International Congress of Mathematicians 2022 July 6–14 Proceedings: Sections 15-20

Front cover
Front matter
Contents
15. Numerical Analysis and Scientific Computing
G. Bao: Inverse scattering problems
1. Introduction
2. Inverse medium problem
2.1. Model problem
2.2. Born approximation
2.3. Recursive linearization
2.4. Numerical experiments
2.5. Stability analysis
3. Inverse source problem
3.1. Model problem
3.2. Uniqueness and stability
3.3. Inverse random source problems
4. Inverse diffraction grating problem
5. Discussions and future directions
References
M. J. Berger and R. J. LeVeque: Towards adaptive simulations of dispersive tsunami propagation from an asteroid impact
1. Introduction
2. Overview of approach
3. Equations and numerical methods
3.1. The shallow water and Boussinesq equations
3.2. Two-dimensional versions
3.3. Numerics
4. Computational results
4.1. Initialization procedure
4.2. Adaptive simulation
4.3. Discussion of results
5. Open problems and future research
References
J. S. Hesthaven, C. Pagliantini, and N. Ripamonti: Structure-preserving model order reduction of Hamiltonian systems
1. Introduction
2. Symplectic geometry and Hamiltonian systems
3. Symplectic Galerkin projection
4. Proper symplectic decomposition
4.1. SVD-based methods for orthonormal symplectic basis generation
4.2. SVD-based methods for nonorthonormal symplectic basis generation
4.3. Greedy approach to symplectic basis generation
5. Dynamical low-rank reduced basis methods for Hamiltonian systems
6. Extensions to more general Hamiltonian problems
6.1. Dissipative Hamiltonian systems
6.2. Noncanonical Hamiltonian systems
7. Conclusion
References
N. J. Higham: Numerical stability of algorithms at extreme scale and low precisions
1. Introduction
2. Blocked algorithms
2.1. Blocked inner products
2.2. Blocked matrix multiplication
2.3. Blocked matrix factorizations
3. Architectural features
3.1. Extended precision registers
3.2. Fused multiply–add
3.3. Mixed precision block fused multiply–add
4. Probabilistic rounding error analysis
4.1. Error analysis for nonrandom data
4.2. Error analysis for random data
4.3. Limitations
5. Other considerations
5.1. Sharpness of error bounds
5.2. Growth factor at low precisions
5.3. Iterative refinement
6. Conclusions
References
G. Kutyniok: The mathematics of artificial intelligence
1. Introduction
1.1. The rise of artificial intelligence
1.2. Impact on mathematics
1.3. Problems of artificial intelligence
1.4. A need for mathematics
1.5. Outline
2. The mathematical setting of artificial intelligence
2.1. Definition of deep neural networks
2.2. Application of a deep neural network
2.3. Relation to a statistical learning problem
2.4. Main research threads
2.4.1. Mathematical foundations for artificial intelligence
2.4.2. Artificial intelligence for mathematical problems
3. Mathematical foundations for artificial intelligence
3.1. Expressivity
3.2. Optimization
3.3. Generalization
3.4. Explainability
4. Artificial intelligence for mathematical problems
4.1. Inverse problems
4.2. Partial differential equations
5. Conclusion: seven mathematical key problems
References
R. Ward: Stochastic gradient descent: where optimization meets machine learning
1. Introduction
1.1. Stochastic gradient descent: Background
1.1.1. Stochastic approximation
1.1.2. The Kaczmarz method for solving linear systems
1.2. Stochastic gradient descent: convergence theory
1.3. Adaptive step-size rules in stochastic gradient descent
1.4. Outlook
References
L. Ying: Solving inverse problems with deep learning
1. Introduction
2. New, mathematically-motivated NN modules
2.1. Pseudodifferential operators
2.2. Fourier integral operators
3. Inverse problems
3.1. Electrical impedance tomography
3.2. Optical tomography
3.3. Inverse acoustic scattering
3.4. Seismic imaging
3.5. Traveltime tomography
4. Concluding remarks
References
16. Control Theory and Optimization – Special lecture
N. Bansal: Discrepancy theory and related algorithms
1. Introduction
1.1. A brief history
1.2. Some examples
1.2.1. Spencer's problem
1.2.2. Beck–Fiala and Komlós problem
1.2.3. Prefix discrepancy
1.2.4. Tusnady's problem
1.2.5. Arithmetic progressions
1.3. Hereditary discrepancy and rounding
2. Classical techniques
2.1. Linear algebraic method
2.2. Partial coloring method
2.2.1. Applications
2.2.2. Proof of the partial coloring lemma
2.3. Banaszczyk method
3. Algorithms for partial coloring
3.1. The SDP-based approach
3.1.1. Algorithm for Theorem 3.1
3.1.2. Algorithmic version of Spencer's result
3.2. The Lovett–Meka algorithm
3.2.1. The algorithm
3.3. Direct approaches
3.3.1. Rothvoss' algorithm
3.3.2. Eldan–Singh algorithm
4. Algorithmic version of Banaszczyks's result
4.1. The Komlós problem
4.1.1. Algorithm
4.1.2. Analysis
4.2. The general setting
4.2.1. Gram–Schmidt walk algorithm
5. Approximating hereditary discrepancy
6. Other recent directions
6.1. Random instances
6.2. Online setting
6.3. Matrix discrepancy
References
16. Control Theory and Optimization
R. S. Burachik: Enlargements: a bridge between maximal monotonicity and convexity
1. Introduction
2. Preliminaries
2.1. Basic facts and tools
2.2. The Fitzpatrick function
3. Enlargements of maximally monotone maps
3.1. The family of enlargements E(T)
3.2. Local boundedness
3.3. Lipschitz continuity
3.4. On graphs, closedness, and convexity
3.5. Brøndsted and Rockafellar property
4. A family of convex functions associated with E(T)
5. A distance induced by H(T)
5.1. Some open problems related to GBDs
6. Final words
References
M. Burger: Nonlinear eigenvalue problems for seminorms and applications
1. Introduction
2. Basic properties and formulations
2.1. Seminorms, duality, and subdifferentials
2.2. Singular values and eigenvalues
2.3. Spectral decomposition
3. Ground states
3.1. p-Laplacian eigenvalues
3.2. Ground states and sparsity
4. Variational problems, iterations, and flows
4.1. Variational regularization methods
4.2. Bregman iterations and inverse scale space flows
4.3. Gradient flows
4.4. Gradient flows and spectral decompositions
5. Applications
5.1. 1-Laplacian graph clustering
5.2. Distance functions from ∞-Laplacians
References
C. Cartis, N. I. M. Gould, and Ph. L. Toint: The evaluation complexity of finding high-order minimizers of nonconvex optimization
1. Introduction
2. A discussion of qth-order necessary optimality conditions
3. The composite problem and its properties
4. An adaptive regularization algorithm for composite optimization
5. Evaluation complexity
6. Sharpness
7. Inexact global minimization
8. Conclusions and perspectives
References
Y. H. Dai: An overview of nonlinear optimization
1. Introduction
2. Unconstrained optimization
2.1. Gradient methods
2.2. Conjugate gradient methods
3. Constrained optimization
3.1. Sequential quadratic programming methods
3.2. Interior-point methods
3.3. Augmented Lagrangian method of multipliers
4. Optimization with least constraint violation
5. Some discussions
References
Q. Lü: Control theory of SDPSs
1. Introduction
2. Exact controllability of stochastic hyperbolic equations
2.1. Formulation of the problem
2.2. Well-posedness of stochastic hyperbolic equations with boundary controls
2.3. The controllability results
2.4. Carleman estimate
3. Pontryagin-type stochastic maximum principle and stochastic transposition method
3.1. Formulation of the optimal control problem
3.2. Transposition solution and relaxed transposition solution to backward stochastic evolution equation
3.3. Pontryagin-type maximum principle
4. Open problems
References
A. Ozdaglar, M. O. Sayin, and K. Zhang: Independent learning in stochastic games
1. Introduction
2. Preliminaries: strategic-form games
2.1. Learning in strategic-form games with repeated play
3. Stochastic games
4. Learning in stochastic games
4.1. Model-free learning in stochastic games
4.2. Radically uncoupled learning in stochastic games
5. Other learning algorithms
5.1. Classical algorithms
5.2. Multiagent reinforcement learning
5.3. Decentralized learning in stochastic games
6. Conclusions and open problems
References
M. Tucsnak: Reachable spaces: old and new
1. Introduction
2. Some background on well-posed linear control systems
3. Single input systems with skew-adjoint generator
4. An example coming from elasticity
5. The heat equation on a half-line
6. The heat equation on an interval
7. Conclusions, remarks, and open questions
7.1. Time reversible systems
7.2. Systems described by parabolic equations
References
17. Statistics and Data Analysis
F. Bach and L. Chizat: Gradient descent on infinitely wide neural networks: global convergence and generalization
1. Introduction
2. Supervised learning
2.1. Statistics and optimization
2.2. Linear predictors and convex optimization
2.3. Neural networks and nonconvex optimization
3. Mean field limit of overparameterized one-hidden layer neural networks
3.1. Reformulation with probability measures
3.2. From gradient descent to gradient flow
3.3. Wasserstein gradient flow
4. Global convergence
4.1. Informal result
4.2. Simplified formal result
4.3. Proof of Theorem 3
4.4. Experiments
5. Generalization guarantees and implicit bias for overparameterized models
5.1. Implicit bias for linear logistic regression
5.2. Extension to two-layer neural networks
5.3. Kernel regime
5.4. Feature learning regime
5.5. Experiments
6. Discussion
References
B. Dong: On mathematical modeling in image reconstruction and beyond
1. Introduction
2. Wavelet frame-based approach for image reconstruction
3. PDE-based approach for image reconstruction
4. Connections between wavelet frame-based and PDE-based approaches
4.1. Wavelet frame transform and differential operators
4.2. Connections between wavelet frame-based analysis model and TV model
4.3. Connections between wavelet shrinkage algorithms and nonlinear evolutional PDEs
5. Going beyond image reconstruction
5.1. ODE-Nets: exploring structural similarity between numerical ODEs and CNNs
5.2. PDE-Nets: exploring structural similarity between numerical PDEs and CNNs
References
S. Jegelka: Theory of graph neural networks
1. Introduction
1.1. Graph Neural Networks (GNNs)
2. Representational power of GNNs
2.1. GNNs and graph isomorphism testing
2.1.1. Representing multiset functions
2.1.2. Implications for graph distinction
2.1.3. Computation trees and structural graph properties
2.2. Node IDs, local algorithms, combinatorial optimization, and lower bounds
2.3. Higher-order GNNs
2.3.1. Higher-order WL networks
2.3.2. Linear equivariant layers
2.3.3. Summary of representational power via WL
2.3.4. Relational pooling
2.3.5. Recursion
2.4. Universal approximation
3. Generalization
3.1. Generalization bounds via complexity of the model class
3.2. Generalization bounds via the Neural Tangent Kernel
3.3. Generalization via algorithmic alignment
4. Extrapolation
5. Conclusion
References
O. V. Lepski: Theory of adaptive estimation
1. Introduction
1.1. Examples of models
1.2. Examples of estimation targets G
2. Minimax adaptive estimation
2.1. Scales of functional classes
2.1.1. Classes of smooth functions
2.1.2. Functional classes with structure
2.2. Existence of adaptive estimators. Fundamental problem
2.3. Adaptive estimation via the oracle approach
3. Universal selection rule and ℓ-oracle inequality
3.1. Assumptions
3.2. Universal selection rule and corresponding ℓ-oracle inequality
4. Examples of estimator collections satisfying Assumption (A^{permute})
4.1. Estimator collections in the density model
4.2. Estimator collections in White Gaussian Noise Model
5. One example of estimator collection satisfying Assumption (A^{upper})
5.1. Functional classes of bandwidths
5.2. Verification of (5.1)
5.3. Verification of (5.2)
References
G. Lugosi: Mean estimation in high dimension
1. Introduction
2. Basic ideas: the one-dimensional case
3. Multivariate sub-Gaussian estimators
3.1. Median-of-means tournaments
3.2. Multivariate trimmed mean
4. Direction-dependent accuracy
5. Conclusion
References
R. Nickl and G. P. Paternain: Non-linear inverse problems
1. Introduction
2. Information geometry in non-linear regression models
2.1. Measurement setup
2.2. The DQM property
2.3. The adjoint score and information operator
2.4. Lower bounds for estimation of functionals
2.5. Non-existence of \sqrt{N}-consistent estimators of linear functionals
3. Application to a divergence form PDE
3.1. Basic setting
3.2. Global injectivity and model examples
3.3. The score operator and its adjoint
3.4. Injectivity of I_θ , I_θ ^* I_θ
3.4.1. Example 1 continued; on the kernel in L^2(\mathcal{O})
3.4.2. Example 2 continued; injectivity on L^2(\mathcal{O})
3.5. The range of I_θ^* and transport PDEs
3.5.1. Example 1 continued; range constraint
3.5.2. Example 2 continued; range constraint
3.6. Concluding remarks
4. Appendix
4.1. Proofs of Theorem 5 and Proposition 3
4.2. Proof of Theorem 2 for I_θ^* I_θ compact
References
B. Schölkopf and J. von Kügelgen: From statistical to causal learning
1. Introduction
2. Statistical learning theory
3. From statistical to causal models
4. Causal modeling frameworks
5. Independent causal mechanisms
6. Levels of causal modeling
7. Causal discovery
8. Implications for machine learning
9. Causal reasoning
10. Current research and open problems
References
C.-H. Zhang: Gaussian anticoncentration
1. Introduction
2. Anticoncentration inequalities for Gaussian maxima
3. Comparison of Gaussian distribution functions
4. Higher-order anticoncentration
References
18. Stochastic and Differential Modelling
J. Bedrossian, A. Blumenthal, and S. Punshon-Smith: Lower bounds on the Lyapunov exponents of stochastic differential equations
1. Lyapunov exponents for stochastic differential equations
Outline
1.1. Lyapunov exponents and their challenges
A discrete-time example
1.2. Lyapunov exponents for SDE
1.2.1. Stationary measures and long-term statistics
1.2.2. Lyapunov exponents
2. Formulae for the Lyapunov exponents
2.1. The projective process
2.2. Sign-definite formulas for Lyapunov exponents
2.3. The best of both worlds: sign-definite and time-infinitesimal
3. Quantitative lower bounds by the Fisher information
3.1. Hypoellipticity
3.2. Uniform hypoellipticity
4. Chaos for 2D Galerkin–Navier–Stokes and related models
4.1. Zero-noise limit and rigidity: proof sketch of Theorem 4.1
4.2. Verifying projective hypoellipticity: a sufficient condition
4.3. Projective hypoellipticity for 2D Galerkin–Navier–Stokes
4.3.1. A distinctness condition on a diagonal subalgebra
4.3.2. Verifying the distinctness condition using computational algebraic geometry
5. Lagrangian chaos in stochastic Navier–Stokes
6. Looking forward
References
N. Champagnat, S. Méléard, and V. C. Tran: Multiscale eco-evolutionary models
1. Introduction and presentation of the individual-based model
2. A general stochastic individual-based model for vertical and horizontal trait transmission
2.1. The model
2.2. Generator
3. Large population limit and rare mutation in the ecological time-scale
3.1. A deterministic approximation
3.2. Particular cases when p=0
4. Rare mutations – fixation probability
5. Very rare mutations in an evolutionary time scale
5.1. Trait substitution sequence
5.2. Canonical equation of the adaptive dynamics
6. Simulations – case of frequency-dependence
7. Stochastic analysis of emergence of evolutionary cyclic behavior – a simple model
7.1. A trait-discretized model
7.2. Some enlightening lemmas
7.3. Dynamics of the exponents
7.4. Reemergence of trait 0
8. Macroscopic Hamilton–Jacobi approximation of the exponents
References
H. Kang: Quantitative analysis of field concentration
1. Introduction
2. The conductivity equation
2.1. Estimates for circular inclusions
2.1.1. Explicit representation of the solution
2.1.2. Optimal estimates for the solution
2.1.3. Optimality of the estimates
2.2. Estimates for inclusions of general shape
2.3. Asymptotic characterizations of the gradient blow-up
3. Lamé system
4. Stokes system
5. Conclusions
References
19. Mathematical Education and Popularization of Mathematics
C. I. Grima: The hug of the scutoid
1. Introduction
2. Who, what, where, when, why, and how (my personal view)
2.1. Why mathematics?
2.2. Where are the women?
3. And, suddenly, flies
3.1. Voronoi diagram and 2-dimensional epithelial tissues
3.2. Scutoids and 3-dimensional epithelial tissues
3.3. Almost famous
4. The show must go on. Conclusions and future work
References
A. Sfard: The long way from mathematics to mathematics education
1. Conundrums that triggered the transformation
1.1. Why doesn't logic suffice to understand mathematics?
1.2. What is so complex about complex numbers?
1.3. Why cannot children see as the same what grownups cannot see as different?
2. What changed on the way from mathematics to mathematics education
2.1. Recognition of the need to elucidate onto-epistemological foundations
2.2. Mathematical object as a mode de parler rather than a part of mind-independent reality
2.3. Mathematics: the activity of telling useful stories about reality rather than a search for the universal truth about it
3. How commognitive insights about learning helped to solve the initial conundrums
3.1. Seeing as the same what so far appeared as different: the paradoxical conditions for objectification
3.2. The complexity of complex numbers: the need to reconcile yourself with the incommensurability between the old and the new discourses of numbers
3.3. The insufficiency of logic for understand mathematics? Some mathematical developments are a matter of choice, not of deductive reasoning
4. Postscript: my personal takeaways from the journey
References
20. History of Mathematics
J. Barrow-Green: George Birkhoff’s forgotten manuscript and his programme for dynamics
1. Birkhoff's work in dynamics
2. Birkhoff's forgotten manuscript
3. Conclusion
References
A. Imhausen: Some uses and associations of mathematics, as seen from a distant historical perspective
1. Introduction
2. Invention of number system and its early uses
3. The Old Kingdom: Early evidence of mathematical concepts and practices
4. The discipline of mathematics in the Middle Kingdom (and after)
5. Further uses of mathematical concepts and practices in ancient Egypt
6. Conclusion
References
K. Ramasubramanian: The history and historiography of the discovery of calculus in India
1. Introduction
2. Developments in the Classical Era of Indian mathematics
2.1. Notion of zero and infinity
2.1.1. Philosophical and cultural context of zero and infinity
2.1.2. The mathematics of zero
2.2. Irrationals and iterative approximations
2.2.1. Approximation for surds in Śulbasūtras
2.2.2. Approximation for π by Āryabhaṭa
2.3. Summation of geometric series
2.4. Āryabhaṭa's computation of Rsine-differences
2.5. Instantaneous velocity of a planet (tātkālika-gati)
3. Kerala School of Mathematics and Astronomy
3.1. Discussion of the sum of an infinite geometric series
3.2. Derivation of binomial series expansion
3.3. Estimation of sums of integral powers of natural numbers
3.4. Mādhava's infinite series for π
3.5. Derivation of end-correction terms (antya-saṃskāra)
4. Historiography of the inception of calculus in India
4.1. Brief note on Charles Whish and his collections
4.2. About Kālasaṅkalita
4.3. Extracts from the exchanges between Whish, Hyne, and Warren
5. Concluding remarks
References
List of contributors
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