Transport coefficients of the gaseous phase were calculated as functions of temperature and concentrations of gaseous species. Thermophysical properties of the liquid phase are assumed to be constant. Using the material balance at the droplet surface we obtained equations for Stefan velocity and the rate of change of the droplet radius taking into account the effect of solvable gas absorption at the gas-liquid interface. We derived also boundary conditions at gas-liquid interface taking into account the effect of gas absorption. The governing equations were solved using a method of lines. Numerical calculations showed essential change of the rates of heat and mass transfer in water droplet-air-water vapor system under the influence of solvable species in a gaseous phase. Consequently, the use of additives of solvable noncondensable gases to enhance the rate of heat and mass transfer in dispersed systems allows to increase the efficiency and reduce the size of gas-liquid contactors.
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Elperin, Tov, Andrew Fominykh, and Boris Krasovitov. "Simultaneous Heat and Mass Transfer During Evaporation/Condensation on the Surface of a Stagnant Droplet in the Presence of Inert Admixtures Containing Non-Condensable Solvable Gas." In ASME 2005 Summer Heat Transfer Conference collocated with the ASME 2005 Pacific Rim Technical Conference and Exhibition on Integration and Packaging of MEMS, NEMS, and Electronic Systems. ASMEDC, 2005. http://dx.doi.org/10.1115/ht2005-72493.

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In this study we investigated numerically simultaneous heat and mass transfer during evaporation/condensation on the surface of a stagnant droplet in the presence of inert admixtures containing non-condensable solvable gas. The performed analysis is pertinent to slow droplet evaporation/condensation when Mach number is small (M≪1). The system of transient conjugate nonlinear energy and mass conservation equations was solved using anelastic approximation. Transport coefficients of the gaseous phase were calculated as functions of temperature and concentrations of gaseous species. Thermophysical properties of the liquid phase are assumed to be constant. Using the material balance at the droplet surface we obtained equations for Stefan velocity and the rate of change of the droplet radius taking into account the effect of solvable gas absorption at the gas-liquid interface. We derived also boundary conditions at gas-liquid interface taking into account the effect of gas absorption. The governing equations were solved using a method of lines. Numerical calculations showed essential change of the rates of heat and mass transfer in water droplet-air-water vapor system under the influence of solvable species in a gaseous phase. Consequently, the use of additives of solvable noncondensable gases to enhance the rate of heat and mass transfer in dispersed systems allows to increase the efficiency and reduce the size of gas-liquid contactors.
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Herkenrath, Maike, Till Fluschnik, Francesco Grothe, and Leon Kellerhals. "Placing Green Bridges Optimally, with Habitats Inducing Cycles." In Thirty-First International Joint Conference on Artificial Intelligence {IJCAI-22}. California: International Joint Conferences on Artificial Intelligence Organization, 2022. http://dx.doi.org/10.24963/ijcai.2022/531.

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Choosing the placement of wildlife crossings (i.e., green bridges) to reconnect animal species' fragmented habitats is among the 17 goals towards sustainable development by the UN. We consider the following established model: Given a graph whose vertices represent the fragmented habitat areas and whose weighted edges represent possible green bridge locations, as well as the habitable vertex set for each species, find the cheapest set of edges such that each species' habitat is connected. We study this problem from a theoretical (algorithms and complexity) and an experimental perspective, while focusing on the case where habitats induce cycles. We prove that the NP-hardness persists in this case even if the graph structure is restricted. If the habitats additionally induce faces in plane graphs however, the problem becomes efficiently solvable. In our empirical evaluation we compare this algorithm as well as ILP formulations for more general variants and an approximation algorithm with another. Our evaluation underlines that each specialization is beneficial in terms of running time, whereas the approximation provides highly competitive solutions in practice.
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Singh, Rituraj, and Krishna M. Singh. "Iterative Solvers for Meshless Petrov Galerkin (MLPG) Method Applied to Large Scale Engineering Problems Challenges." In ASME 2015 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2015. http://dx.doi.org/10.1115/imece2015-53343.

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In recent years, significant research effort has been invested in development of mesh-free methods for different types of continuum problems. Prominent amongst these methods are element free Galerkin (EFG) method, RKPM, and mesh-less Petrov Galerkin (MLPG) method. Most of these methods employ a set of nodes for discretization of the problem domain, and use a moving least squares (MLS) approximation to generate shape functions. Of these methods, MLPG method is seen as a pure meshless method since it does not require any background mesh. Accuracy and flexibility of MLPG method is well established for a variety of continuum problems. However, most of the applications have been limited to small scale problems solvable on serial machines. Very few attempts have been made to apply it to large scale problems which typically involve many millions (or even billions) of nodes and would require use of parallel algorithms based on domain decomposition. Such parallel techniques are well established in context of mesh-based methods. Extension of these algorithms in conjunction with MLPG method requires considerable further research. Objective of this paper is to spell out these challenges which need urgent attention to enable the application of meshless methods to large scale problems. We specifically address the issue of the solution of large scale linear problems which would necessarily require use of iterative solvers. We focus on application of BiCGSTAB method and an appropriate set of preconditioners for the solution of the MLPG system.
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Zhong, Mingyuan, and Emanuel Todorov. "Moving least-squares approximations for linearly-solvable MDP." In 2011 Ieee Symposium On Adaptive Dynamic Programming And Reinforcement Learning. IEEE, 2011. http://dx.doi.org/10.1109/adprl.2011.5967383.

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Manko, D. J., and W. L. Whittaker. "Inverse Dynamic Models Used for Force Control of Compliant, Closed-Chain Mechanisms." In ASME 1989 Design Technical Conferences. American Society of Mechanical Engineers, 1989. http://dx.doi.org/10.1115/detc1989-0106.

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Abstract A general inverse dynamic model is presented that is applicable to mechanisms incorporating member, joint and base compliance. Previous approaches for defining inverse dynamic models of compliant mechanisms have been approximations or limited to simple mechanism geometries and open-chain mechanisms. Hence, the motivation for a more general approach. Inverse dynamic equations for compliant mechanisms modeled with and without constraint equations are shown to be solvable sets of differential/algebraic equations (DAE’s); relevant characteristics and solutions of DAE systems are discussed. An important application for inverse dynamic models of compliant mechanisms is model-based force control of closed-chain mechanisms. The formulation and solution procedures discussed in this paper have been successfully applied to model legged locomotion on natural terrain.
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Wilde, Douglass J. "Monotonicity Analysis of Taguchi’s Robust Circuit Design Problem." In ASME 1990 Design Technical Conferences. American Society of Mechanical Engineers, 1990. http://dx.doi.org/10.1115/detc1990-0052.

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Abstract A recent article showed how to formulate Taguchi’s robust circuit design problem rigorously as an optimization problem. A necessary condition for optimality was found to be that the control range be centered about the target value. This generates a constraint on the two design variables which cannot be solved for either variable. The present article shows that by approximating this unsolvable constraint with a simpler constraint that is solvable, one variable can be eliminated and the problem reduced to an unconstrained one in a single variable. Since this reduced objective turns out to be monotonic in the remaining design variable, its optimum value must be at the limit of its range. The corresponding optimum value of the other variable is then determined exactly from the true, not approximate, constraint. Since no model construction, experimentation, statistical analysis or numerical iteration is needed, this procedure is recommended whenever the input-output relation is known to be a monotonic algebraic function.
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Xiros, Nikolaos I. "Investigation of a Nonlinear Control Model for Marine Propulsion Power-Plants." In ASME 2011 International Mechanical Engineering Congress and Exposition. ASMEDC, 2011. http://dx.doi.org/10.1115/imece2011-63797.

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A cycle-mean-value, quasi-steady, thermodynamic model of slow-speed, two-stroke marine Diesel engines, used for performance prediction and engine-propeller/turbocharger matching, is converted into a power-plant analytic model. The dynamic part of the cycle-mean model consists of the two first-order differential equations for the cycle-mean crankshaft and turbocharger shaft rotational accelerations. This form implies a state-space formulation of the power-plant modeling approach. However, engine, turbine and compressor torques have to be calculated through the solution of the algebraic part of the model, which consists of a nonlinear, perplexed algebraic system of equations not analytically solvable. This inhibits the formulation of the power-plant state-space description. By approximating the torque maps, generated by the thermodynamic model, with neural nets, explicit functional relationships are obtained. Identification of the power-plant operating regimes through linearization and decomposition is performed. In effect, a supervisory power-plant controller structure, applicable to real-time control and diagnostics, is proposed, incorporating the nonlinear state-space description of the plant.
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