**Contemporary Mathematics**

Volume: 754;
2020;
364 pp;
Softcover

MSC: Primary 11; 16; 35; 41; 65;

**Print ISBN: 978-1-4704-5163-9
Product Code: CONM/754**

List Price: $120.00

AMS Member Price: $96.00

MAA Member Price: $108.00

**Electronic ISBN: 978-1-4704-5637-5
Product Code: CONM/754.E**

List Price: $120.00

AMS Member Price: $96.00

MAA Member Price: $108.00

# 75 Years of Mathematics of Computation

Share this page *Edited by *
*Susanne C. Brenner; Igor Shparlinski; Chi-Wang Shu; Daniel B. Szyld*

The year 2018 marked the 75th anniversary of
the founding of Mathematics of Computation, one of the four primary
research journals published by the American Mathematical Society and
the oldest research journal devoted to computational mathematics. To
celebrate this milestone, the symposium “Celebrating 75 Years of
Mathematics of Computation” was held from November 1–3, 2018, at the
Institute for Computational and Experimental Research in Mathematics
(ICERM), Providence, Rhode Island.

The sixteen papers in this volume, written by the symposium
speakers and editors of the journal, include both survey articles and
new contributions.

On the discrete side, there are four papers covering topics in
computational number theory and computational algebra. On the
continuous side, there are twelve papers covering topics in machine
learning, high dimensional approximations, nonlocal and fractional
elliptic problems, gradient flows, hyperbolic conservation laws,
Maxwell's equations, Stokes's equations, a posteriori error
estimation, and iterative methods. Together they provide a snapshot of
significant achievements in the past quarter century in computational
mathematics and also in important current trends.

#### Readership

Graduate students and research mathematicians interested in computational mathematics.

# Table of Contents

## 75 Years of Mathematics of Computation

- Cover Cover11
- Title page iii4
- Contents v6
- Preface vii8
- Invited presentations at the symposium ix10
- Adaptive low-rank approximations for operator equations: Accuracy control and computational complexity 112
- The Prager–Synge theorem in reconstruction based a posteriori error estimation 4556
- 1. Introduction 4556
- 2. The Prager–Synge theory and its application to error estimates 4758
- 3. The Braess–Schöberl construction 5061
- 4. Equivalence with the residual error estimator 5263
- 5. Optimal convergence rate for 𝜂^{\patchsymbol} 5566
- 6. Optimal convergence rate for 𝜂^{Δ} 6273
- 7. Conclusion 6475
- Acknowledgments 6475
- References 6475

- Linear and nonlinear fractional elliptic problems 6980
- Lattice algorithms for multivariate approximation in periodic spaces with general weight parameters 93104
- Multiscale modeling, homogenization and nonlocal effects: Mathematical and computational issues 115126
- The Stokes complex: A review of exactly divergence–free finite element pairs for incompressible flows 141152
- Finite element methods for Maxwell’s equations 159170
- Partial differential equation regularization for supervised machine learning 177188
- 1. Introduction 177188
- 2. Machine learning problem definition 178189
- 3. Function approximation and regularization 180191
- 4. Kernel methods 181192
- 5. Training and SGD 182193
- 6. Statistical learning theory 183194
- 7. Deep neural networks 185196
- 8. Adversarial attacks 188199
- 9. Regularization in DNN 189200
- 10. PDE regularization interpretation 190201
- 11. Conclusions 193204
- References 193204

- Iterative methods for linear systems of equations: A brief historical journey 197208
- 1. It all started with Gauss 197208
- 2. Solution by relaxation 199210
- 3. Early 20th century 201212
- 4. 1930s–1940s: Southwell 202213
- 5. The SOR era 202213
- 6. A turning point: The Forsythe article 203214
- 7. In brief: Chaotic iterations 204215
- 8. Meanwhile, on the opposite camp 205216
- 9. One-dimensional projection processes 207218
- 10. Krylov methods take off: The CG algorithm 209220
- 11. Krylov methods: The nonsymmetric period 210221
- 12. Present and future 210221
- Acknowledgments 211222
- References 211222

- The IEQ and SAV approaches and their extensions for a class of highly nonlinear gradient flow systems 217228
- A class of bound-preserving high order schemes: The main ideas and recent developments 247258
- BDDC domain decomposition algorithms 261272
- 1. Introduction 261272
- 2. Some problems to be considered 262273
- 3. Finite element subdomains, equivalence classes, and BDDC 263274
- 4. Averaging operators: Muliplicity- and 𝜌-scalings and BDDC deluxe 269280
- 5. Adaptive choice of primal spaces 274285
- 6. Two ways of decreasing the dimension of large primal spaces 275286
- 7. Almost incompressible elasticity 276287
- Acknowledgments 278289
- References 278289

- Computing modular polynomials and isogenies of rank two Drinfeld modules over finite fields 283294
- 1. Introduction and motivation 283294
- 2. Background on Drinfeld modules 285296
- 3. Endomorphism rings and isogeny graphs in rank 2 289300
- 4. Analyzing algorithms for Drinfeld modules 291302
- 5. Computing the 𝑗-function over \fc 292303
- 6. Computing Drinfeld modular polynomials 295306
- 7. Applications: Computing isogeny volcanoes and endomorphism rings 301312
- 8. Computing explicit isogenies 304315
- 9. Computing dual isogenies 307318
- 10. Current and future work 309320
- Acknowledgment 311322
- References 311322

- A new ranking function for polynomial selection in the number field sieve 315326
- A nilpotent quotient algorithm for finitely presented associative ℤ-algebras and its application to integral group rings 327338
- A tale of two omegas 343354
- 1. Introduction 343354
- 2. Some results on 𝜔(𝑛) and Ω(𝑛) 347358
- 3. The parity of Ω(𝑛)-𝜔(𝑛) 349360
- 4. Weak independence and oscillations 351362
- 5. The Dirichlet series ℎ(𝑠)=∑_{𝑛≥1}(-1)^{𝜔(𝑛)}𝑛^{-𝑠} 354365
- 6. Qualitative analysis 356367
- 7. Calculations 358369
- 8. Open problems 360371
- Acknowledgments 361372
- References 362373

- Back Cover Back Cover1378