Richard Ernest BellmanRichard Bellman's Biography
/ref> (August 26, 1920 – March 19, 1984) was an American

applied mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History One ...

, who introduced dynamic programming in 1953, and made important contributions in other fields of mathematics, such as biomathematics. He founded the leading biomathematical journal Mathematical Biosciences, as well as the

Journal of Mathematical Analysis and Applications The ''Journal of Mathematical Analysis and Applications'' is an academic journal in mathematics, specializing in mathematical analysis and related topics in applied mathematics. It was founded in 1960 by Richard Bellman, as part of a series of new ...

Biography

Bellman was born in 1920 in

New York City New York, often called New York City (NYC), is the most populous city in the United States, located at the southern tip of New York State on one of the world's largest natural harbors. The city comprises five boroughs, each coextensive w ...

to non-practising Jewish parents of Polish and Russian descent, Pearl (née Saffian) and John James Bellman, who ran a small grocery store on Bergen Street near Prospect Park, Brooklyn. He was an

atheist Atheism, in the broadest sense, is an absence of belief in the existence of deities. Less broadly, atheism is a rejection of the belief that any deities exist. In an even narrower sense, atheism is specifically the position that there no ...

. He attended Abraham Lincoln High School, Brooklyn in 1937,Salvador Sanabria
Richard Bellman profile at http://www-math.cudenver.edu
retrieved October 3, 2008. and studied

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

Brooklyn College Brooklyn College is a public university in Brooklyn in New York City, United States. It is part of the City University of New York system and enrolls nearly 14,000 students on a campus in the Midwood and Flatbush sections of Brooklyn as of fall ...

where he earned a BA in 1941. He later earned an MA from the

University of Wisconsin A university () is an institution of tertiary education and research which awards academic degrees in several academic disciplines. ''University'' is derived from the Latin phrase , which roughly means "community of teachers and scholars". Uni ...

. During

World War II World War II or the Second World War (1 September 1939 – 2 September 1945) was a World war, global conflict between two coalitions: the Allies of World War II, Allies and the Axis powers. World War II by country, Nearly all of the wo ...

, he worked for a

Theoretical Physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental p ...

Division group in Los Alamos. In 1946, he received his Ph.D. at

Princeton University Princeton University is a private university, private Ivy League research university in Princeton, New Jersey, United States. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial ...

under the supervision of

Solomon Lefschetz Solomon Lefschetz (; 3 September 1884 – 5 October 1972) was a Russian-born American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear ordinary differential equatio ...

. Beginning in 1949, Bellman worked for many years at

RAND corporation The RAND Corporation, doing business as RAND, is an American nonprofit global policy think tank, research institute, and public sector consulting firm. RAND engages in research and development (R&D) in several fields and industries. Since the ...

, and it was during this time that he developed dynamic programming. Later in life, Richard Bellman's interests began to emphasize biology and medicine, which he identified as "the frontiers of contemporary science". In 1967, he became founding editor of the journal '' Mathematical Biosciences'', which rapidly became (and remains) one of the most important journals in the field of Mathematical Biology. In 1985, the Bellman Prize in Mathematical Biosciences was created in his honor, being awarded biannually to the journal's best research paper. Bellman was diagnosed with a brain tumor in 1973, which was removed but resulted in complications that left him severely disabled. He was a professor at the

University of Southern California The University of Southern California (USC, SC, or Southern Cal) is a Private university, private research university in Los Angeles, California, United States. Founded in 1880 by Robert M. Widney, it is the oldest private research university in ...

, a Fellow in the

American Academy of Arts and Sciences The American Academy of Arts and Sciences (The Academy) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, and other ...

(1975), a member of the

National Academy of Engineering The National Academy of Engineering (NAE) is an American Nonprofit organization, nonprofit, NGO, non-governmental organization. It is part of the National Academies of Sciences, Engineering, and Medicine (NASEM), along with the National Academ ...

(1977), and a member of the National Academy of Sciences (1983). He was awarded the

IEEE Medal of Honor The IEEE Medal of Honor is the highest recognition of the American Institute of Electrical and Electronics Engineers (IEEE). It has been awarded since 1917, and is presented to an individual or team of up to three who have made exceptional contri ...

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 A Bellman equation, named after Richard E. Bellman, is a necessary condition for optimality associated with the mathematical Optimization (mathematics), optimization method known as dynamic programming. It writes the "value" of a decision problem ...

Work

Bellman equation

, also known as the ''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 Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criteria, from some set of available alternatives. It is generally divided into two subfiel ...

can also be solved by analyzing the appropriate Bellman equation. The Bellman equation was first applied to engineering

control theory Control theory is a field of control engineering and applied mathematics that deals with the control system, control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the applic ...

and to other topics in applied mathematics, and subsequently became an important tool in

economic theory Economics () is a behavioral science that studies the production, distribution, and consumption of goods and services. Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics anal ...

Hamilton–Jacobi–Bellman equation

The Hamilton–Jacobi–Bellman equation (HJB) is a

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to ho ...

which is central to

optimal control Optimal control theory is a branch of control theory that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering and operations ...

theory. The solution of the HJB equation is the 'value function', which gives the optimal cost-to-go for a given

dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...

with an associated cost function. Classical variational problems, for example, the

brachistochrone problem In physics and mathematics, a brachistochrone curve (), or curve of fastest descent, is the one lying on the plane between a point ''A'' and a lower point ''B'', where ''B'' is not directly below ''A'', on which a bead slides frictionlessly under ...

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

. In continuous time, the result can be seen as an extension of earlier work in

classical physics Classical physics refers to physics theories that are non-quantum or both non-quantum and non-relativistic, depending on the context. In historical discussions, ''classical physics'' refers to pre-1900 physics, while '' modern physics'' refers to ...

on the

Hamilton–Jacobi equation In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mecha ...

William Rowan Hamilton Sir William Rowan Hamilton (4 August 1805 – 2 September 1865) was an Irish astronomer, mathematician, and physicist who made numerous major contributions to abstract algebra, classical mechanics, and optics. His theoretical works and mathema ...

and Carl Gustav Jacob Jacobi.

Curse of dimensionality

The ''curse of dimensionality'' is an expression coined by Bellman to describe the problem caused by the exponential increase in

volume Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch) ...

associated with adding extra dimensions to a (mathematical) space. One implication of the curse of dimensionality is that some methods for numerical solution of the Bellman equation require vastly more computer time when there are more state variables in the value function. For example, 100 evenly spaced sample points suffice to sample a

unit interval In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysi ...

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.)

Bellman–Ford algorithm

Though discovering the algorithm after Ford he is referred to in the

Bellman–Ford algorithm The Bellman–Ford algorithm is an algorithm that computes shortest paths from a single source vertex (graph theory), vertex to all of the other vertices in a weighted digraph. It is slower than Dijkstra's algorithm for the same problem, but more ...

, also sometimes referred to as the Label Correcting Algorithm, computes single-source shortest paths in a weighted digraph where some of the

edge Edge or EDGE may refer to: Technology Computing * Edge computing, a network load-balancing system * Edge device, an entry point to a computer network * Adobe Edge, a graphical development application * Microsoft Edge, a web browser developed by ...

weights may be negative.

Dijkstra's algorithm Dijkstra's algorithm ( ) is an algorithm for finding the shortest paths between nodes in a weighted graph, which may represent, for example, a road network. It was conceived by computer scientist Edsger W. Dijkstra in 1956 and published three ...

accomplishes the same problem with a lower running time, but requires edge weights to be non-negative.

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 brain surgery (Dreyfus, 2003). A selection: * 1957. ''Dynamic Programming'' * 1959. ''Asymptotic Behavior of Solutions of Differential Equations'' * 1961. ''An Introduction to Inequalities'' * 1961. ''Adaptive Control Processes: A Guided Tour'' * 1962. ''Applied Dynamic Programming'' * 1967. ''Introduction to the Mathematical Theory of Control Processes'' * 1970. ''Algorithms, Graphs and Computers'' * 1972. ''Dynamic Programming and Partial Differential Equations'' * 1982. ''Mathematical Aspects of Scheduling and Applications'' * 1983. ''Mathematical Methods in Medicine'' * 1984. ''Partial Differential Equations'' * 1984. ''Eye of the Hurricane: An Autobiography,'' World Scientific Publishing. * 1985. ''Artificial Intelligence'' * 1995. ''Modern Elementary Differential Equations'' * 1997. ''Introduction to Matrix Analysis'' * 2003. ''Dynamic Programming'' * 2003. ''Perturbation Techniques in Mathematics, Engineering and Physics'' * 2003. ''Stability Theory of Differential Equations'' (originally publ. 1953)

References

External links

*
Harold J. Kushner's speech on Richard Bellman, when accepting the Richard E. Bellman Control Heritage Award
(click on "2004: Harold J. Kushner")
IEEE biography
*
Author profile
in the database

zbMATH zbMATH Open, formerly Zentralblatt MATH, is a major reviewing service providing reviews and abstracts for articles in pure and applied mathematics, produced by the Berlin office of FIZ Karlsruhe – Leibniz Institute for Information Infrastru ...

Biography of Richard Bellman
from the Institute for Operations Research and the Management Sciences (INFORMS) {{DEFAULTSORT:Bellman, Richard 1920 births 1984 deaths 20th-century American mathematicians Jewish American atheists American atheists American people of Polish-Jewish descent American people of Russian-Jewish descent Brooklyn College alumni American applied mathematicians American control theorists American game theorists Fellows of the American Academy of Arts and Sciences Members of the United States National Academy of Engineering IEEE Medal of Honor recipients John von Neumann Theory Prize winners Princeton University alumni University of Wisconsin–Madison alumni University of Southern California faculty Richard E. Bellman Control Heritage Award recipients Abraham Lincoln High School (Brooklyn) alumni Mathematicians from New York (state)