Method of random directions and steepest descent

Let us minimize the sum of the squares of the components of the residual vector given by

$$\begin{eqnarray} {\rm residual} &=& {\rm \hbox{transform} \ \ \ \ \hbox{model s... ...t[ \matrix { \matrix { \cr \cr \bold d \cr \cr \cr} } \right]\end{eqnarray}$$ (39) (40)

A contour plot is based on an altitude function of space. The altitude is the dot product $\bold r \cdot \bold r$.By finding the lowest altitude, we are driving the residual vector $\bold r$ as close as we can to zero. If the residual vector $\bold r$ reaches zero, then we have solved the simultaneous equations $\bold d= \bold F \bold x$.In a two-dimensional world the vector $\bold x$ has two components, (x₁ , x₂ ). A contour is a curve of constant $\bold r \cdot \bold r$ in (x₁ , x₂ )-space. These contours have a statistical interpretation as contours of uncertainty in (x₁ , x₂ ), with measurement errors in $\bold d$.

Let us see how a random search-direction can be used to reduce the residual $0\approx \bold r= \bold F \bold x - \bold d$.Let $\Delta \bold x$ be an abstract vector with the same number of components as the solution $\bold x$,and let $\Delta \bold x$ contain arbitrary or random numbers. We add an unknown quantity $\alpha$of vector $\Delta \bold x$ to the vector $\bold x$,and thereby create $\bold x_{\rm new}$: 

$$\bold x_{\rm new} \eq \bold x+\alpha \Delta \bold x$$ (41)

This gives a new residual:

$$\begin{eqnarray} \bold r_{\rm new} &=&\bold F\ \bold x_{\rm new} -\bold d \\ \bo... ...Delta\bold r&=&\bold F\bold x-\bold d +\alpha\bold F\Delta\bold x\end{eqnarray}$$ (42) (43) (44)

which defines $\Delta \bold r = \bold F \Delta \bold x$.

Next we adjust $\alpha$ to minimize the dot product: $\bold r_{\rm new} \cdot \bold r_{\rm new}$ 

$$(\bold r+\alpha\Delta \bold r)\cdot (\bold r+\alpha\Delta \b... ...lta \bold r) \ +\ \alpha^2 \Delta \bold r \cdot \Delta \bold r$$ (45)

Set to zero its derivative with respect to $\alpha$ using the chain rule  

$$0\eq (\bold r+\alpha\Delta \bold r) \cdot \Delta \bold r \ ... ...ld r) \eq 2 (\bold r+\alpha\Delta \bold r) \cdot \Delta \bold r$$ (46)

which says that the new residual $\bold r_{\rm new} = \bold r + \alpha \Delta \bold r$ is perpendicular to the ``fitting function'' $\Delta \bold r$.Solving gives the required value of $\alpha$. 

$$\alpha \eq - \ { (\bold r \cdot \Delta \bold r ) \over ( \Delta \bold r \cdot \Delta \bold r ) }$$ (47)

A ``computation template'' for the method of random directions is

 

		 $\bold r \quad\longleftarrow\quad\bold F \bold x - \bold d$ 
		 iterate { 
		 		  $\Delta \bold x \quad\longleftarrow\quad{\rm random\ numbers}$ 
		 		  $\Delta \bold r\ \quad\longleftarrow\quad\bold F \ \Delta \bold x$ 
		 		 $\alpha \quad\longleftarrow\quad
 -( \bold r \cdot \Delta\bold r )/
 (\Delta \bold r \cdot \Delta\bold r )$		 		 $\bold x \quad\longleftarrow\quad\bold x + \alpha\Delta \bold x$ 
		 		 $\bold r\ \quad\longleftarrow\quad\bold r\ + \alpha\Delta \bold r$ 
		 		 } 

A nice thing about the method of random directions is that you do not need to know the adjoint operator $\bold F'$.

In practice, random directions are rarely used. It is more common to use the gradient direction than a random direction. Notice that a vector of the size of $\Delta \bold x$ is

$$\bold g \eq \bold F' \bold r$$ (48)

Notice also that this vector can be found by taking the gradient of the size of the residuals:

$${\partial \over \partial \bold x' } \ \bold r \cdot \bold r ... ...') \, ( \bold F \, \bold x \ -\ \bold d) \eq \bold F' \ \bold r$$ (49)

Choosing $\Delta \bold x$ to be the gradient vector $\Delta\bold x = \bold g = \bold F' \bold r$is called ``the method of steepest descent.''

Starting from a model $\bold x = \bold m$ (which may be zero), below is a template of pseudocode for minimizing the residual $\bold 0\approx \bold r = \bold F \bold x - \bold d$by the steepest-descent method:  

		 $\bold r \quad\longleftarrow\quad\bold F \bold x - \bold d$ 
		 iterate { 
		 		  $\Delta \bold x \quad\longleftarrow\quad\bold F'\ \bold r$ 
		 		  $\Delta \bold r\ \quad\longleftarrow\quad\bold F \ \Delta \bold x$ 
		 		 $\alpha \quad\longleftarrow\quad
 -( \bold r \cdot \Delta\bold r )/
 (\Delta \bold r \cdot \Delta\bold r )$		 		 $\bold x \quad\longleftarrow\quad\bold x + \alpha\Delta \bold x$ 
		 		 $\bold r\ \quad\longleftarrow\quad\bold r\ + \alpha\Delta \bold r$ 
		 		 }