Exploring potential energy surfaces with gentlest ascent dynamics in combination with the shrinking dimer method and Newtonian dynamics

Albareda, Guillermo; Bofill, Josep María; Moreira, Ibério de P. R.; Quapp, Wolfgang; Rubio-Martínez, Jaime

doi:10.1007/s00214-018-2246-8

Cited by 3 publications

(9 citation statements)

References 36 publications

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“…Since throughout the iterative process we often do not know in advance the type of the final stationary solution, and since the Eigenray method is normally applied in cases where conventional two-point kinematic ray tracing fails (these cases involve complex wave phenomena), the target function that includes the traveltime gradient squared, valid for both the minimum and the saddle-point solutions (equation 39), is normally the default choice for the minimization. Numerical procedures for locating constrained saddle points have been studied by Du (2012a, 2012b), Ren and Vanden-Eijnden (2013), Gao et al (2015), Albareda et al (2018), and .…”

Section: O P T I M I Z At I O N O F T H E Ta R G E T F U N C T I O Nmentioning

confidence: 99%

Eigenrays in 3D heterogeneous anisotropic media, Part I: Kinematics

Ravve

Koren

2020

Geophysical Prospecting

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We present a new ray bending approach, referred to as the Eigenray method, for solving two-point boundary-value kinematic and dynamic ray tracing problems in 3D smooth heterogeneous general anisotropic elastic media. The proposed Eigenray method is aimed to provide reliable stationary ray path solutions and their dynamic characteristics, in cases where conventional initial-value ray shooting methods, followed by numerical convergence techniques, become challenging. The kinematic ray bending solution corresponds to the vanishing first traveltime variation, leading to a stationary path between two fixed endpoints (Fermat's principle), and is governed by the nonlinear second-order Euler-Lagrange equation. The solution is based on a finite-element approach, applying the weak formulation that reduces the Euler-Lagrange second-order ordinary differential equation to the first-order weightedresidual nonlinear algebraic equation set. For the kinematic finite-element problem, the degrees of freedom are discretized nodal locations and directions along the ray trajectory, where the values between the nodes are accurately and naturally defined with the Hermite polynomial interpolation. The target function to be minimized includes two essential penalty (constraint) terms, related to the distribution of the nodes along the path and to the normalization of the ray direction. We distinguish between two target functions triggered by the two possible types of stationary rays: a minimum traveltime and a saddle-point solution (due to caustics). The minimization process involves the computation of the global (all-node) traveltime gradient vector and the traveltime Hessian matrix. The traveltime Hessian is used for the minimization process, analysing the type of the stationary ray, and for computing the geometric spreading of the entire resolved stationary ray path. The latter, however, is not a replacement for the dynamic ray tracing solution, since it does not deliver the geometric spreading for intermediate points along the ray, nor the analysis of caustics. Finally, we demonstrate the efficiency and accuracy of the proposed method along three canonical examples.

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Section: O P T I M I Z At I O N O F T H E Ta R G E T F U N C T I O Nmentioning

confidence: 99%

Eigenrays in 3D heterogeneous anisotropic media, Part I: Kinematics

Ravve

Koren

2020

Geophysical Prospecting

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“…In 2011, E and Zhou [58] proposed the so-called Gentlest Ascent Dynamics (GAD) method that reformulates the procedure of uphill walking from a minimum basin through a set of ordinary differential equations whose solutions converge to first index saddle points [14,[59][60][61][62][63]65]. The set of equations that governs the GAD is…”

Section: Gentlest Ascent Dynamicsmentioning

confidence: 99%

“…We note that at the starting point the norm of the v(t 0 )-vector should be equal to 1. The GAD algorithm can be seen as a Zermelo like navigation model on the PES to reach TSs in some optimal way, see [14,63,65]. The Zermelo navigation model is an example of a Mayer-Bolza problem of calculus of variations [64].…”

Section: Gentlest Ascent Dynamicsmentioning

confidence: 99%

“…The GAD method, as mentioned above, can be formulated as a Mayer-Bolza problem of the calculus of variations related to the solution of a problem of optimal control theory [14,63,65]. In the present context this problem can be formulated as follows: determine the vector function q(t) of dimension N , satisfying the Eq.…”

Section: Spmentioning

confidence: 99%

“…(50a) with initial condition q 0 = q(t 0 ) and determine the control vector function, v(t), of dimension N , restricted to the normalization relation, v(t) T v(t) = 1, in such a way that the functional I(q(t)) = t − t 0 (51) assumes an extremal value with the boundary condition, g(q(t f )) = 0, being satisfied at t = t f . It can be proved that this variational problem defined in this way has an associated Hamiltonian [14,65]. The Hamiltonian has the following form, 2H(q, y) = (2v T g(q)) 2 (y T y) − (1 + y T g(q)) 2 = 0 (52) where y is the co-vector of q.…”

Section: Spmentioning

confidence: 99%

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Calculus of variations as a basic tool for modelling of reaction paths and localisation of stationary points on potential energy surfaces

Bofill

Quapp

2019

Molecular Physics

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The theory of calculus of variations is a mathematical tool which is widely used in different scientific areas in particular in physics and chemistry. This theory is strongly related with optimization. In fact the former seeks to optimize an integral related with some physical magnitude over some space to an extremum by varying a function of the coordinates. On the other hand reaction paths and potential energy surfaces, in particular their stationary points, are the basis of many chemical theories, in particular reactions rate theories. We present a review where it is gathered together the variational nature of many types of reaction paths: steepest descent, Newton trajectories, artificial force induced reaction (AFIR) paths, gradient extremals, and gentlest ascent dynamics (GAD) curves. The variational basis permits to select the best optimization technique in order to locate important theoretical objects on a potential energy surface.

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Interplay between the Gentlest Ascent Dynamics Method and Conjugate Directions to Locate Transition States

Bofill

Ribas‐Ariño

Valero

et al. 2019

J. Chem. Theory Comput.

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An algorithm to locate transition states on a potential energy surface (PES) is proposed and described. The technique is based on the GAD method where the gradient of the PES is projected into a given direction and also perpendicular to it. In the proposed method, named GAD-CD, the projection is not only applied to the gradient but also to the Hessian matrix. Then, the resulting Hessian matrix is block diagonal. The direction is updated according to the GAD method. Furthermore, to ensure stability and to avoid a high computational cost, a trust region technique is incorporated and the Hessian matrix is updated at each iteration. The performance of the algorithm in comparison with the standard ascent dynamics is discussed for a simple two dimensional model PES. Its efficiency for describing the reaction mechanisms involving small and medium size molecular systems is demonstrated for five molecular systems of interest.

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Exploring potential energy surfaces with gentlest ascent dynamics in combination with the shrinking dimer method and Newtonian dynamics

Cited by 3 publications

References 36 publications

Eigenrays in 3D heterogeneous anisotropic media, Part I: Kinematics

Eigenrays in 3D heterogeneous anisotropic media, Part I: Kinematics

Calculus of variations as a basic tool for modelling of reaction paths and localisation of stationary points on potential energy surfaces

Interplay between the Gentlest Ascent Dynamics Method and Conjugate Directions to Locate Transition States

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