Richard E. Bellman: Difference between revisions

Richard E. Bellman
Richard E. Bellman
	File:Richard E. Bellman.jpg
Born	26 August 1920
Died	19 March 1984 (aged 63)
Nationality	American
Alma mater	Princeton University; University of Wisconsin-Madison; Brooklyn College
	Scientific career
Fields	Mathematics
Institutions	University of Southern California

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Richard Ernest Bellman (August 26, 1920 – March 19, 1984) was an applied mathematician, celebrated for his invention of dynamic programming in 1953, and important contributions in other fields of mathematics.

Biography

Bellman was born in 1920 and studied mathematics at Brooklyn College (B.A. 1941) and the University of Wisconsin-Madison (M.A.). He then went to work for a Theoretical Physics Division group in Los Alamos. In 1946 he received his Ph.D. at Princeton under the supervision of Solomon Lefschetz^[1].

He was a professor at the University of Southern California, a Fellow in the American Academy of Arts and Sciences (1975), and a member of the National Academy of Engineering (1977).

He was awarded the IEEE Medal of Honor in 1979, "For contributions to decision processes and control system theory, particularly the creation and application of dynamic programming". His key work is the Bellman-Equation.

Work

Bellman equation

A Bellman equation also known as a dynamic programming equation, is a necessary condition for optimality associated with the mathematical optimization method known as dynamic programming. Almost any problem which can be solved using optimal control theory can also be solved by analyzing the appropriate Bellman equation. The Bellman equation was first applied to engineering control theory and to other topics in applied mathematics, and subsequently became an important tool in economic theory.

Bellman–Ford algorithm

The Bellman-Ford algorithm sometimes referred to as the Label Correcting Algorithm, computes single-source shortest paths in a weighted digraph (where some of the edge weights may be negative). Dijkstra's algorithm accomplishes the same problem with a lower running time, but requires edge weights to be non-negative. Thus, Bellman–Ford is usually used only when there are negative edge weights.

Curse of dimensionality

The Curse of dimensionality", is a term coined by Bellman to describe the problem caused by the exponential increase in volume associated with adding extra dimensions to a (mathematical) space.

For example, 100 evenly-spaced sample points suffice to sample a unit interval with no more than 0.01 distance between points; an equivalent sampling of a 10-dimensional unit hypercube with a lattice with a spacing of 0.01 between adjacent points would require 10²⁰ sample points: thus, in some sense, the 10-dimensional hypercube can be said to be a factor of 10¹⁸ "larger" than the unit interval. (Adapted from an example by R. E. Bellman, see below.)

Hamilton-Jacobi-Bellman

The Hamilton-Jacobi-Bellman equation (HJB) equation is a partial differential equation which is central to optimal control theory. The solution of the HJB equation is the 'value function', which gives the optimal cost-to-go for a given dynamical system with an associated cost function. Classical variational problems, for example, the brachistochrone problem can be solved using this method as well.

The equation is a result of the theory of dynamic programming which was pioneered in the 1950s by Richard Bellman and coworkers. The corresponding discrete-time equation is usually referred to as the Bellman equation. In continuous time, the result can be seen as an extension of earlier work in classical physics on the Hamilton-Jacobi equation by William Rowan Hamilton and Carl Gustav Jacob Jacobi.

Publications

Over the course of his career he published 619 papers and 39 books. During the last 11 years of his life he published over 100 papers despite suffering from crippling complications of a brain surgery (Dreyfus, 2003).

References

^ Mathematics Genealogy Project http://genealogy.math.ndsu.nodak.edu/id.php?id=12968

Bellman, Richard, Eye of the Hurricane, an Autobiography, World Scientific Publishing, 1984.
IEEE History Center - Legacies
Int. Trans. in Op. Res. Biographical Article on Bellman
Harold J. Kushner's speech when accepting the Richard E. Bellman Control Heritage Award
S. Dreyfus, Richard Bellman on the Birth of Dynamic Programming
Sanabria, Salvador. [1] Richard Bellman's Biography.
J.J. O'Connor and E.F. Robertson. Biography of Richard Bellman from the MacTutor History of Mathematics.

Template:S-awards

Preceded by Robert Noyce	IEEE Medal of Honor 1979	Succeeded by William Shockley

This article about an American mathematician is a stub. You can help Wikipedia by expanding it.

[1] Mathematics Genealogy Project http://genealogy.math.ndsu.nodak.edu/id.php?id=12968

[1]

@@ Line 24: / Line 24: @@
 He was awarded the [[IEEE Medal of Honor]] in 1979, "For contributions to decision processes and control system theory, particularly the creation and application of dynamic programming". His key work is the [[Bellman-Equation]].
-==See also==
+== Work ==
-*[[Bellman equation]]
+===Bellman equation===
+A [[Bellman equation]] also known as a ''dynamic programming equation'', is a necessary condition for optimality associated with the mathematical optimization method known as [[dynamic programming]]. Almost any problem which can be solved using [[optimal control theory]] can also be solved by analyzing the appropriate Bellman equation. The Bellman equation was first applied to engineering [[control theory]] and to other topics in applied mathematics, and subsequently became an important tool in [[economic theory]].
-*[[Bellman-Ford algorithm]]
-*[[Hamilton-Jacobi-Bellman equation]]
+=== Bellman–Ford algorithm ===
-*"[[Curse of dimensionality]]", a term coined by Bellman
+The [[Bellman-Ford algorithm]] sometimes referred to as the Label Correcting Algorithm, computes single-source shortest paths in a [[weighted digraph]] (where some of the [[edge (graph theory)|edge]] weights may be negative).  [[Dijkstra's algorithm]] accomplishes the same problem with a lower running time, but requires edge weights to be non-negative.  Thus, Bellman–Ford is usually used only when there are negative edge weights.
+=== Curse of dimensionality ===
+The [[Curse of dimensionality]]", is a term coined by Bellman to describe the problem caused by the exponential increase in [[volume]] associated with adding extra dimensions to a (mathematical) space.
+For example, 100 evenly-spaced sample points suffice to sample a [[unit interval]] with no more than 0.01 distance between points; an equivalent sampling of a 10-dimensional [[unit hypercube]] with a lattice with a spacing of 0.01 between adjacent points would require 10<sup>20</sup> sample points: thus, in some sense, the 10-dimensional hypercube can be said to be a factor of 10<sup>18</sup> "larger" than the unit interval. (Adapted from an example by R. E. Bellman, see below.)
+=== Hamilton-Jacobi-Bellman ===
+The [[Hamilton-Jacobi-Bellman equation]] (HJB) equation is a [[partial differential equation]] which is central to [[optimal control]] theory. The solution of the HJB equation is the 'value function', which gives the optimal cost-to-go for a given [[dynamical system]] with an associated cost function. Classical variational problems, for example, the [[brachistochrone problem]] can be solved using this method as well.
+The equation is a result of the theory of [[dynamic programming]] which was pioneered in the 1950s by Richard Bellman and coworkers. The corresponding discrete-time equation is usually referred to as the [[Bellman equation]]. In continuous time, the result can be seen as an extension of earlier work in [[classical physics]] on the [[Hamilton-Jacobi equation]] by [[William Rowan Hamilton]] and [[Carl Gustav Jacob Jacobi]].
 == Publications ==