Quasi-linearization

A collection of methods for the numerical solution of non-linear problems by reducing them to a sequence of linear problems. Lying at the basis of the apparatus of quasi-linearization is the Newton method and its generalization to function spaces, the theory of differential inequalities (cf. Differential inequality) and the method of dynamic programming. The simplest example illustrating a method of quasi-linearization is the use of the Newton–Raphson method for finding the root $ r $ of a scalar monotone-decreasing strictly-convex function $ f $. In case the original non-linear function $ f $ is approximated at each stage of the iteration process by a linear function $ \phi $, the root of $ \phi $ is found, which serves as the next approximation, so that

$$ x _ {n+} 1 = x _ {n} - \frac{f ( x _ {n} ) }{f ^ { \prime } ( x _ {n} ) } ,\ \ n = 0 , 1 ,\dots . $$

Under fairly general conditions the sequence so constructed has the property of monotonicity $ ( x _ {0} < x _ {1} < \dots < r ) $ and quadratic convergence:

$$ | x _ {n+} 1 - r | \leq k | x _ {n} - r | ^ {2} . $$

The application of quasi-linearization to solving the Riccati equation

$$ v ^ \prime + v ^ {2} + p ( t) v + q ( t) = 0 ,\ \ v ( 0) = c $$

(it is assumed that a solution exists on an interval $ [ 0 , t _ {0} ] $), is as follows. The original equation is replaced by the equivalent equation

$$ v ^ \prime = \min _ { u } [ u ^ {2} - 2 u v - p ( t) v - q ( t) ] , $$

where the minimum is taken over all functions $ u $ defined on $ [ 0 , t _ {0} ] $. This equation has a number of properties inherent to linear equations, and in order to solve it one uses the linear differential equation

$$ w ^ \prime = u ^ {2} - 2 u w - p ( t) w - q ( t) ,\ \ w ( 0) = c , $$

where $ u $ is a fixed function. By appealing to the property $ v ( t) \leq w ( t) $( equality holding when $ u ( t) = v ( t) $), one can construct a system of successive approximations

$$ v _ {1} \geq v _ {2} \geq \dots $$

satisfying the linear equations

$$ v _ {n+} 1 ^ \prime = v _ {n} ^ {2} - 2 v _ {n} v _ {n+} 1 - p ( t) v _ {n+} 1 - q ( t) ,\ \ v _ {n+} 1 ( 0) = c . $$

The same recurrence relation can be obtained by applying the Newton–Kantorovich method (cf. Kantorovich process) to the original non-linear equation.

The employment of a quasi-linearization scheme in the solution of boundary value problems for the non-linear second-order differential equation

$$ u ^ {\prime\prime} = f ( u ^ \prime , u , t ) ,\ \ t _ {1} \leq t \leq t _ {2} , $$

$$ g _ {1} ( u ( t _ {1} ) , u ^ \prime ( t _ {1} ) ) = 0 ,\ \ g _ {2} ( u ( t _ {2} ) , u ^ \prime ( t _ {2} ) ) = 0, $$

leads to a sequence of functions $ \{ u _ {n} \} $ satisfying the linear equations

$$ u _ {n+} 1 ^ {\prime\prime} = f ( u _ {n} ^ \prime , u _ {n} , t ) + f _ {u ^ \prime } ( u _ {n} ^ \prime , u _ {n} , t ) ( u _ {n+} 1 ^ \prime - u _ {n} ^ \prime ) + $$

$$ + f _ {n} ( u _ {n} ^ \prime , u _ {n} , t ) ( u _ {n+} 1 - u _ {n} ) , $$

with the linearized boundary conditions

$$ g _ {i} ( u _ {n} ( t _ {i} ) ,\ u _ {n} ^ \prime ( t _ {i} ) ) + g _ {iu} ( u _ {n} ( t _ {i} ) , u _ {n} ^ \prime ( t _ {i} ) ) ( u _ {n+} 1 ( t _ {i} ) - u _ {n} ( t _ {i} ) ) + $$

$$ + g _ {i u ^ \prime } ( u _ {n} ( t _ {i} ) , u _ {n} ^ \prime ( t _ {i} ) ) ( u _ {n+} 1 ^ \prime ( t _ {i} ) - u _ {n} ^ \prime ( t _ {i} ) ) = 0 . $$

The existence, uniqueness and quadratic convergence of the sequence follows from the corresponding convexity of the functions $ f , g _ {1} , g _ {2} $ over a sufficiently small interval $ [ t _ {1} , t _ {2} ] $.

The method of quasi-linearization finds application in the solution of two-point and multi-point boundary value problems for linear and non-linear ordinary differential equations, boundary value problems for elliptic and parabolic partial differential equations, variational problems, differential-difference and functional-differential equations, etc. As with every iteration scheme, the method of quasi-linearization is suitable for computer implementation and has various modifications enabling one to accelerate the convergence for narrower classes of problems. There exists a variety of examples of its use as a heuristic method for solving a number of problems in physics, technology and economy.

References

[1]	R.E. Bellman, R.E. Kalaba, "Quasilinearization and nonlinear boundary-value problems" , Elsevier (1965) (Translated from Russian)

References

[a1]	R. Bellman, G. Adomian, "Partial differential equations" , Reidel (1985) pp. Chapt. IV
[a2]	R. Bellman, R. Vasudevan, "Wave propagation. An invariant imbedding approach" , Reidel (1986)