2015 Tsinghua University Chia-Chiao Lin Distinguished Lecture
Speaker: Gilbert Strang
Professor of Massachusetts Institute of Technology
Fellow of American Academy of Arts and Sciences
Member of US National Academy of Sciences
Time: 16:30-17:30, August 6 (Thu.), August 7 (Fri.)
Place: Lecture Hall, Floor 3, Jin Chun Yuan West Building, Tsinghua University
Lecture 1: Linear Algebra Online and Offline
Abstract: At some time in the 20th century I was asked to reorganize two courses—“Linear Algebra”, and “Computational Science and Engineering”. At that time linear algebra was studied by a small group of math majors and the equivalent course to Computational Science and Engineering never saw a computer. This had to change — without losing the mathematical content!
In this talk I will share my experience in redesigning these courses, writing new textbooks, getting the courses online and making it available to MIT students and to the world.
Lecture 2: Banded Matrices and Fast Inverses
Abstract: The inverse of a banded matrix A has a special form with low rank submatrices except at the main diagonal. That form comes directly from the "Nullity Theorem." Then the inverse of that matrix A-1 is the original A ----- which can be found by a remarkable \local" inverse formula. This formula uses only the banded part of A-1 and it oers a very fast algorithm to produce A.
That fast algorithm has a potentially valuable application. Start now with a banded matrix B. (Possibly B is the positive denite beginning of a covariance matrix C ----- but covariances outside the band are unknown or too expensive to compute). It is a poor idea to assume that those unknown covariances are zero. Much better to complete B to C by a rule of maximum entropy ----- for Gaussians this rule means maximum determinant.
As statisticians and also linear algebraists discovered, the optimally completed matrix C is the inverse of a banded matrix. Best of all, the matrix actually needed in computations is that banded C-1 (which is not B !). And C-1 comes quickly and eciently from the local inverse formula.
A very special subset of banded matrices contains those whose inverses are also banded. These arise in studying orthogonal polynomials and also in wavelet theory — the wavelet transform and its inverse are both banded ( = FIR lters). We describe a factorization for all banded matrices that have banded inverses.
Introduction of Speaker
Gilbert Strang is an American mathematician, with contributions to finite element theory, the calculus of variations, wavelet analysis and linear algebra. He was an undergraduate at MIT and a Rhodes Scholar at Balliol College, Oxford. His Ph.D. was from UCLA and since then he has taught at MIT. He has been a Sloan Fellow and a Fairchild Scholar and is a Fellow of the American Academy of Arts and Sciences. He is also an Honorary Fellow of Balliol College, and a member of the National Academy of Sciences.
He has made many contributions to mathematics education, including publishing eleven books. He teaches Introduction to Linear Algebra and Computational Science and Engineering and his lectures are freely available through MIT Open Course Ware.
He was the President of SIAM during 1999 and 2000, and Chair of the Joint Policy Board for Mathematics. He received the von Neumann Medal of the US Association for Computational Mechanics, and the Henrici Prize for applied analysis. The first Su Buchin Prize from the International Congress of Industrial and Applied Mathematics, and the Haimo Prize from the Mathematical Association of America, were awarded for his contributions to teaching around the world.
Source: http://www-math.mit.edu/~gs/bio.html
