Gabriel Pinochet-Soto

Me in a glance

I am Gabriel Pinochet-Soto, a 28-year-old graduate student at Portland State University. I'm pursuing my Ph.D. in Mathematical Sciences. I began in Fall 2021, and I am expected to graduate in June 2026.

Education

Previously, I obtained my Master and Bachelor degrees in Mathematics at Pontificia Universidad Católica de Valparaíso. I also participated in IMPA's summer school in 2019 and attended the Second NGSolve user meeting in 2018. I have completed internships at ANL and LLNL.

Research Interests

I like math.

More specifically, I am interested in numerical analysis, (applied) functional analysis, (applied) differential equations, scientific computing, and machine learning.

My current research focuses on the development and analysis of numerical methods for partial differential equations in the context of the finite element method, and their applications to real-world problems. More specifically, I am working on cluster-based eiganvalue error estimation methods and mesh adaptivity techniques.

Current research

• Mesh adaptivity and error estimation.
• Dual-weighted residual (DWR) methods for non-self-adjoint problems.
• (Time-harmonic) Maxwell equations.
• Helmholtz equation.
• PML technique.
• Residual-based discretizations.
• DLS methods.
• DPG methods.
• FOSLS methods.
• Eigenvalue problems.
• Filtered subspace methods.
• Spectral mapping for rational filters.
• Applications to photonics and microstructured fibers.

Other topics of interest

• FEM and implementation.
• Block solvers, saddle-point problems.
• Matrix-free methods and preconditioners.
• Reconstruction from broken spaces methods.
• Multilevel methods and application.
• Encoder-decoder-based projectors/interpolators.
• Machine learning.
• JAX, Flax, PyTorch usage.
• Optimizers
• Auto-differentiation frameworks (Enzyme AD, JAX, etc).
• Other "other" topics.
• Theory of computation, complexity classes, formal verification, and languages.
• Quantum computing and algorithms.
• Convex analysis, operator theory, and more functional analysis.

Personal interests

Here are some interests and hobbies of mine, beyond my academic life, in no particular order or importance.

Favorite Theorems and Proofs

My current favorite Theorems are

• Grothendieck's characterization of compact sets in normed spaces,
• Banach's fixed point principle, and
• Arzelà-Ascoli Theorem.

My favorite proof techniques are

• Cantor's diagonal argument for the undecidability of the Halting problem, and
• the use of dyadic numbers to index separating sets in Urysohn's Lemma,

Side projects and hobbies

I would like to learn more about

• Complexity theory, quantum computing, and machine learning,
• Programming languages such as C/C++, Julia, and Python, and
• Writing "super useful" software such as a shell, a FEM library, or a programming language.

Personal life

Outside of academia, I enjoy reading philosophy, watching movies and series, running, and biking. I also like playing guitar and writing songs and poetry.

I have been living in Portland, Oregon for more than three years. Before moving to Portland, I lived in Viña del Mar, Rio de Janeiro, Hayward, and Livermore.