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Gabriel Peyré

@gabrielpeyre • 102,535 subscribers

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

Shorts

Mathematical modeling: part 1/2.

Mathematical modeling: part 1/2.

231,120 Aufrufe

Taylor series expansion of sine.

Taylor series expansion of sine.

305,048 Aufrufe

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).

225,217 Aufrufe

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.

232,236 Aufrufe

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.

206,420 Aufrufe

Heat diffusion vs. wave equation on a surface.

Heat diffusion vs. wave equation on a surface.

205,129 Aufrufe

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

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

168,783 Aufrufe

Poljak heavy ball and Nesterov methods speed up gradient descent by introducing momentum. Corresponding to second order ODEs.

Poljak heavy ball and Nesterov methods speed up gradient descent by introducing momentum. Corresponding to second order ODEs.

141,154 Aufrufe

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 Aufrufe

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.

125,323 Aufrufe

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 Aufrufe

Power iterations converge to the leading eigenvector of a matrix.

Power iterations converge to the leading eigenvector of a matrix.

114,114 Aufrufe

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.

74,712 Aufrufe

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 Aufrufe

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.

85,333 Aufrufe

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.

67,799 Aufrufe

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.

78,837 Aufrufe

Möbius transforms are the bijective conformal maps from the Riemann sphere (complex numbers with a point at infinity) to itself. They are rational functions, form a group under composition, and have, in general, two fixed points (in black).

Möbius transforms are the bijective conformal maps from the Riemann sphere (complex numbers with a point at infinity) to itself. They are rational functions, form a group under composition, and have, in general, two fixed points (in black).

78,030 Aufrufe

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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.

187,988 Aufrufe • vor 3 Jahren

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.

89,305 Aufrufe • vor 2 Jahren

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

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

73,268 Aufrufe • vor 2 Jahren

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 Aufrufe • vor 2 Jahren

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 Aufrufe • vor 3 Jahren

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.

14,846 Aufrufe • vor 1 Jahr

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 Aufrufe • vor 1 Jahr

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 Aufrufe • vor 1 Jahr

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