Discrete mathematics is the study of mathematical structures that can be considered "discrete" rather than "continuous". Objects studied in discrete mathematics include integers, graphs, and statements in logic. By contrast, discrete mathematics excludes topics in "continuous mathematics" such as real numbers, calculus or Euclidean geometry. Discrete objects can often be enumerated by integers; more formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets. However, there is no exact definition of the term "discrete 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 with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline.
In mathematics and computer science, the Krohn–Rhodes theory is an approach to the study of finite semigroups and automata that seeks to decompose them in terms of elementary components. These components correspond to finite aperiodic semigroups and finite simple groups that are combined in a feedback-free manner.
Logic in computer science covers the overlap between the field of logic and that of computer science. The topic can essentially be divided into three main areas:
Computational science, also known as scientific computing, technical computing or scientific computation (SC), is an area of science that uses advanced computing capabilities to understand and solve complex physical problems. This includes
Calyampudi Radhakrishna Rao,, commonly known as C. R. Rao, is an Indian-American mathematician and statistician. He is currently professor emeritus at Pennsylvania State University and Research Professor at the University at Buffalo. Rao has been honoured by numerous colloquia, honorary degrees, and festschrifts and was awarded the US National Medal of Science in 2002. The American Statistical Association has described him as "a living legend whose work has influenced not just statistics, but has had far reaching implications for fields as varied as economics, genetics, anthropology, geology, national planning, demography, biometry, and medicine." The Times of India listed Rao as one of the top 10 Indian scientists of all time. In 2023, Rao was awarded the International Prize in Statistics, an award often touted as the "statistics’ equivalent of the Nobel Prize". Rao is also a Senior Policy and Statistics advisor for the Indian Heart Association non-profit focused on raising South Asian cardiovascular disease awareness.
Numerical methods for partial differential equations is the branch of numerical analysis that studies the numerical solution of partial differential equations (PDEs).
In mathematics, in particular numerical analysis, the FETI method is an iterative substructuring method for solving systems of linear equations from the finite element method for the solution of elliptic partial differential equations, in particular in computational mechanics In each iteration, FETI requires the solution of a Neumann problem in each substructure and the solution of a coarse problem. The simplest version of FETI with no preconditioner in the substructure is scalable with the number of substructures but the condition number grows polynomially with the number of elements per substructure. FETI with a preconditioner consisting of the solution of a Dirichlet problem in each substructure is scalable with the number of substructures and its condition number grows only polylogarithmically with the number of elements per substructure. The coarse space in FETI consists of the nullspace on each substructure.
In numerical analysis, BDDC (balancing domain decomposition by constraints) is a domain decomposition method for solving large symmetric, positive definite systems of linear equations that arise from the finite element method. BDDC is used as a preconditioner to the conjugate gradient method. A specific version of BDDC is characterized by the choice of coarse degrees of freedom, which can be values at the corners of the subdomains, or averages over the edges or the faces of the interface between the subdomains. One application of the BDDC preconditioner then combines the solution of local problems on each subdomains with the solution of a global coarse problem with the coarse degrees of freedom as the unknowns. The local problems on different subdomains are completely independent of each other, so the method is suitable for parallel computing. With a proper choice of the coarse degrees of freedom (corners in 2D, corners plus edges or corners plus faces in 3D) and with regular subdomain shapes, the condition number of the method is bounded when increasing the number of subdomains, and it grows only very slowly with the number of elements per subdomain. Thus the number of iterations is bounded in the same way, and the method scales well with the problem size and the number of subdomains.
In mathematics, numerical analysis, and numerical partial differential equations, domain decomposition methods solve a boundary value problem by splitting it into smaller boundary value problems on subdomains and iterating to coordinate the solution between adjacent subdomains. A coarse problem with one or few unknowns per subdomain is used to further coordinate the solution between the subdomains globally. The problems on the subdomains are independent, which makes domain decomposition methods suitable for parallel computing. Domain decomposition methods are typically used as preconditioners for Krylov space iterative methods, such as the conjugate gradient method, GMRES, and LOBPCG.
The FETI-DP method is a domain decomposition method that enforces equality of the solution at subdomain interfaces by Lagrange multipliers except at subdomain corners, which remain primal variables. The first mathematical analysis of the method was provided by Mandel and Tezaur. The method was further improved by enforcing the equality of averages across the edges or faces on subdomain interfaces which is important for parallel scalability for 3D problems. FETI-DP is a simplification and a better performing version of FETI. The eigenvalues of FETI-DP are same as those of BDDC, except for the eigenvalue equal to one, and so the performance of FETI-DP and BDDC is essentially same.
In numerical analysis, the balancing domain decomposition method (BDD) is an iterative method to find the solution of a symmetric positive definite system of linear algebraic equations arising from the finite element method. In each iteration, it combines the solution of local problems on non-overlapping subdomains with a coarse problem created from the subdomain nullspaces. BDD requires only solution of subdomain problems rather than access to the matrices of those problems, so it is applicable to situations where only the solution operators are available, such as in oil reservoir simulation by mixed finite elements. In its original formulation, BDD performs well only for 2nd order problems, such elasticity in 2D and 3D. For 4th order problems, such as plate bending, it needs to be modified by adding to the coarse problem special basis functions that enforce continuity of the solution at subdomain corners, which makes it however more expensive. The BDDC method uses the same corner basis functions as, but in an additive rather than multiplicative fashion. The dual counterpart to BDD is FETI, which enforces the equality of the solution between the subdomain by Lagrange multipliers. The base versions of BDD and FETI are not mathematically equivalent, though a special version of FETI designed to be robust for hard problems has the same eigenvalues and thus essentially the same performance as BDD.
In numerical analysis, mortar methods are discretization methods for partial differential equations, which use separate finite element discretization on nonoverlapping subdomains. The meshes on the subdomains do not match on the interface, and the equality of the solution is enforced by Lagrange multipliers, judiciously chosen to preserve the accuracy of the solution. Mortar discretizations lend themselves naturally to the solution by iterative domain decomposition methods such as FETI and balancing domain decomposition In the engineering practice in the finite element method, continuity of solutions between non-matching subdomains is implemented by multiple-point constraints.
Lawrence Jay Rosenblum is an American mathematician, and Program Director for Graphics and Visualization at the National Science Foundation.
Charbel Farhat is the Vivian Church Hoff Professor of Aircraft Structures in the School of Engineering and the inaugural James and Anna Marie Spilker Chair of the Department of Aeronautics and Astronautics, at Stanford University. He is also Professor of Mechanical Engineering, Professor in the Institute for Computational and Mathematical Engineering, and Director of the Stanford-King Abdulaziz City for Science and Technology Center of Excellence for Aeronautics and Astronautics. He currently serves on the Space Technology Industry-Government-University Roundtable.
In numerical analysis, coarse problem is an auxiliary system of equations used in an iterative method for the solution of a given larger system of equations. A coarse problem is basically a version of the same problem at a lower resolution, retaining its essential characteristics, but with fewer variables. The purpose of the coarse problem is to propagate information throughout the whole problem globally.
Jinchao Xu is an American-Chinese mathematician. He is currently the Verne M. Willaman Professor in the Department of Mathematics at the Pennsylvania State University, University Park. He is known for his work on multigrid methods, domain decomposition methods, finite element methods, and more recently deep neural networks.
Yousef Saad is an I.T. Distinguished Professor of Computer Science in the Department of Computer Science and Engineering at the University of Minnesota. He holds the William Norris Chair for Large-Scale Computing since January 2006. He is known for his contributions to the matrix computations, including the iterative methods for solving large sparse linear algebraic systems, eigenvalue problems, and parallel computing. He is listed as an ISI highly cited researcher in mathematics, is the most cited author in the journal Numerical Linear Algebra with Applications, and is the author of the highly cited book Iterative Methods for Sparse Linear Systems. He is a SIAM fellow and a fellow of the AAAS (2011).
Data science is an interdisciplinary academic field that uses statistics, scientific computing, scientific methods, processes, algorithms and systems to extract or extrapolate knowledge and insights from noisy, structured, and unstructured data.
Radhia Cousot was a French computer scientist known for inventing abstract interpretation.
