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Gabriel Peyré

@gabrielpeyre • 101,059 subscribers

@CNRS researcher at @ENS_ULM. One tweet a day on computational mathematics.

Shorts

Mathematical modeling: part 1/2.

Mathematical modeling: part 1/2.

230,903 views

Taylor series expansion of sine.

Taylor series expansion of sine.

305,030 views

Fokker-Planck equation equivalently describes the movement of a random particle with a drift (as a stochastic ODE) and the evolution of its density (as a PDE).

Fokker-Planck equation equivalently describes the movement of a random particle with a drift (as a stochastic ODE) and the evolution of its density (as a PDE).

225,076 views

Heat, Wave and Schrodinger equations are fundamental linear partial differential equations of physics.

Heat, Wave and Schrodinger equations are fundamental linear partial differential equations of physics.

232,163 views

Comparison of the Wasserstein, Hellinger, Kullback-Leibler and reverse KL on the space of Gaussian distributions.

Comparison of the Wasserstein, Hellinger, Kullback-Leibler and reverse KL on the space of Gaussian distributions.

206,365 views

Heat diffusion vs. wave equation on a surface.

Heat diffusion vs. wave equation on a surface.

205,112 views

$Lorenz attractor is the set of limit trajectories for a simplified mathematical model for atmospheric convection. It has a fractal geometry.$

Lorenz attractor is the set of limit trajectories for a simplified mathematical model for atmospheric convection. It has a fractal geometry.

168,765 views

The marching squares/cubes is the standard algorithm to extract iso-curves/surfaces from a discretized image/volume.

The marching squares/cubes is the standard algorithm to extract iso-curves/surfaces from a discretized image/volume.

130,317 views

Impact of the interpolation method on the level set of the resulting function.

Impact of the interpolation method on the level set of the resulting function.

125,312 views

Oldies but goldies: L. Greengard and V. Rokhlin, A Fast Algorithm for Particle Simulations, 1987. Evaluates in O(n) in place of O(n^2) sums involving long-range interaction kernels.

Oldies but goldies: L. Greengard and V. Rokhlin, A Fast Algorithm for Particle Simulations, 1987. Evaluates in O(n) in place of O(n^2) sums involving long-range interaction kernels.

68,683 views

Power iterations converge to the leading eigenvector of a matrix.

Power iterations converge to the leading eigenvector of a matrix.

114,104 views

Large-scale N-body simulations require the evaluation of sums involving long-range interaction kernels. Fast multipoles and Barnes–Hut methods approximate these sums in n*log(n) operations instead of n^2.

Large-scale N-body simulations require the evaluation of sums involving long-range interaction kernels. Fast multipoles and Barnes–Hut methods approximate these sums in n*log(n) operations instead of n^2.

74,702 views

The Cucker-Smale system of ODEs is the simplest model of particles exhibiting flocking behaviors. Diffusion of speeds is governed by a Laplacian on the fully connected graph of points.

The Cucker-Smale system of ODEs is the simplest model of particles exhibiting flocking behaviors. Diffusion of speeds is governed by a Laplacian on the fully connected graph of points.

77,155 views

The three ways to solve Fokker-Planck equations: PDE evolution, stochastic (independent) or deterministic (coupled) particle systems.

The three ways to solve Fokker-Planck equations: PDE evolution, stochastic (independent) or deterministic (coupled) particle systems.

85,288 views

Laplacian eigenmaps perform non-linear dimensionality reduction ("manifold learning") by embedding data points using the eigenvectors of a graph Laplacian as coordinates.

Laplacian eigenmaps perform non-linear dimensionality reduction ("manifold learning") by embedding data points using the eigenvectors of a graph Laplacian as coordinates.

67,757 views

Interpolating at uniformly sampled point can suffer from Runge oscillations. Using Chebyshev nodes resolves this issue.

Interpolating at uniformly sampled point can suffer from Runge oscillations. Using Chebyshev nodes resolves this issue.

78,818 views

Photo-realistic texture synthesis methods perform pixels, patches or more complicated features recopy from the input to the output.

Photo-realistic texture synthesis methods perform pixels, patches or more complicated features recopy from the input to the output.

56,340 views

A ReLu network computes a piecewise linear approximation of a target function.

A ReLu network computes a piecewise linear approximation of a target function.

75,639 views

Oldies but goldies: Y Nesterov, A method for solving a convex programming problem with convergence rate O(1/k^2). Improves the 1/k rate of vanilla gradient descent to 1/k^2.

Oldies but goldies: Y Nesterov, A method for solving a convex programming problem with convergence rate O(1/k^2). Improves the 1/k rate of vanilla gradient descent to 1/k^2.

47,616 views

Oldies but goldies: R. Keys, Cubic convolution interpolation for digital image processing, 1980. Introduces bicubic interpolation, the most frequently used image interpolation method.

Oldies but goldies: R. Keys, Cubic convolution interpolation for digital image processing, 1980. Introduces bicubic interpolation, the most frequently used image interpolation method.

37,791 views

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Hopfield networks are recurrent networks minimizing an Ising-type energy parameterized by its weights. Learning weights means encoding patterns of +1/-1 as local minimizers.

Hopfield networks are recurrent networks minimizing an Ising-type energy parameterized by its weights. Learning weights means encoding patterns of +1/-1 as local minimizers.

187,883 views • 3 years ago

Mirror descent selects among the minimizers the one closest to initialization for the associated Bregman divergence (its "implicit bias"). Euclidean descent computes an orthogonal projection on the minimizers.

Mirror descent selects among the minimizers the one closest to initialization for the associated Bregman divergence (its "implicit bias"). Euclidean descent computes an orthogonal projection on the minimizers.

89,293 views • 2 years ago

Through five points in generic position passes a single conic curve.

Through five points in generic position passes a single conic curve.

73,268 views • 2 years ago

Oldies but goldies: Richard von Mises and Hilda Geiringer, Praktische Verfahren der Gleichungsauflösung, 1929. Introduces the power iterations, which converge to the leading eigenvector of a matrix.

Oldies but goldies: Richard von Mises and Hilda Geiringer, Praktische Verfahren der Gleichungsauflösung, 1929. Introduces the power iterations, which converge to the leading eigenvector of a matrix.

59,883 views • 1 year ago

For small step sizes, mirror descent is equivalent to a gradient descent on a Hessian manifold.

For small step sizes, mirror descent is equivalent to a gradient descent on a Hessian manifold.

61,868 views • 2 years ago

Oldies but goldies: A. Barron, Universal Approximation Bounds for Superpositions of a Sigmoidal Function, 1993. Proves that 1 hidden layer perceptrons break the curse of dimensionality to approximate a class of smooth functions.

Oldies but goldies: A. Barron, Universal Approximation Bounds for Superpositions of a Sigmoidal Function, 1993. Proves that 1 hidden layer perceptrons break the curse of dimensionality to approximate a class of smooth functions.

52,682 views • 2 years ago

Oldies but goldies: J. Bertrand, Theoreme relatif au mouvement d'un point attire vers un centre fixe, 1873. Only harmonic spring and gravity central forces produce periodic motions.

Oldies but goldies: J. Bertrand, Theoreme relatif au mouvement d'un point attire vers un centre fixe, 1873. Only harmonic spring and gravity central forces produce periodic motions.

14,092 views • 1 year ago

Oldies but goldies: A Witkin, Scale-space filtering, 1984. Introduces (with J. Koenderink) the notion of scale space, i.e., the multi-scales analysis of signals and images, which is at the heart of many vision architectures.

Oldies but goldies: A Witkin, Scale-space filtering, 1984. Introduces (with J. Koenderink) the notion of scale space, i.e., the multi-scales analysis of signals and images, which is at the heart of many vision architectures.

14,339 views • 1 year ago

Oldies but goldies: Maurice G. Cox, The Numerical Evaluation of B-Splines, 1972. Introduced (with De Boor) an efficient algorithm for the evaluation of B-splines, which revolutionized CAGD.

Oldies but goldies: Maurice G. Cox, The Numerical Evaluation of B-Splines, 1972. Introduced (with De Boor) an efficient algorithm for the evaluation of B-splines, which revolutionized CAGD.

14,115 views • 1 year ago

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