PPT-Local Linear Approximation

Math 200 Week 6 Friday Math 200 Goals Be able to compute the local linear approximation for a function of two or more variables at a given point Be able to use a local linear approximation to estimate a given quantity. Numerical Solutions of Equations. Dr J Frost (jfrost@tiffin.kingston.sch.uk) . Last modified: 2. nd. January 2014. y. = x. y = . cos. (x). Iterative Methods. Suppose we wanted to find solutions to . Matt Weinberg. MIT .  Princeton  MSR. References: . . http. ://arxiv.org/abs/. 1305.4002. http. ://arxiv.org/abs/. 1405.5940. http. ://arxiv.org/abs/. 1305.4000. Recap. Costis. ’ Talk: . Optimal multi-dimensional mechanism: additive bidders, no constraints. Differentials and Linear Approximation. BC Calculus. Related Rates :. How . fast . is . y . changing as . x . is changing?. -. Differentials:. How . much. does . y . change as . x . changes?. M. Matsui.. Linear Cryptanalysis Method for DES Cipher. . EUROCRYPT 93, 1994. . The first experimental cryptanalysis of the Data Encryption Standard. . CRYPT0 94, 1994. . Linear Approximations. A function with one bit output is a linear function over if output is XOR of input bits.. Alexandr. . Andoni. . (Simons Institute). Robert . Krauthgamer. . (. Weizmann. . Institute). Ilya Razenshteyn . (CSAIL MIT). 1. Sketching. Compress a massive object to a . small. . sketch. Rich theories: . Sometimes we can handle NP problems with polynomial time algorithms which are guaranteed to return a solution within some specific bound of the optimal solution. within a constant . c. . of the optimal. Local algebraic approximations. Variants on Taylor series. Local-Global approximations. Variants on “fudge factor”. Local algebraic approximations. Linear Taylor series. Intervening variables. Transformed approximation. δ. -Timeliness. Carole . Delporte-Gallet. , . LIAFA . UMR 7089. , Paris VII. Stéphane Devismes. , VERIMAG UMR 5104, Grenoble I. Hugues Fauconnier. , . LIAFA . UMR 7089. , Paris VII. LIAFA. Motivation. . for Binary . Pairwise. Energies. Lena . Gorelick. . joint work with. . O. . Veksler. I. Ben . Ayed. A. Delong. . Y. . Boykov. Example of Simple Binary Energy. 2. Potts Model. Binary . Pairwise. in mechanism design. Shuchi Chawla. University of Wisconsin – Madison. FOCS 2012. So far today…. Revenue & Social Welfare. This talk:. Non-linear functions of type & allocation. Question: how well can we optimize in strategic settings?. Local algebraic approximations. Variants on Taylor series. Local-Global approximations. Variants on “fudge factor”. Local algebraic approximations. Linear Taylor series. Intervening variables. Transformed approximation. Grigory. . Yaroslavtsev. . Penn State + AT&T Labs - Research (intern). Joint work with . Berman (PSU). , . Bhattacharyya (MIT). , . Makarychev. (IBM). , . Raskhodnikova. (PSU). Directed. Spanner Problem. How accurate is your estimate?. Differential Notation. The Linear Approximation to . y. = . f. (. x. ) is often written using the “differentials” . dx. and . dy. . In this notation, . dx. is used instead of . Rayleigh-Ritz method approximate solution in the entire beam. Difficult to find good approximate solution (discontinuities in derivatives). Finite element approximates solution in an element. In a small element simple functions are acceptably accurate.