DOCSLIB.ORG

Algebra Policy in California Great Expectations and Serious Challenges

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Algebra Policy in California Great Expectations and Serious Challenges M A Y 2 0 0 9 REPORT Algebra Policy in California Great Expectations and Serious Challenges HIGHLIGHTS Most California policymakers and educators would likely agree that all students need to take and master algebra—and that they need to do so n California has used its academic content standards and accountability as early as possible in their school careers. policies to push for earlier student Agreement on this goal represents or advanced on the state’s Algebra I test in participation and success in algebra ......2 important progress in a state where—less 2008 than in 2003. Nearly 26,000 more low- n The state has seen large increases than 10 years ago—the notion of algebra income 8th graders did so. However, too in 8th grade algebra participation as a high school graduation requirement many California students still struggle to and proficiency ......................................5 seemed like a radical idea. That said, the get through the Algebra I gateway leading to same Californians are much more divided more rigorous math and science courses in n Test scores, “repeater” data, and regarding when exactly students should high school. Participation for all clearly does the NAEP all provide evidence that take algebra and how to assure them a not translate automatically into success for all. math performance falls well short reasonable chance for success. That divi- Many call for strengthening math of the state’s expectations for many students ......................................6 sion was reflected in strong reactions to a instruction, particularly in the late elemen- State Board of Education decision made tary and middle grades, as the crucial next n Critics are raising hard questions last July—blocked by a California court step in moving California toward its goals. about the state’s algebra policies ruling that is currently under appeal— Some also say this is an opportune moment and pushing for clarification about that would have increased pressure for all for California to consider its mathematics what to teach and when ...................... 11 students to take Algebra I by 8th grade. policies more broadly, including its academic n The debate sheds light on shortcomings Since the late 1990s, state officials have content standards and annual assessments. in California’s teacher credentialing used assessment and accountability poli- Any of these would require investments of policies and a lack of support for cies as powerful levers to encourage schools time and funding. What combination of state professional development ................... 14 to enroll more 8th and 9th grade students policies and school practices would be most in Algebra I courses. In the process, schools effective to help students succeed? What have raised expectations and afforded greater investments in educator capacity would be opportunity to thousands of previously most strategic? These are complex but impor- EdSource thanks the Noyce Foundation for underserved students. Nearly 45,000 more tant issues for California to explore, and underwriting the research, development, and California 8th graders scored proficient they are the focus of this EdSource report. dissemination of this publication. EdSource® is a not-for-profit 501(c)(3) organization established in California in 1977. Independent and impartial, EdSource strives to advance the common good by developing and widely distributing trustworthy, useful information that clarifies complex K–12 education issues and promotes thoughtful decisions about California’s public school system. EDS O U R C E R eport California continues to reach toward earlier student success in algebra When California adopted academic content standards in mathematics in 1997, no one assumed that large-scale changes in math achievement and course-taking—including in algebra success in grade 8— would happen overnight. Early decisions by California policymakers establishing the content standards in math and the 2004 requirement that students complete Algebra I to graduate from high school left many debates for the future. A brief history of the State Board of Education’s algebra decision In July 2008, the California State Board of Education SBE approval, the California Department of Education The board’s motion directed state education officials (SBE) passed a motion—now blocked by a state (CDE) also explored a second option: the development to negotiate with the ED for: court—that called for the state’s Algebra I test of a blueprint for a new grade 8 math test. This test, if n An agreement to make the Algebra I CST the sole to become the “sole test of record” in grade 8 approved by SBE and accepted by the U.S. Department test of record in California for grade 8 math; mathematics for federal accountability purposes. of Education, would replace the General Math CST and be taken by students not yet enrolled in algebra. n A transition period for building the state’s California and federal education policies make It would be based on a subset of the state’s Algebra I capacity to ensure all 8th graders can succeed different assumptions about testing in grade 8 math. standards, excluding more rigorous content such as on the test; and n California policy assumes that the California factoring and quadratic equations. n In the long term, better accommodation of Standards Test (CST) an 8th grader takes in This new CST blueprint sparked some intense California’s education policies under federal law. math is connected to the math course she or he disagreement: takes. Most commonly, 8th graders enrolled in In September 2008, CSBA and ACSA filed a lawsuit— n Algebra I take the Algebra I CST and 8th graders Some—including CDE, the California School later joined by the California Teachers Association not yet enrolled in an algebra course take the Boards Association (CSBA), and the Association (CTA) and State Superintendent of Public Instruction General Mathematics CST. of California School Administrators (ACSA)— Jack O’Connell—to block the state board’s decision. supported the blueprint as an improvement on They claimed that SBE (1) violated the Bagley-Keene n In contrast, the federal No Child Left Behind the General Math CST that would allow local Open Meeting Act by not providing adequate public law (NCLB) assumes that math testing in educators to make judgments about proper notice of the action it took that day; and (2) exceeded grade 8 is based on a state’s content standards student placements. its legal authority by implicitly changing California’s for that grade. n Other state leaders saw the blueprint as lowering academic content standards to require Algebra I in On Feb. 6, 2008, the U.S. Department of Education the state’s academic standards for what counts grade 8, despite the fact that the math standards (ED) notified California that its General Math CST does as grade-level math in grade 8. Three former state for grades 8–12 are organized by discipline rather not comply with federal requirements because it tests secretaries of education and four former SBE than grade level. 8th graders on content defined by California’s math presidents signed a letter arguing that “anything In January 2009, the Sacramento County Superior standards as intended for grades 6 and 7. Failure to less than Algebra I” should not be considered Court ruled in favor of the plaintiffs on both grounds enter an agreement with the ED to align California’s “grade-level proficient at eighth grade.” and issued a preliminary injunction preventing assessment system with federal expectations would One day before the state board was to consider the state board from taking any further steps to result in $1 million in Title I Part A administrative the new blueprint further, in July 2008, Gov. Arnold implement its algebra decision. SBE is appealing funds being withheld from the state and provided Schwarzenegger also went public with his opposition the court’s ruling that the board exceeded its legal instead to California’s local education agencies. (At to it. He sent a letter encouraging the state board to authority but is not disputing the ruling pertaining to the time of this writing, California and ED officials pursue the Algebra I CST as the sole test of record the Bagley-Keene Act. were discussing a settlement of this matter.) in grade 8 math for federal accountability purposes. One option for complying with federal expectations By a vote of 8-1, the board passed a motion to was to have all 8th graders take the Algebra I CST. With this effect. n Algebra Policy in California n May 2009 © Copyright 2009 by EdSource, Inc. EDS O U R C E R eport A July 2008 decision by the State Board of figure 1 How California’s academic content standards in mathematics are organized in Education (SBE) has raised issues not re- grades K–7 versus grades 8–12 solved 10 years ago. This decision—blocked In grades K–7, the mathematics content In grades 8–12, the mathematics content by a Sacramento County court ruling that is standards were established for each grade level, standards were established for nine specialized currently under appeal—directed the state using five common strands: math disciplines: to enter an agreement with the U.S. Depart- ment of Education (ED) to make the Algebra I n Number sense n Algebra I California Standards Test (CST) the “sole n Algebra and functions n Geometry test of record” in grade 8 math for federal n Measurement and geometry n Algebra II accountability purposes. Depending on the n Statistics, data analysis, and probability n Probability and Statistics kind
Load more

Recommended publications
  • Randomized Linear Algebra for Model Order Reduction Oleg Balabanov
    Randomized linear algebra for model order reduction Oleg Balabanov To cite this version: Oleg Balabanov. Randomized linear algebra for model order reduction. Numerical Analysis [math.NA]. École centrale de Nantes; Universitat politécnica de Catalunya, 2019. English. NNT : 2019ECDN0034. tel-02444461v2 HAL Id: tel-02444461 https://tel.archives-ouvertes.fr/tel-02444461v2 Submitted on 14 Feb 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. THÈSE DE DOCTORAT DE École Centrale de Nantes COMUE UNIVERSITÉ BRETAGNE LOIRE ÉCOLE DOCTORALE N° 601 Mathématiques et Sciences et Technologies de l’Information et de la Communication Spécialité : Mathématiques et leurs Interactions Par Oleg BALABANOV Randomized linear algebra for model order reduction Thèse présentée et soutenue à l’École Centrale de Nantes, le 11 Octobre, 2019 Unité de recherche : Laboratoire de Mathématiques Jean Leray, UMR CNRS 6629 Thèse N° : Rapporteurs avant soutenance : Bernard HAASDONK Professeur, University of Stuttgart Tony LELIÈVRE Professeur, École des Ponts ParisTech Composition du Jury : Président : Albert COHEN Professeur, Sorbonne Université Examinateurs : Christophe PRUD’HOMME Professeur, Université de Strasbourg Laura GRIGORI Directeur de recherche, Inria Paris, Sorbonne Université Marie BILLAUD-FRIESS Maître de Conférences, École Centrale de Nantes Dir.
    [Show full text]
  • The Limits of Depth Reduction for Arithmetic Formulas: It’S All About the Top Fan-In
    The Limits of Depth Reduction for Arithmetic Formulas: It's all about the top fan-in Mrinal Kumar∗ Shubhangi Sarafy Abstract In recent years, a very exciting and promising method for proving lower bounds for arithmetic circuits has been proposed. This method combines the method of depth reduction developed in the works of Agrawal-Vinay [AV08], Koiran [Koi12] and Tavenas [Tav13], and the use of the shifted partial derivative complexity measure developed in the works of Kayal [Kay12] and Gupta et al [GKKS13a]. These results inspired a flurry of other beautiful results and strong lower bounds for various classes of arithmetic circuits, in particular a recent work of Kayal et al [KSS13] showing superpolynomial lower bounds for regular arithmetic formulas via an improved depth reduction for these formulas. It was left as an intriguing question if these methods could prove superpolynomial lower bounds for general (homogeneous) arithmetic formulas, and if so this would indeed be a breakthrough in arithmetic circuit complexity. In this paper we study the power and limitations of depth reduction and shifted partial derivatives for arithmetic formulas. We do it via studying the class of depth 4 homogeneous arithmetic circuits. We show: (1) the first superpolynomial lower bounds for the class of homoge- neous depth 4 circuits with top fan-in o(log n). The core of our result is to show improved depth reduction for these circuits. This class of circuits has received much attention for the problem of polynomial identity testing. We give the first nontrivial lower bounds for these circuits for any top fan-in ≥ 2.
    [Show full text]
  • History of Algebra
    History of Algebra The term algebra usually denotes various kinds of mathematical ideas and techniques, more or less directly associated with formal manipulation of abstract symbols and/or with finding the solutions of an equation. The notion that in mathematics there is such a sepa- rate sub-discipline, as well as the very use of the term “algebra” to denote it, are them- selves the outcome of historical evolution of ideas. The ideas to be discussed in this article are sometimes put under the same heading due to historical circumstances no less than to any “essential” mathematical reason. Part I. The long way towards the idea of “equation” Simple and natural as the notion of “equation” may appear now, it involves a great amount of mutually interacting, individual mathematical notions, each of which was the outcome of a long and intricate historical process. Not before the work of Viète, in the late sixteenth century, do we actually find a fully consolidated idea of an equation in the sense of a sin- gle mathematical entity comprising two sides on which operations can be simultaneously performed. By performing such operations the equation itself remains unchanged, but we are led to discovering the value of the unknown quantities appearing in it. Three main threads in the process leading to this consolidation deserve special attention here: (1) attempts to deal with problems devoted to finding the values of one or more unknown quantities. In Part I, the word “equation” is used in this context as a short-hand to denote all such problems,
    [Show full text]
  • Gottlob Frege's Place in the History of Mathematics
    University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Transactions of the Nebraska Academy of Sciences and Affiliated Societies Nebraska Academy of Sciences 1976 Gottlob Frege's Place In The History Of Mathematics Eckehart Kohler University of Nebraska-Lincoln Follow this and additional works at: https://digitalcommons.unl.edu/tnas Kohler, Eckehart, "Gottlob Frege's Place In The History Of Mathematics" (1976). Transactions of the Nebraska Academy of Sciences and Affiliated Societies. 405. https://digitalcommons.unl.edu/tnas/405 This Article is brought to you for free and open access by the Nebraska Academy of Sciences at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Transactions of the Nebraska Academy of Sciences and Affiliated Societiesy b an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. HISTORY AND PHILOSOPHY OF SCIENCE GOTTLOB FREGE'S PLACE IN THE HISTORY OF MATHEMATICS ECKEHART KOHLER Dept. of Philosophy University of Nebraska-Lincoln 68588 The frontiers of mathematics lie in two directions: higher towards more and more sophisticated and complicated constructions based on accepted principles and methods; and lower towards deeper and deeper foundations of the superstructure of accepted mathematics. Frege was a great pioneer on the latter frontier and single-handedly opened up a deep and rich area of research from which mathematicians and philosophers still profit. In this paper I will describe Frege's contributions to the foundations of number theory and analysis and then discuss the tenability of his program. In a nutshell, Frege's principle goal was to reduce mathematics to logic. His greatest obstacle lay in the weakness of the logical systems prevalent in his day, consisting as they did mainly of syllogistics and smatterings of propositional and modal logics.
    [Show full text]
  • Arithmetic, Set Theory, Reduction and Explanation
    Arithmetic, Set Theory, Reduction and Explanation William D’Alessandro May 20, 2017 Forthcoming in Synthese. Abstract Philosophers of science since Nagel have been interested in the links between intertheoretic reduc- tion and explanation, understanding and other forms of epistemic progress. Although intertheoretic reduction is widely agreed to occur in pure mathematics as well as empirical science, the relation- ship between reduction and explanation in the mathematical setting has rarely been investigated in a similarly serious way. This paper examines an important and well-known case: the reduction of arithmetic to set theory. I claim that the reduction is unexplanatory. In defense of this claim, I offer some evidence from mathematical practice, and I respond to contrary suggestions due to Steinhart, Maddy, Kitcher and Quine. I then show how, even if set-theoretic reductions are gener- ally not explanatory, set theory can nevertheless serve as a legitimate and successful foundation for mathematics. Finally, some implications of my thesis for philosophy of mathematics and philosophy of science are discussed. In particular, I suggest that some reductions in mathematics are proba- bly explanatory, and I propose that differing standards of theory acceptance might account for the apparent lack of unexplanatory reductions in the empirical sciences. [I]n the philosophy of science the notions of explanation and reduction have been extensively discussed, even in formal frameworks, but there exist few successful and exact applications of the notions to actual theories, and, furthermore, any two philosophers of science seem to think differently about the question of how the notions should be reconstructed. On the other hand, philosophers of mathematics and mathematicians have been successful in defining and applying various exact notions of reduction (or interpretation), but they have not seriously studied the questions of explanation and understanding.1 1[Rantala 1992], 47.
    [Show full text]
  • The History of Mathematics: Alternative Perspectives
    Copyrighted Material Chapter One The History of Mathematics: Alternative Perspectives A Justification for This Book An interest in history marks us for life. How we see ourselves and others is shaped by the history we absorb, not only in the classroom but also from the Internet, films, newspapers, television programs, novels, and even strip cartoons. From the time we first become aware of the past, it can fire our imagination and excite our curiosity: we ask questions and then seek an- swers from history. As our knowledge develops, differences in historical perspectives emerge. And, to the extent that different views of the past affect our perception of ourselves and of the outside world, history becomes an important point of reference in understanding the clash of cultures and of ideas. Not surprisingly, rulers throughout history have recognized that to control the past is to master the present and thereby consolidate their power. During the last four hundred years, Europe and its cultural dependen- cies1 have played a dominant role in world affairs. This is all too often reflected in the character of some of the historical writing produced by Eu- ropeans in the past. Where other people appear, they do so in a transitory fashion whenever Europe has chanced in their direction. Thus the history of the Africans or the indigenous peoples of the Americas often appears to begin only after their encounter with Europe. An important aspect of this Eurocentric approach to history is the man- ner in which the history and potentialities of non-European societies were represented, particularly with respect to their creation and development of science and technology.
    [Show full text]
  • Key Moments in the History of Numerical Analysis
    Key Moments in the History of Numerical Analysis Michele Benzi Department of Mathematics and Computer Science Emory University Atlanta, GA http://www.mathcs.emory.edu/ benzi ∼ Outline Part I: Broad historical outline • 1. The prehistory of Numerical Analysis 2. Key moments of 20th Century Numerical Analysis Part II: The early history of matrix iterations • 1. Iterative methods prior to about 1930 2. Mauro Picone and Italian applied mathematics in the Thirties 3. Lamberto Cesari's work on iterative methods 4. Gianfranco Cimmino and his method 5. Cimmino's legacy PART I Broad historical outline The prehistory of Numerical Analysis In contrast to more classical fields of mathematics, like Analysis, Number Theory or Algebraic Geometry, Numerical Analysis (NA) became an independent mathematical disci- pline only in the course of the 20th Century. This is perhaps surprising, given that effective methods of computing approximate numerical solutions to mathematical problems are already found in antiquity (well before Euclid!), and were especially prevalent in ancient India and China. While algorithmic mathematics thrived in ancient Asia, classical Greek mathematicians showed relatively little interest in it and cultivated Ge- ometry instead. Nevertheless, Archimedes (3rd Century BCE) was a master calculator. The prehistory of Numerical Analysis (cont.) Many numerical methods studied in introductory NA courses bear the name of great mathematicians including Newton, Euler, Lagrange, Gauss, Jacobi, Fourier, Chebyshev, and so forth. However, it should be kept in mind that well into the 19th Century, the distinction between mathematics and natural philosophy (including physics, chemistry, astronomy etc.) was almost non-existent. Scientific specialization is a modern phenomenon, and nearly all major mathematicians were also physicists and astronomers.
    [Show full text]
  • History of Algebraic Ideas and Research on Educational Algebra
    History of Algebraic Ideas and Research on Educational Algebra Luis Puig Departmento de Didáctica de la Matemática Universidad de Valencia (España) 0. INTRODUCTION The history of algebraic ideas is nowadays frequently used on research on educational algebra. As we have stated in Puig and Rojano (2004), there is no need to further discuss the necessity or usefulness of studying the history of mathematics for mathematics education. ICMI established a group more than 20 years ago to study the relationship between the history and pedagogy of mathematics, and during the last few years important publications have appeared that summarise the work done outside and inside this group, especially the ICMI Study History in Mathematics Education (Fauvel & van Maanen, 2000), but also Katz (2000), and Jahnke, Knoche, and Otte (1996). What we are going to present here is a way of using history in educational mathematics pioneered by Eugenio Filloy some 25 years ago 1. I have been working with him since some 15 years2, so my work owes him the starting idea and is tangled with his. All good ideas comes from him, the mistakes are my own responsibility. 1. FEATURES OF OUR USE OF HISTORY Our use of history has two fundamental features. On the one hand, it is concerned with an analysis of algebraic ideas. As a result, there is very little interest for us in, for example, questions of dating or of priority in developing the concepts of algebra. What we are basically interested in is identifying the algebraic ideas that are brought into play in a specific text and the evolution of those ideas, which can be seen by comparing texts; in this context we can consider historical texts as cognitions and analyse them as we analyse the performance of pupils, whose output also constitutes mathematical texts.
    [Show full text]
  • HISTORY of MATH CONCEPTS Essentials, Images and Memes
    HISTORY OF MATH CONCEPTS Essentials, Images and Memes Sergey Finashin Modern Mathematics having roots in ancient Egypt and Babylonia, really flourished in ancient Greece. It is remarkable in Arithmetic (Number theory) and Deductive Geometry. Mathematics written in ancient Greek was translated into Arabic, together with some mathematics of India. Mathematicians of Islamic Middle East significantly developed Algebra. Later some of this mathematics was translated into Latin and became the mathematics of Western Europe. Over a period of several hundred years, it became the mathematics of the world. Some significant mathematics was also developed in other regions, such as China, southern India, and other places, but it had no such a great influence on the international mathematics. The most significant for development in mathematics was giving it firm logical foundations in ancient Greece which was culminated in Euclid’s Elements, a masterpiece establishing standards of rigorous presentation of proofs that influenced mathematics for many centuries till 19th. Content 1. Prehistory: from primitive counting to numeral systems 2. Archaic mathematics in Mesopotamia (Babylonia) and Egypt 3. Birth of Mathematics as a deductive science in Greece: Thales and Pythagoras 4. Important developments of ideas in the classical period, paradoxes of Zeno 5. Academy of Plato and his circle, development of Logic by Aristotle 6. Hellenistic Golden Age period, Euclid of Alexandria 7. Euclid’s Elements and its role in the history of Mathematics 8. Archimedes, Eratosthenes 9. Curves in the Greek Geometry, Apollonius the Great Geometer 10. Trigonometry and astronomy: Hipparchus and Ptolemy 11. Mathematics in the late Hellenistic period 12. Mathematics in China and India 13.
    [Show full text]
  • Notes on Mathematics - 1021
    Notes on Mathematics - 1021 Peeyush Chandra, A. K. Lal, V. Raghavendra, G. Santhanam 1Supported by a grant from MHRD 2 Contents I Linear Algebra 7 1 Matrices 9 1.1 DefinitionofaMatrix .................................. .... 9 1.1.1 SpecialMatrices ................................... .. 10 1.2 OperationsonMatrices ............................... ...... 10 1.2.1 Multiplication of Matrices . .. 12 1.3 SomeMoreSpecialMatrices.. .. .. .. .. .. .. .. .. .. .. .. ....... 13 1.3.1 SubmatrixofaMatrix................................ .. 14 1.3.1 BlockMatrices ..................................... 15 1.4 MatricesoverComplexNumbers . ....... 17 2 Linear System of Equations 19 2.1 Introduction...................................... ...... 19 2.2 Definition and a Solution Method . ...... 20 2.2.1 ASolutionMethod................................... 21 2.3 RowOperationsandEquivalentSystems. ......... 21 2.3.1 Gauss Elimination Method . 24 2.4 RowReducedEchelonFormofaMatrix . ....... 26 2.4.1 Gauss-Jordan Elimination . ... 27 2.4.2 ElementaryMatrices................................ ... 29 2.5 RankofaMatrix..................................... .... 30 2.6 Existence of Solution of Ax = b ................................. 33 2.6.1 Example.......................................... 33 2.6.2 MainTheorem ...................................... 34 2.6.3 Exercises ........................................ 35 2.7 InvertibleMatrices .................................. ...... 35 2.7.1 InverseofaMatrix................................. ... 35 2.7.2 Equivalent conditions for
    [Show full text]
  • Math 2331 – Linear Algebra 1.2 Row Reduction and Echelon Forms
    1.2 Echelon Forms Math 2331 { Linear Algebra 1.2 Row Reduction and Echelon Forms Jiwen He Department of Mathematics, University of Houston [email protected] math.uh.edu/∼jiwenhe/math2331 Jiwen He, University of Houston Math 2331, Linear Algebra 1 / 19 1.2 Echelon Forms Definition Reduction Solution Theorem 1.2 Row Reduction and Echelon Forms Echelon Form and Reduced Echelon Form Uniqueness of the Reduced Echelon Form Pivot and Pivot Column Row Reduction Algorithm Reduce to Echelon Form (Forward Phase) then to REF (Backward Phase) Solutions of Linear Systems Basic Variables and Free Variable Parametric Descriptions of Solution Sets Final Steps in Solving a Consistent Linear System Back-Substitution General Solutions Existence and Uniqueness Theorem Using Row Reduction to Solve Linear Systems Consistency Questions Jiwen He, University of Houston Math 2331, Linear Algebra 2 / 19 1.2 Echelon Forms Definition Reduction Solution Theorem Echelon Forms Echelon Form (or Row Echelon Form) 1 All nonzero rows are above any rows of all zeros. 2 Each leading entry (i.e. left most nonzero entry) of a row is in a column to the right of the leading entry of the row above it. 3 All entries in a column below a leading entry are zero. Examples (Echelon forms) 2 3 2 3 ∗ ∗ ∗ ∗ ∗ ∗ 6 0 ∗ ∗ ∗ 7 6 0 ∗ 7 (a) 6 7 (b) 6 7 4 0 0 0 0 0 5 4 0 0 5 0 0 0 0 0 0 0 0 2 3 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 6 0 0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ 7 6 7 (c) 6 0 0 0 0 ∗ ∗ ∗ ∗ ∗ ∗ 7 6 7 4 0 0 0 0 0 0 0 ∗ ∗ ∗ 5 0 0 0 0 0 0 0 0 ∗ ∗ Jiwen He, University of Houston Math 2331, Linear Algebra 3 / 19 1.2 Echelon Forms Definition Reduction Solution Theorem Reduced Echelon Form Reduced Echelon Form Add the following conditions to conditions 1, 2, and 3 above: 4.
    [Show full text]
  • Randomized Linear Algebra for Model Reduction. Part I: Galerkin Methods and Error Estimation
    Randomized linear algebra for model reduction. Part I: Galerkin methods and error estimation. Oleg Balabanov∗y and Anthony Nouy∗z Abstract We propose a probabilistic way for reducing the cost of classical projection-based model order reduction methods for parameter-dependent linear equations. A reduced order model is here approximated from its random sketch, which is a set of low-dimensional random pro- jections of the reduced approximation space and the spaces of associated residuals. This ap- proach exploits the fact that the residuals associated with approximations in low-dimensional spaces are also contained in low-dimensional spaces. We provide conditions on the dimension of the random sketch for the resulting reduced order model to be quasi-optimal with high probability. Our approach can be used for reducing both complexity and memory require- ments. The provided algorithms are well suited for any modern computational environment. Major operations, except solving linear systems of equations, are embarrassingly parallel. Our version of proper orthogonal decomposition can be computed on multiple workstations with a communication cost independent of the dimension of the full order model. The re- duced order model can even be constructed in a so-called streaming environment, i.e., under extreme memory constraints. In addition, we provide an efficient way for estimating the error of the reduced order model, which is not only more efficient than the classical approach but is also less sensitive to round-off errors. Finally, the methodology is validated on benchmark problems. Keywords| model reduction, reduced basis, proper orthogonal decomposition, random sketching, subspace embedding 1 Introduction arXiv:1803.02602v4 [math.NA] 31 Oct 2019 Projection-based model order reduction (MOR) methods, including the reduced basis (RB) method or proper orthogonal decomposition (POD), are popular approaches for approximating large-scale parameter-dependent equations (see the recent surveys and monographs [8,9, 23, 31]).
    [Show full text]