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Moving least-squares enhanced Shepard interpolation for the fast marching and string methods
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10.1063/1.2996579
/content/aip/journal/jcp/130/2/10.1063/1.2996579
http://aip.metastore.ingenta.com/content/aip/journal/jcp/130/2/10.1063/1.2996579

Figures

Image of FIG. 1.
FIG. 1.

The four-well potential with . Starting from minimum and ending at minimum with . The black curve is the backtrace path. The dots show points where the energy and gradient are calculated by FMM explicitly; at all other points the interpolated energy is used.

Image of FIG. 2.
FIG. 2.

The MB potential with . The starting point for the FMM is , with backtrace path (black curve) starting from the minimum at . The dots show points where the energy and gradient are calculated. For this example with a large , most points are calculated.

Image of FIG. 3.
FIG. 3.

Log of the convergence to the correct barrier height for a 15 point path on the M-B potential using different orders of the Shepard interpolation and the damped BFGS Hessian update. The maximum of the cubic spline interpolated energy was used to compare against the exact energy .

Image of FIG. 4.
FIG. 4.

AM1 potential energy surface for alanine dipeptide. Small dark regions are areas where GAUSSIAN03 had trouble converging to the correct energy. For FMM, , , , and . FMM was started from , axial. The QSM path is given, as an initial guess, the shortest linear path between the endpoints, using eight points and the initial trust radius set to 11.5°. The QSM path after ten steps is shown with stars . The backtrace path is the black curve.

Image of FIG. 5.
FIG. 5.

The energy profile for alanine dipeptide (inset molecule), with the energy values obtained by the Shepard interpolation from the backtrace path. The sharp point at an arc length of about 80° is due to an ill converged result from GUASSIAN03 close to the path.

Image of FIG. 6.
FIG. 6.

The alanine dipeptide barrier from the, cubic spline interpolated, energy profile of the QSM path using eight points.

Image of FIG. 7.
FIG. 7.

Methyl chloride reaction with . The path with large points is the QSM path after five iterations. The solid black curve is the backtrace path and the dots are the points calculated with GAUSSIAN03 (Ref. 47). Here with endpoints at and . The -axis is the Cl–C distance and the -axis is the C–F distance.

Tables

Generic image for table
Table I.

The efficiency of interpolation with the four-well potential for different orders of Shepard interpolation and different grid spacings. “Total” is the total number of FMM points used while “interpolated” is the total number of points where interpolation was accurate enough to be used. For this table .

Generic image for table
Table II.

The efficiency of interpolation with the MB potential. The same labeling is used as in Table I. Here .

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/content/aip/journal/jcp/130/2/10.1063/1.2996579
2009-01-12
2014-04-18
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Moving least-squares enhanced Shepard interpolation for the fast marching and string methods
http://aip.metastore.ingenta.com/content/aip/journal/jcp/130/2/10.1063/1.2996579
10.1063/1.2996579
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