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 Dynamic programming is both a mathematical optimization method and a computer programming method. The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering to economics. I ...

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

Biography

Bellman was born in 1920 in

New York City New York, often called New York City or NYC, is the List of United States cities by population, most populous city in the United States. With a 2020 population of 8,804,190 distributed over , New York City is also the L ...

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 Prospect Park is an urban park in Brooklyn, New York City. The park is situated between the neighborhoods of Park Slope, Prospect Heights, Prospect Lefferts Gardens, Flatbush, and Windsor Terrace, and is adjacent to the Brooklyn Museum, Gran ...

. On his religious views, 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 an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

Brooklyn College Brooklyn College is a public university in Brooklyn, Brooklyn, New York. It is part of the City University of New York system and enrolls about 15,000 undergraduate and 2,800 graduate students on a 35-acre campus. Being New York City's first publ ...

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

University of Wisconsin A university () is an institution of higher (or tertiary) education and research which awards academic degrees in several academic disciplines. Universities typically offer both undergraduate and postgraduate programs. In the United States, t ...

. During

World War II World War II or the Second World War, often abbreviated as WWII or WW2, was a world war that lasted from 1939 to 1945. It involved the vast majority of the world's countries—including all of the great powers—forming two opposin ...

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 natural phenomena. This is in contrast to experimental physics, which uses experim ...

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

Princeton Princeton University is a private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the nine ...

under the supervision of

Solomon Lefschetz Solomon Lefschetz (russian: Соломо́н Ле́фшец; 3 September 1884 – 5 October 1972) was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear o ...

. Beginning 1949 Bellman worked for many years at

RAND corporation The RAND Corporation (from the phrase "research and development") is an American nonprofit global policy think tank created in 1948 by Douglas Aircraft Company to offer research and analysis to the United States Armed Forces. It is financed ...

and it was during this time that he developed

. 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 C ...

, a Fellow in the

American Academy of Arts and Sciences The American Academy of Arts and Sciences (abbreviation: AAA&S) 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 ...

(1975), a member of the

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

(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 Institute of Electrical and Electronics Engineers (IEEE). It has been awarded since 1917, when its first recipient was Major Edwin H. Armstrong. It is given for an exceptional contribution ...

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 method known as dynamic programming. It writes the "value" of a decision problem at a certain point in time ...

Work

Bellman equation

, also known as a ''dynamic programming equation'', is a necessary condition for optimality associated with the mathematical optimization method known as

. 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 criterion, from some set of available alternatives. It is generally divided into two subfi ...

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 mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...

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

economic theory Economics () is the social 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 analyzes ...

Hamilton–Jacobi–Bellman equation

The

Hamilton–Jacobi–Bellman equation In optimal control theory, the Hamilton-Jacobi-Bellman (HJB) equation gives a necessary and sufficient condition for optimality of a control with respect to a loss function. It is, in general, a nonlinear partial differential equation in the value ...

(HJB) is a

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...

which is central to

optimal control Optimal control theory is a branch of mathematical optimization 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 ...

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. Examples include the mathematical models that describe the swinging of a ...

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 unde ...

can be solved using this method as well. The equation is a result of the theory of

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 is a group of physics theories that predate modern, more complete, or more widely applicable theories. If a currently accepted theory is considered to be modern, and its introduction represented a major paradigm shift, then the ...

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 mechan ...

William Rowan Hamilton Sir William Rowan Hamilton Doctor of Law, LL.D, Doctor of Civil Law, DCL, Royal Irish Academy, MRIA, Royal Astronomical Society#Fellow, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the ...

and

Carl Gustav Jacob Jacobi Carl Gustav Jacob Jacobi (; ; 10 December 1804 – 18 February 1851) was a German mathematician who made fundamental contributions to elliptic functions, dynamics, differential equations, determinants, and number theory. His name is occasiona ...

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 occupied 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). The de ...

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 analysis, ...

with no more than 0.01 distance between points; an equivalent sampling of a 10-dimensional

unit hypercube A unit cube, more formally a cube of side 1, is a cube whose sides are 1 unit long.. See in particulap. 671. The volume of a 3-dimensional unit cube is 1 cubic unit, and its total surface area is 6 square units.. Unit hypercube The term ' ...

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 to all of the other vertices in a weighted digraph. It is slower than Dijkstra's algorithm for the same problem, but more versatile, as it is ...

, also sometimes referred to as the Label Correcting Algorithm, computes single-source shortest paths in a

weighted digraph In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a Graph (discrete mathematics), graph that is made up of a set of Vertex (graph theory), vertices connected by directed Edge (graph theory), edges, often call ...

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 graph, which may represent, for example, road networks. It was conceived by computer scientist Edsger W. Dijkstra in 1956 and published three years ...

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 mathematics, pure and applied mathematics, produced by the Berlin office of FIZ Karlsruhe – Leibniz Institute for Informa ...

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 people of Polish-Jewish descent American people of Russian-Jewish descent Brooklyn College alumni Applied mathematicians Control theorists 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)