幻灯片16Conjugate gradientDisadvantage of steepest descent is that itloses information about previous steps.Notion of"conjugate gradient"search is toaugment steepest descent with requirementtokeepeachsteporthogonaltosomenumberof previous steps..Generally performs much better than steepestdescent,andisoftenthefirstchoiceiffarfromanequilibriumgeometry.https:/len.wikipedia.org/wiki/Conjugate_gradient_methoda.最速下降法的缺点是没有利用上一步的信息。b.改进的方法称为共轭梯度法。每一步与上一步或者上几步的移动方向共轭。比最速下降收敛快，且适合计算原理平衡（极小值）的结构。C.d.图中对比了这两种方法红色（共轭梯度法）绿色（最速下降）
幻灯片 16 Conjugate gradient 16 • Disadvantage of steepest descent is that it loses information about previous steps. • Notion of “conjugate gradient” search is to augment steepest descent with requirement to keep each step orthogonal to some number of previous steps. • Generally performs much better than steepest descent, and is often the first choice if far from an equilibrium geometry. https://en.wikipedia.org/wiki/Conjugate_gradient_method a. 最速下降法的缺点是没有利用上一步的信息。 b. 改进的方法称为共轭梯度法。每一步与上一步或者上几步的移动方向共轭。 c. 比最速下降收敛快，且适合计算原理平衡（极小值）的结构。 d. 图中对比了这两种方法红色（共轭梯度法）绿色（最速下降）
幻灯片17TheNewton-RaphsonmethodThe two above only use/require information aboutg;.If we also knew something about the second-derivative of the energy, H, that should speed upoptimizationevenfurther;In one dimension, if function was quadratic, can9-g-8la)exactly write (dE/dq'=0):H(g)If not quadratic, make Taylor expansion of energy in qup to second-order,arrive atsame expression forpredicting q';.Apply iteratively;In well-behaved cases this algorithm rapidly convergestolocalminimumhttps:/len.wikipedia.org/wiki/Newton%27s method上述两方法依赖于g，如果我们知道h，优化还能加快。--维，对于q附近的势能曲线可以用二次方程描述，求dE/dg=o即极小点，则新坐标q可以准确写为：如果势能面在g附近不能用二次方程描述，但是可以Taylor展开并截断为q的二价导数，那么同样可以求q此方法可以选代使用，对于表现良好的体系（q附近近似二次方程的势能面），这个方法收敛很快
幻灯片 17 The Newton-Raphson method 17 • The two above only use/require information about g; • If we also knew something about the secondderivative of the energy, H, that should speed up optimization even further; • In one dimension, if function was quadratic, can exactly write (dE/dq’=0): • If not quadratic, make Taylor expansion of energy in q up to second-order, arrive at same expression for predicting q′; • Apply iteratively; • In well-behaved cases this algorithm rapidly converges to local minimum https://en.wikipedia.org/wiki/Newton%27s_method 上述两方法依赖于g，如果我们知道h，优化还能加快。 一维，对于q附近的势能曲线可以用二次方程描述，求dE/dq’=0即极小点，则新 坐标q’可以准确写为： 如果势能面在q附近不能用二次方程描述，但是可以Taylor展开并截断为q的二价 导数，那么同样可以求q’ 此方法可以迭代使用，对于表现良好的体系（q附近近似二次方程的势能面）， 这个方法收敛很快
幻灯片18TheNewton-Raphsonmethodforfinding a minimum.Taylor expand about q:(E(0)+(α-)+(-0)2 dg第5α--0dqdgdg-qq.Can generalize this approach to multiple dimensions;.Make second-order Taylorexpansion of energyabout R, can showoptimal R'=R-H-(R)g(R):Again can perform line-search in direction R'-Rifyou want;.Apply iteratively;The Newton-Raphson method works really well;18这里在q附近对E进行Taylor展开，极小值对应的q'可以通过g和h计算得到。这个方法可以扩展至多维。表达式相同；同样可以使用线搜索，送代
幻灯片 18 The Newton-Raphson method for finding a minimum 18 • Taylor expand about q: • Can generalize this approach to multiple dimensions; R′ = R −H−1(R)g(R) • Make second-order Taylor expansion of energy about R, can show optimal • Again can perform line-search in direction R′− R if you want; • Apply iteratively; • The Newton-Raphson method works really well; 这里在q附近对E进行Taylor展开，极小值对应的q’可以通过g和h计算得到。 这个方法可以扩展至多维。 表达式相同； 同样可以使用线搜索，迭代