Rafael Monteiro's website

Rafael A. Monteiro

Former Postdoctoral Researcher at the Mathematics for Advanced Materials - Open Innovation Laboratory (MathAm-OIL/AIMR-Tohoku Univ.), Japan

Former Postdoctoral Researcher at the University of Minnesota, USA

Ph.D. in Mathematics at Indiana University, USA. Advisor: Kevin Zumbrun

M. S. in Computational Methods and Applied Mathematics - Instituto de Matematica Pura e Aplicada - IMPA - Brazil. Advisor: Andre Nachbin

B.sc. in Applied Mathematics - Instituto de Matematica e Estatistica, University of Sao Paulo, Brazil

Research interests:

I am an applied mathematician doing research in the field of Pattern Formation and Statistical Pattern Recognition.

My last research work was done at MathAM-OIL, a subdivision of the Advanced Institute for Materials Research (AIMR-Tohoku University) focused on research and development in material sciences. Due to the complexity and interdisciplinarity of the projects I am involved in, most teams I work with are made up of researchers with different backgrounds who self-organize around problems to solve them. These projects require techniques that range from Partial Differential Equations (PDEs) and dynamical systems to statistical and computational modeling (at least those are the lenses I see things through).

My research interests are in a few overlapping fields:

Pattern-Formation, focusing on the approximation or reduction of infinite dimensional models to others that are much simpler (and, sometimes, finite dimensional);
Statistical Pattern Recognition/Prediction, where one combines tools from abstract mathematics (e.g., Harmonic and Spectral analysis) with computer science (design of algorithms and efficient numerical implementations), Statistics, and ML techniques;
A.I./data-driven material sciences, where we use statistical data to optimize/calibrate models, predict/understand structural formation, or do spatio-temporal forecasting. Some of these problems differ from the previous item because they involve decision-making and interaction with an environment.

Recent works encompass the use of applied mathematics methods in computational chemistry/material informatics, the discovery of a new Recurrent Neural Network based on a Reaction-Diffusion Equations, and applications of Reinforcement Learning + Deep Learning in the field of Molecular Dynamics (in preparation).

Curriculum vitae * Google scholar * Git-hub * LinkedIn

Preprints and work in progress:

Landscape-Sketch-Step: An AI/ML-Based Metaheuristic for Surrogate Optimization Problems, joint with Kartik Sau. preprint arXiv, October 2023. (Code available on Git-hub.)

Binary classification as a phase separation process, preprint arXiv, September 2021. (Code available on Git-hub; see also the data repository (at Zenodo) and short tutorial.)

[UPDATE: the results in this paper have been upgraded. Accuracy now is in the range 99% for the digits 0 and 1 of the MNIST database. I have also extended the result to all pairs of digits, besides giving a new implementation of the code in Tensorflow/Keras.

Overall, the paper is now much shorter, and the efficiency attained by the new numerical parameterization is impressive. The code runs extremely fast: where I once needed days of computation on a supercomputer with 22 cores now a 4 core laptop suffices, with results available in a matter of minutes.]

Publications:

Transverse bifurcation of viscous slow MHD shocks (joint with Blake Barker and Kevin Zumbrun, 2021; Physica D)

[Code for symbolic computations available on Git-hub.]

[One of the plots in this work was featured in the poster of the conference "Shock waves and beyond", held in 2015 at the Institut Henri Poincaré, in Paris (see the poster on the right). Further information here.]

[Needless to say that, back when I wrote the code, SAGE and sympy were equally strong. I'm not sure about that nowadays, for I have switched to sympy. In summary: if you want to use the code for something I would recommend you to first convert it to sympy, which is not too hard.]

The Rising Sun Envelope Method: an automatic and accurate peak location technique for XANES measurements

(J. Phys. Chem. A 2020, joint work with K. Takahashi and I. Miyazato; open-access!)

[Code available on Git-hub; read more about the method in this tutorial.]

The Swift-Hohenberg Equation under directional-quenching: finding heteroclinic connections using spatial and spectral decompositions, joint with Natsuhiko Yoshinaga (Archive for Rational Mechanics and Analysis, 2019; open-access!)

Horizontal patterns from finite speed directional quenching, (Discrete and Continuous Dynamical Systems - Series B, 2018)
Contact angle selection for interfaces in growing domain, joint work with Arnd Scheel (ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift fur Angewandte Mathematik und Mechanik-2018)
Phase separation from directional quenching , joint work with Arnd Scheel (Journal of Nonlinear Science- 2017)

[Even though the statement of Proposition 3.7 in the above paper is correct, there are some minor errors and misprints in the Lemmas 3.8, 3.9, and 3.10 used in its proof. For a fixed proof, please check Lemma 3.8 in the subsequent paper Horizontal patterns from finite speed directional quenching.]

Transverse steady bifurcation of viscous shock solutions of a system of parabolic conservation laws in a strip (Journal of Differential Equations - 2014)

Teaching (at Indiana University and University of Minnesota)

Some slides, posters, and presentations

Notes and old stuff

Math blogs, recommendations, and others:

Terence Tao: interesting math discussions, in general in a wide range of subjects.
The relativity of wrong, by Isaac Asimov. A MUST-READ, especially in days like this, where a growing amount of flat-Earth-like theories (and their believers) are around.
MATEMATECA, an amazing project by the Applied Mathematics Institute at the University of Sao Paulo-Brazil. They design mathematical toys and other related things to show people the beauty and omnipresence of Mathematics in our lives. (They accept donations!!!)
Other things that matter to me