T. F. COLEMAN AND A. VERMA, Structure and efficient Jacobian calculation, in Computational Differentiation: Techniques, Applications, and Tools, M. Berz, C. Bischof, G. Corliss, and A. Griewank, eds., SIAM, Philadelphia, Penn., 1996, pp. 149--159.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....require the calculation (or approximation) of the Jacobian matrix J ; however, they typically converge very slowly we have not investigated those methods in this work. In this paper we explore two possibilities in the framework of our optimization approach: 1. Use of automatic differentiation [8] and or finite differencing to compute J (k) def = J(# (k) 2. Use of a secant update to approximate J (k) when p = m. 3.1. The Problem Structure. The evaluation of f(#) requires the evaluation of each component of F , i.e. w 1 2 j [v j (c(s, t) #) v j ] for j = 1, m. ....

T. F. COLEMAN AND A. VERMA, Structure and efficient Jacobian calculation, in Computational Differentiation: Techniques, Applications, and Tools, M. Berz, C. Bischof, G. Corliss, and A. Griewank, eds., SIAM, Philadelphia, Penn., 1996, pp. 149--159.

....mesh in Table 6, computing the 306 Theta 306 Jacobian block J (5) average bandwidth of 10) required that we first assemble the 44064 Theta 44064 Jacobian matrix J. A very promising topic for future research is to make use of recent advances in using automatic differentiation (AD) 51] [52], to compute the symmetry adapted Jacobian blocks J ( in a much more direct manner. It can be shown that the orthogonal blocks J ( can be directly computed as J ( Q T f u (49) where f u is an n Theta m matrix whose columns are defined as f u i = ....

.... u i = lim h 0 f (u hQ ( i) Gamma f (u) h (i = 1; 2; m ) 50) Note that eq: 50) only requires that f be computed in the standard coordinate system. Equations (49) and (50) can be used for the basis of a numerical differentiation procedure, as discussed in [51] and [52], to efficiently compute the orthogonal blocks J ( 5.4 Symmetry Boundary Conditions As discussed above, the symmetry adapted Jacobian matrices are found by first computing the Jacobian of the full matrix and then performing similarity transformations. For large problems, computing the full ....

T. F. Coleman and A. Verma. Structure and efficient jacobian calculation. In M. Berz, C. Bischof, G. Corliss, and A. Griewank, editors, Computational Differentiation: Techniques, Applications and Tools, pages 149--159, Philadelphia, 1996. SIAM.

....require the calculation (or approximation) of the Jacobian matrix J ; however, they typically converge very slowly we have not investigated those methods in this work. In this paper we explore two possibilities in the framework of our optimization approach: 1. Use of automatic differentiation [8] and or finite differencing to compute J (k) def J(# ) 2. Use of a secant update to approximate J when p = m. 3.1. The Problem Structure. The evaluation of f(#) requires the evaluation of each component of F , i.e. w [v j (c(s, t) #) m. These are generalized BlackScholes ....

T. F. COLEMAN AND A. VERMA, Structure and efficient Jacobian calculation, in Computational Differentiation: Techniques, Applications, and Tools, M. Berz, C. Bischof, G. Corliss, and A. Griewank, eds., SIAM, Philadelphia, Penn., 1996, pp. 149--159.

....require the calculation (or approximation) of the Jacobian matrix J ; however, they typically converge very slowly we have not investigated those methods in this work. In this paper we explore two possibilities in the framework of our optimization approach: 1. Use of automatic differentiation [8] and or finite differencing to compute J (k) def = J(# (k) 2. Use of a secant update to approximate J (k) when p = m. 3.1. The Problem Structure. The evaluation of f(#) requires the evaluation of each component of F , i.e. w 1 2 j [v j (c(s, t) #) v j ] for j = 1, m. These ....

T. F. COLEMAN AND A. VERMA, Structure and efficient Jacobian calculation, in Computational Differentiation: Techniques, Applications, and Tools, M. Berz, C. Bischof, G. Corliss, and A. Griewank, eds., SIAM, Philadelphia, Penn., 1996, pp. 149--159.

....the amount of work. The algorithm described in [2] illustrates the use of one sided coloring. Two sided coloring [3] combines the powers of both forward and reverse mode, by constructing thin matrices V and W so that the Jacobian J can be determined from the pair (JV , W T J) Coleman and Verma [4, 6] show how sparsity and structure can be exploited to compute Jacobian and Hessian matrices efficiently using AD. In the scalar valued case they define a structured computation, evaluate z = f(x) f : R n R 1 , as follows: Solve for y 1 : F 1 (x; y 1 ) 0 Solve for y 2 : F 2 (x; y 1 ; y 2 ) ....

....mode: Reverse mode is used to take the derivative of the function f(x) The code that is differentiated computes f(x) using a sparse solver for symmetric and positive definite systems to solve the system Ay = F . 2. Extended Jacobian: Following the structural ideas of Coleman and Verma [4] we form the extended function FE : x y 7 A(x)y Gamma F (x) f(y) 6) 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 Figure 2: The sparsity pattern for the extended Jacobian We use AD with graph coloring techniques to compute the extended Jacobian JE = A x y Gamma ....

T. F. Coleman and A. Verma, Structure and efficient Jacobian Calculation, in Computational Differentiation: Techniques, Applications and Tools, M. Berz, C. Bischof, G. Corliss and A. Griewank, editors, SIAM, Philadelphia, Penn, 1996, pp. 149-159.

....is the Jacobian of F (y) w.r.t. y. The Jacobians J i might be sparse individually but the product is most likely dense. So if we want to compute the Newton step by solving a linear system in J , it will be very expensive, specially for large scale problems. However, as shown by Coleman and Verma [3], sparsity can be preserved if we use extended functions. The idea is to construct an extended function, GE (x; y 1 ; y 2 ; y n ) which is a function of input variable x as well as all the intermediate variables as follows: Solve for y 1 : y 1 Gamma x Gamma h Delta F (t 0 ; x) 0 ....

....of the section, we demonstrate the usefulness of AD in stiff solvers. In particular, we present three case studies in the next three subsections. 4.1 Ease and Effectiveness of Forming Jacobians Automatically For most large scale problems, the Jacobian J is sparse. Recent work by Coleman and Verma [4, 3] provide techniques to exploit sparsity and structure of Jacobian matrices effectively using AD. The sparsity structure of the 10 Jacobian matrix can be automatically detected via AD. On the other hand, for sparse finite differencing, the sparsity structure must be provided by the user. This ....

T. F. Coleman and A. Verma, Structure and efficient Jacobian calculation, Tech. Rep. CTC96TR238, Theory Center, Cornell University, 1996.

....simulation the wave phenomena will lead to memory problems. As we will show in the next section, the wave propagation can be modeled effectively 3 using time stepping finite difference schemes. The time stepping nature of the scheme can be exploited using the general Extended Jacobian framework [3, 4]. The spatial discretization by finite differences reveal further structure. Each finite difference stencil encodes the dependence of a computed intermediate variable on other variables. In particular, it shows that there is an inherent sparsity in the Jacobian. A combination of these structure ....

....on given vectors. The desired directional derivative (Jacobian times vector dc) is dh = dh 1 ; dh 2 ; Delta Delta Delta ; dhm ) T . 3. 1 Adjoint computation via linear algebra The above calculation can be defined as a set of matrix equations through the use of the extended Jacobian framework [3, 4] (see also [5] Let dU = 2 6 6 6 6 4 du 1 du 2 . du m 3 7 7 7 7 5 Define the m(n 1) Theta m(n 1) matrix M = 2 6 6 6 6 6 6 6 6 6 6 4 GammaI 0 . 0 0 F 2 (c; u 1 ; u 0 ) GammaI 0 . 0 F 3 (c; u 2 ; u 1 ) F 2 (c; u 2 ; u 1 ) GammaI . ....

T. Coleman and A. Verma, Structure and efficient Jacobian calculation, in Computational Differentiation: Techniques, Applications, and Tools, Berz, et al eds., SIAM, Philadelphia, 1996, pp. 149-159.

....by typing help ADMIT. 12 Other documents Here are some useful related references: ffl Automatic differentiation [10, 9] ffl Single sided determination of sparse Jacobians [5, 2] ffl Determination of sparse Hessian matrices [1, 4, 3] ffl Bi coloring [8] ffl Structured Jacobians, Hessians [7, 6] ffl ADOL C [11] 28 ....

T. F. Coleman and A. Verma, Structure and efficient Jacobian calculation, in Computational Differentiation: Techniques, Applications, and Tools, M. Berz, C. Bischof, G. Corliss, and A. Griewank, eds., SIAM, Philadelphia, Penn., 1996, pp. 149--159.

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