Jacobian correction

Fig. 3.14. Comparison of the torsion angle density distribution for a MC simulation of an athermal Ci0 phantom chains with Jacobian correction (the horizontal noisy line) and without Jacobian correction (the noisy curve showing two maxima at ca. 100° and 260° and a sharp minimum at 180°)...

The N relations given by (15-40) form a tridiagonal matrix equation that is linear in The form of the matrix equation is identical to (15-12) where, for example, A = (dH2ldT f 82 = (dH dTiTK C2 = ( Hz/aTa)" , x,2 AT and D2 = —H K The matrix of partial derivatives is called the Jacobian correction matrix. The Thomas algorithm can be employed to solve for the set of corrections AT) New guesses of 7 are then determined from... 

Another feature of general interest is the Jacobian correction to AE, which proves to be inversely proportional to R and thus is a significant factor. It is extremely easy to evaluate this factor in the large-D limit. Many treatments of tunneling assume or approximate... 

The development of an SC procedure involves a number of important decisions (1) What variables should be used (2) What equations should be used (3) How should variables be ordered (4) How should equations be ordered (5) How should flexibility in specifications be provided (6) Which derivatives of physical properties should be retained (7) How should equations be linearized (8) If Newton or quasi-Newton hnearization techniques are employed, how should the Jacobian be updated (9) Should corrections to unknowns that are computed at each iteration be modified to dampen or accelerate the solution or be kept within certain bounds (10) What convergence criterion should be applied ... 

Calculations of relative free energies of binding often involve the alteration of bond lengths in the course of an alchemical simulation. When the bond lengths are subject to constraints, a correction is needed for variation of the Jacobian factor in the expression for the free energy. Although a number of expressions for the correction formula have been described in the literature, the correct expressions are those presented by Boresch and Karplus.21... 

Like Newton s method, the Newton-Raphson procedure has just a few steps. Given an estimate of the root to a system of equations, we calculate the residual for each equation. We check to see if each residual is negligibly small. If not, we calculate the Jacobian matrix and solve the linear Equation 4.19 for the correction vector. We update the estimated root with the correction vector,... 

At each step in the Newton-Raphson iteration, we evaluate the residual functions and Jacobian matrix. We then calculate a correction vector as the solution to the matrix equation... 

Note that the Jacobian matrix dh/dx on the left-hand side of Equation (A.26) is analogous to A in Equation (A.20), and Ax is analogous to x. To compute the correction vector Ax, dh/dx must be nonsingular. However, there is no guarantee even then that Newton s method will converge to an x that satisfies h(x) = 0. 

In most cases, the structural procedure is able to determine whether the measurements can be corrected and whether they enable the computation of all of the state variables of the process. In some configurations this technique, used alone, fails in the detection of indeterminable variables. This situation arises when the Jacobian matrix used for the resolution is singular. 

Note that, in this system of coordinates, the correction force includes a term involving the Jacobian of the transformation from generalized to Cartesian coordinates, which arises from the In contribution to In / in Eq. (2.157) for V. ... 

Because of large scale disparity the numerical solution to the locally linear problem at every iteration (Eq. 15.46) is highly sensitive to small errors. In other words very small variations in the trial solution y(m> or the Jacobian J cause very large variations in the correction vector Ay(m). From the linear algebra perspective, scale disparity can be measured by the condition number1 of the Jacobian matrix. As the condition number increases the... 

In 3D we also need the two transformations used with the 2D isoparametric element. In the first place, the global derivatives of the formulation, dNi/dx, must be expresses in terms of local derivatives, dNi/d . Second, the integration of volume (or surface) needs to be performed in the appropriate coordinate system with the correct limits of integration. The global and local derivatives are related through a Jacobian transformation matrix as follows... 

For a moment, assume that Eq. (3) is correct for all t where Eq. (4) gives the true solution. Then the response modes of the system are given by the eigenvalues (A ) of the Jacobian J0. For a stable system, the eigenvalues have the property that... 

With the Jacobian X prepared as indicated and the vector of variables p limited to the independent coordinates, X has maximum rank and the problem can be solved by the iterated least-squares treatment. After each iteration step, p should be expanded to obtain p by means of Eq. 56b for the correction of the independent and the dependent coordinates. Due to the presence of E in Eq. 58, p has the required zero component wherever a coordinate has to be kept fixed and must not be changed. The corrected coordinates are required to recalculate y and X (the quantities 77 w(.s)(0) and (377gw(s)/3/i 1f ty0 for the next iteration step. In contrast to most applications of the least-squares procedure, the covariance matrix of the (effective) observations, 0- (Eq. 55) must also be recalculated because 0 depends on U which changes (though probably very little) with each step (Eq. 53). 

To interchange the variables Vi and the following procedure can be used. The Jacobian matrix is first modified by setting to zero the ith column of each of the submatrices ]ki for k = 1 to m. The ith column contains the effect of changing of Vi on each element of E because qj does not enter into any of the material balance equations, this column becomes zero. The ith column of Jm+i,i is set equal to the th column of an n by n identity matrix because qj enters only the one energy balance equation. Upon solving for the correction vector [(C)v+i — (C)v] the ith element is now the correction for The correction to is zero. 

Many variations of the correction method we have proposed can be used. Among these are the use of the same Jacobian for several iterations and the use of modified Newton methods such as Marquardt s method (8). We have tried Marquardt s method on some of these problems without observing any significant improvement, but this is only a tentative evaluation. Improved methods for generating starting conditions would be helpful. 

Equation [43] was first derived in Ref. 15, where we represented V (r) by a generic Jacobian function/(F), and it represents a correct generalization of the Liouville equation to account for the nonvanishing compressibility of phase space. Equation [43] can be derived in many ways. A general approach starts with a statement of continuity valid for a space with any metric (see Appendix 2). One can examine the transformation from one set of phase space coordinates r to another F. The metric determinant transforms according to... 

These equations are linear and can be solved by a linear equation solver to get the next reference point (ah, A21). Iteration is continued until a solution of satisfectory precison is reached. Of course, a solution may not be reached, as illustrated in Fig. L,6c, or may not be reached because of round-off or truncation errors. If the Jacobian matrix [see Eq. (L.ll) below] is singular, the linearized equations may have no solution or a whole family of solutions, and Newton s method probably will fail to obtain a solution. It is quite common for the Jacobian matrix to become ill-conditioned because if ao is far from the solution or the nonlinear equations are badly scaled, the correct solution will not be obtained. 

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