Nonlinear Computation and Analysis


Key words: Nonlinear computation and analysis, matrix computation and analysis, linearization, numerical linear algebra, Hadamard product, SJT product, special matrix product, uncoupling, Jacobian matrix, polynomial equations, Newton method, iteration, inversion, nonlinear stability analysis, linear iteration methods, nonlinear function vector, pseudo-Jacobian, ill-posed linear equation, generalized linearization, singular value decomposition.

 

Although great endeavour has been devoted to nonlinear computation and analysis, it seems very difficult to attack nonlinear problems directly. The linearization procedure such as the Newton method and its variants is now commonly used to transform a nonlinear system to a linear system in a point-wise approximate way so that the standard numerical linear algebra approach can be employed for computation and analysis. It is noted that the strategy of linerization is not an intrinsically nonlinear approach and often leads to a very huge amount of computing cost and encounters great difficulty in nonlinear stability analysis. A truly nonlinear strategy should be desirable to handle nonlinear problems more naturally. Recently we introduced the Hadamard matrix product and SJT product of matrix and vector to nonlinear analysis. These two special matrix product operations possess distinctive inherently nonlinear merit but are very mathematically simple concepts. Some primary works show that this new framework is very promising in nonlinear stability analysis, uncoupling computation, evaluation of Jacobian matrix, and construction of simple, efficient and robust iteration formulas. It is still open to find other special matrix opeartions essentially applicable for nonlinear algebra analysis and computation.

On the other hand, in comparison to nonlinear problems, a vast amount of computing and analysis tools of linear problems has been quite well developed today. It is natural desire to extend these linear methods to nonlinear problems. Traditionally, this is done via the linearization procedure. However, this approach has some notorious drawbacks as mentioned above. Rather recently we found and proved a straightforward relationship theorem between nonlinear polynomial equations and its Jacobian matrix. The theorem has leaded to several significant results of fundamental importance and promised a competitive alternative to the standard linearization approach.

In order to avoid the evaluation of the Jacobian matrix and its inverse, we introduced the pseudo-Jacobian matrix concept with a general applicability for any nonlinear algebraic systems.

Matrix computations constitute very essential part of nonlinear numerical analysis and engineering computations. My contributions are general enough to be applicable to nonlinear simulations of various discretization techniques such as the FEM, FDM, BEM, finite volume, collocation, and spectral methods. Experimenting these novel concepts and strategies with some nonlinear computational mechanics problems is very encouraging. Further extension to nonlinear control problems is also under active study. For some details see below:

#1: We first used the Hadamard product, a special matrix product, to express the nonlinear numerical discretization of all point-approximation numerical techniques (FDM, collocation, pseudo-spectral, various time integrators, finite volume, differential quadrature, and dual reciprocity BEM, etc.) and radial basis function methods. By using Hadamard product power concept, we can easily construct some very simple and efficient iteration formulas of the simple iteration method for the solution of the Hadamard product representation of nonlinear formulations without the use of linearization.

#2: We also defined a new special matrix and vector product, SJT product, for easy, efficient and accurate calculation of the Jacobian matrix of the nonlinear analogous equations of the Hadamard product representation. The approach produces the exact Jacobian matrix in the chain rules similar to those in differentiation of a scalar function. The computational effort of a SJT product is only n^2 scalar multiplications less than using the finite difference method, which is often employed to calculate the approximate Jacobian matrix for large complex system. The work also leads to a general approach of uncoupling the numerical formulation of the coupled nonlinear partial differential systems. Consequently, a considerable saving of computer resources was achieved. For example, the computational effort and storage requirements in the solution of geometrically nonlinear bending of orthotropic plates were reduced by about one twenty-seventh and one-ninth, respectively.

#3: We found and proved that a simple and underlying relationship formula between the nonlinear polynomial equations and the corresponding Jacobian matrix, which provides a new elementary approach to compute and analyze nonlinear problems without the standard linearization procedure. This relationship formula is in fact a straightforward extension of differentiation of a scalar power function to the Jacobian derivative matrix of a polynomial function vector. The result leads to a great simplification of nonlinear numerical stability analysis, especially for nonlinear dynamic problem. For example, Burger's equation.

#4: By using the above-mentioned relationship formula, we derived a modified BFGS updating formula of quasi-Newton method. Moreover, the linear iteration methods such as the Jacobi, Gauss-Sideal, and SOR methods can be directly applied to solve nonlinear equations without the linearization. A Newton-Raphson iteration formula was derived without the evaluation of nonlinear function vector. Also a very simple formula was given to accurately evaluate the relative error of the approximate Jacobian matrix without the need of the exact Jacobian matrix but with no loss of accuracy.

#5: The time-consuming effort in computation and analysis of nonlinear problem is the repeated evaluation of nonlinear function vector, Jacobian matrix and its inverse. Very recently we proposed a new concept of the pseudo-Jacobian matrix for stability analysis of nonlinear initial value problems. Also a new pseudo-Newton method was derived without the evaluation of Jacobian matrix and its inversion by using this original concept. However, the potential of this new concept need be further demonstrated.


Related Publications

 

Ph.D. dissertation:

Differential Quadrature Method and Its Applications to Engineering Problems - Application special matrix product to nonlinear computations (in English, defense 1997).

International Journals

  1. W. Chen and T. Zhong, The study on the nonlinear computations of the DQ and DC methods, Numerical Methods for Partial Differential Equations, 13, 57-75, 1997.

 

  1. W. Chen, S. Liang and T. Zhong, On the DQ analysis of geometrically nonlinear vibration of immovably simply supported beams, J. of Sound & Vibration, 206(5), 745-748, 1997.

 

  1. W. Chen, C. Shu, and W. He, The DQ solution of geometrically nonlinear bending of orthotropic rectangular plates by using Hadamard and SJT product, Computers & Structures, 74(1), 65-74 2000.

 

  1. W. Chen, Relationship Theorem between Nonlinear Polynomial Equation and the Corresponding Jacobian Matrix, Int. J. of Nonlinear Sciences and Numerical Simulation, 1, 5-14, 2000.



Chinese Journals

  1. W. Chen, T. Zhong and Y. Yu, "Applying special matrix product to nonlinear numerical computations", J. Appl. Comp. Math. Vol. 12, No. 1, 51-58, 1998.



Preprints

  1. W. Chen, A modified BFGS quasi-Newton iterative formula, CoRR preprint, July, 1999.

 

  1. W. Chen, Pseudo-Newton method for nonlinear equations, CoRR preprint, July, 1999.

 

  1. W. Chen, Generalized linearization in nonlinear modeling of data, CoRR preprint, July, 1999.






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Last updated 28/12/99