Notes and essays
Blog
Writing on mathematics, research, computation, and academic life.
Writing
Mathematics in theory and practice
Notes on inverse problems, scientific computing, and the process of doing mathematical research. I also write about teaching, academic work, and applications that matter beyond the university.
Primal-Dual Methods for Convex Optimization
How saddle-point formulations and primal-dual splitting algorithms handle the non-smooth objectives that arise in imaging and inverse problems; total variation regularization requires going beyond gradient descent.
Electrical Impedance Tomography in Clinical Practice: From Research to Bedside
How EIT supports continuous bedside monitoring of regional lung ventilation, what current evidence supports, and which reconstruction problems remain open.
Inverse Problems
Krylov Methods: Solving Large Linear Systems Without Forming the Matrix
An introduction to Krylov subspace methods: the iterative solvers that make large-scale scientific computing possible, and why they work so well for the linear systems that arise in inverse problems and PDE discretizations.
Mathematics
Full Waveform Inversion: The Mathematics Behind Finding Oil
How a PDE-constrained optimization problem sits at the heart of modern seismic exploration; solving it is one of the most computationally demanding inverse problems in industry.
Inverse Problems
Mathematics After Proof Scarcity: What Terence Tao's ICM 2026 Lecture Means for Mathematicians
Reflections on Terence Tao's ICM 2026 lecture on mathematics in the age of AI: proof abundance, mathematical understanding, authorship, and what researchers should preserve as AI becomes more capable.
Mathematics
Reading Tikhonov (1963): The Paper That Made Ill-Posed Problems Solvable
A close reading of Tikhonov's 1963 paper on the regularization of ill-posed problems: what it actually says, what it assumed, and why it changed how we think about inverse problems.
Mathematics
Archive
Earlier essays
- Stochastic PDEs: When the Equations Themselves Are Random
- Spectral Methods: Exponential Accuracy from Smooth Solutions
- Optimal Experimental Design: Choosing Measurements That Matter
- Optimal Transport: Moving Mass as Efficiently as Possible
- Model Order Reduction: Making Big Problems Small
- On Working Across Disciplines
- Compressed Sensing: Recovering Signals from Few Measurements
- The Adjoint Method: Gradients Independent of Parameter Dimension
- Gaussian Processes: Priors Over Functions
- Deep Learning Meets Differential Equations: Neural ODEs and Beyond
- Presenting Mathematics So People Actually Listen
- Bayesian Inverse Problems: Quantifying What We Do Not Know
- Variational Methods in Image Reconstruction
- What Makes a Good Research Problem?
- Scientific Computing with Python: The Tools I Actually Use
- Neural Operators: Learning the Solution Map
- The Finite Element Method: Building Solutions Piece by Piece
- From PhD to Postdoc: What Changes and What Doesn't
- Electrical Impedance Tomography: Seeing Through the Body with Mathematics
- Regularization: The Art of Choosing What to Believe
- Physics-Informed Neural Networks: What Works, What Doesn't, and What Nobody Tells You
- Inverse Problems: The Art of Working Backwards
- On the Quiet Erosion of Deep Thinking