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COMPUTING SEMICLASSICAL QUANTUM DYNAMICS WITH HAGEDORN
"... Abstract. We consider the approximation of multiparticle quantum dynamics in the semiclassical regime by Hagedorn wavepackets, which are products of complex Gaussians with polynomials that form an orthonormal L 2 basis and preserve their type under propagation in Schrödinger equations with quadrati ..."
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Cited by 134 (4 self)
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Abstract. We consider the approximation of multiparticle quantum dynamics in the semiclassical regime by Hagedorn wavepackets, which are products of complex Gaussians with polynomials that form an orthonormal L 2 basis and preserve their type under propagation in Schrödinger equations with quadratic potentials. We build a fully explicit, timereversible timestepping algorithm to approximate the solution of the Hagedorn wavepacket dynamics. The algorithm is based on a splitting between the kinetic and potential part of the Hamiltonian operator, as well as on a splitting of the potential into its local quadratic approximation and the remainder. The algorithm is robust in the semiclassical limit. It reduces to the Strang splitting of the Schrödinger equation in the limit of the full basis set, and it advances positions and momenta by the Störmer–Verlet method for the classical equations of motion. The algorithm allows for the treatment of multiparticle problems by thinning out the basis according to a hyperbolic cross approximation, and of highdimensional problems by Hartreetype approximations in a moving coordinate frame.
IMPLICITEXPLICIT VARIATIONAL INTEGRATION OF HIGHLY OSCILLATORY PROBLEMS
, 808
"... ABSTRACT. In this paper, we derive a variational integrator for certain highly oscillatory problems in mechanics. To do this, we take a new approach to the splitting of fast and slow potential forces: rather than splitting these forces at the level of the differential equations or the Hamiltonian, w ..."
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Cited by 19 (1 self)
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ABSTRACT. In this paper, we derive a variational integrator for certain highly oscillatory problems in mechanics. To do this, we take a new approach to the splitting of fast and slow potential forces: rather than splitting these forces at the level of the differential equations or the Hamiltonian, we split the two potentials with respect to the Lagrangian action integral. By using a different quadrature rule to approximate the contribution of each potential to the action, we arrive at a geometric integrator that is implicit in the fast force and explicit in the slow force. This can allow for significantly longer time steps to be taken (compared to standard explicit methods, such as Störmer/Verlet) at the cost of only a linear solve rather than a full nonlinear solve. We also analyze the stability of this method, in particular proving that it eliminates the linear resonance instabilities that can arise with explicit multipletimestepping methods. Next, we perform some numerical experiments, studying the behavior of this integrator for two test problems: a system of coupled linear oscillators, for which we compare against the resonance behavior of the rRESPA method; and slow energy exchange in the Fermi–Pasta–Ulam problem, which couples fast linear oscillators with slow nonlinear oscillators. Finally, we prove that this integrator accurately preserves the slow energy exchange between the fast oscillatory components, which explains the numerical behavior observed for the Fermi–Pasta–Ulam problem. 1.
Computational geometric mechanics and control of rigid bodies
, 2008
"... It has been a blessing to me that I could do what I really like to do under encouragements and supports from my family and friends. As everybody says, a graduation implies a new challenge, which makes me excited and thrilled. But, before setting out on my new journey of academia, I would like to exp ..."
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Cited by 17 (11 self)
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It has been a blessing to me that I could do what I really like to do under encouragements and supports from my family and friends. As everybody says, a graduation implies a new challenge, which makes me excited and thrilled. But, before setting out on my new journey of academia, I would like to express my gratitude to who have influenced on me. From the very beginning, my parents have inspired me with courage: when I made any decision, they always supported me and deduced several reasons that made the decision more appropriate. Now, I understand a little bit about raising a child with absolute trust, instead of having expectations or concerning about him. It is easy to tell an instructive story to a child, but it requires a tremendous effort to teach a child by showing a real model of what he ought to be. My parents have been doing it for me ever since I was born, and I am really proud of my parents. I have had another exclusive privilege of being advised by Professor N. Harris McClamroch and Professor Melvin Leok. It has been an honor to have an academic guidance, instruction, encouragement, and insight from both of them, throughout my doctoral study. I am aware that, in some cases, the relationship between an advisor and a graduate student is similar to that of an employer and an employee. My relationship with them can be described as the exact opposite: they have treated me
OPTIMAL ATTITUDE CONTROL OF A RIGID BODY USING GEOMETRICALLY EXACT COMPUTATIONS ON SO(3)
"... Abstract. An efficient and accurate computational approach is proposed for a nonconvex optimal attitude control for a rigid body. The problem is formulated directly as a discrete time optimization problem using a Lie group variational integrator. Discrete time necessary conditions for optimality are ..."
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Cited by 17 (10 self)
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Abstract. An efficient and accurate computational approach is proposed for a nonconvex optimal attitude control for a rigid body. The problem is formulated directly as a discrete time optimization problem using a Lie group variational integrator. Discrete time necessary conditions for optimality are derived, and an efficient computational approach is proposed to solve the resulting twopoint boundaryvalue problem. This formulation wherein the optimal control problem is solved based on discretization of the attitude dynamics and derivation of discrete time necessary conditions, rather than development and discretization of continuous time necessary conditions, is shown to have significant advantages. In particular, the use of geometrically exact computations on SO(3) guarantees that this optimal control approach has excellent convergence properties even for highly nonlinear large angle attitude maneuvers. 1.
Introduction to molecular dynamics simulation
 COMPUTATIONAL SOFT MATTER: FROM SYNTHETIC POLYMERS TO PROTEINS, LECTURE NOTES, NORBERT ATTIG , KURT BINDER , HELMUT GRUBMÜLLER , KURT KREMER (EDS .)
, 2004
"... ..."
Divided edge bundling for directional network data
 IEEE TVCG
"... Highlighted edges fade from blue (source) to red (target) to indicate direction. Divided edge bundling separates antiparallel edges into emergent “traffic lanes”, enabling inspection of network asymmetries, such as the connections between Berlin and London. (235 nodes, 2101 edges; 18.1 seconds to bu ..."
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Cited by 15 (0 self)
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Highlighted edges fade from blue (source) to red (target) to indicate direction. Divided edge bundling separates antiparallel edges into emergent “traffic lanes”, enabling inspection of network asymmetries, such as the connections between Berlin and London. (235 nodes, 2101 edges; 18.1 seconds to bundle) Abstract—The nodelink diagram is an intuitive and venerable way to depict a graph. To reduce clutter and improve the readability of nodelink views, Holten & van Wijk’s forcedirected edge bundling employs a physical simulation to spatially group graph edges. While both useful and aesthetic, this technique has shortcomings: it bundles spatially proximal edges regardless of direction, weight, or graph connectivity. As a result, highlevel directional edge patterns are obscured. We present divided edge bundling to tackle these shortcomings. By modifying the forces in the physical simulation, directional lanes appear as an emergent property of edge direction. By considering graph topology, we only bundle edges related by graph structure. Finally, we aggregate edge weights in bundles to enable more accurate visualization of total bundle weights. We compare visualizations created using our technique to standard forcedirected edge bundling, matrix diagrams, and clustered graphs; we find that divided edge bundling leads to visualizations that are easier to interpret and reveal both familiar and previously obscured patterns. Index Terms—Graph visualization, aggregation, nodelink diagrams, edge bundling, physical simulation. 1
GEOMETRIC STRUCTUREPRESERVING OPTIMAL CONTROL OF A RIGID BODY
"... Abstract. In this paper, we study a discrete variational optimal control problem for a rigid body. The cost to be minimized is the external torque applied to move the rigid body from an initial condition to a prespecified terminal condition. Instead of discretizing the equations of motion, we use t ..."
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Cited by 14 (7 self)
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Abstract. In this paper, we study a discrete variational optimal control problem for a rigid body. The cost to be minimized is the external torque applied to move the rigid body from an initial condition to a prespecified terminal condition. Instead of discretizing the equations of motion, we use the discrete equations obtained from the discrete Lagrange–d’Alembert principle, a process that better approximates the equations of motion. Within the discretetime setting, these two approaches are not equivalent in general. The kinematics are discretized using a natural Liealgebraic formulation that guaranteesthattheflowremainsontheLiegroupSO(3) and its algebra so(3). We use the Lagrange method for constrained problems in the calculus of variations to derive the discretetime necessary conditions. We give a numerical example for a threedimensional rigid body maneuver. 1.
Numerical Integrators for Highly Oscillatory Hamiltonian Systems: A Review
"... Summary. Numerical methods for oscillatory, multiscale Hamiltonian systems are reviewed. The construction principles are described, and the algorithmic and analytical distinction between problems with nearly constant high frequencies and with time or statedependent frequencies is emphasized. Trig ..."
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Cited by 12 (1 self)
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Summary. Numerical methods for oscillatory, multiscale Hamiltonian systems are reviewed. The construction principles are described, and the algorithmic and analytical distinction between problems with nearly constant high frequencies and with time or statedependent frequencies is emphasized. Trigonometric integrators for the first case and adiabatic integrators for the second case are discussed in more detail. 1
Numerical Methods for Quantum Monte Carlo Simulations of the Hubbard Model
, 2009
"... One of the core problems in materials science is how the interactions between electrons in a solid give rise to properties like ..."
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Cited by 8 (3 self)
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One of the core problems in materials science is how the interactions between electrons in a solid give rise to properties like
Explicit Newmark/Verlet algorithm for time integration of the rotational dynamics of rigid bodies
 International Journal for Numerical Methods in Engineering
, 2004
"... We consider the problem of integration of the initial value problem of the rotational rigid body dynamics. We are motivated by a very practical problem: time integration ..."
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Cited by 8 (2 self)
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We consider the problem of integration of the initial value problem of the rotational rigid body dynamics. We are motivated by a very practical problem: time integration