## Jonas A. Actor

Ph.D. Student, Rice University

I am currently a Ph.D. student in the Department of Computational and Applied Mathematics (CAAM) at Rice University in Houston, Texas. My current research, with Dr. Beatrice Riviere (CAAM) and Dr. David Fuentes (MD Anderson), focuses on automated detection and measurement of liver cancer from medical imaging data. For these tasks, I am developing novel image segmentation techniques that combine partial differential equations and neural networks.

I am a predoctoral fellow of the Gulf Coast Consortia, as part of the Keck Center's National Library of Medicine Training Grant in Biomedical Informatics and Data Science.

In addition to my studies, I serve as a Graduate Writing Consultant at the Center for Academic and Professional Communication . In this position, I advise undergraduate and graduate students on their academic writing and presentations. I have also served two years on the Rice University SIAM Student Chapter executive board, first as secretary and more recently as president.

My CV contains more information about me and my research background. I can be reached at jonasactor@rice.edu.

##### Education
• Ph.D., Computational and Applied Mathematics, Rice University, May 2021 (expected)
• M.A., Computational and Applied Mathematics, Rice University, August 2018
• B.S., Mathematics, University of Chicago, June 2016
##### Experience
• Student Assistant, Lawrence Berkeley National Laboratory, May - August 2018
• Consultant, NanoEar Technologies, September 2017 - May 2018
##### Teaching
• Grader, CAAM 336, Differential Equations in Science and Engineering, Fall 2018
• Course Assistant, CAAM 536, Numerical Methods for PDE's, Spring 2018
• Grader, CAAM 453/550, Numerical Analysis, Fall 2017
• Grader, CAAM 335, Matrix Analysis, Spring 2017
• Grader, CAAM 335, Matrix Analysis, Fall 2016
##### Awards
• Keck Fellowship, Gulf Coast Consortia, August 2018 - June 2021 (subject to renewal)
• NSF GRFP, Honorable Mention, April 2018
• Alan Weiser Travel Award, Travel to SGA 2018, TU Munich, July 23-28 2018
• Computer Science and Engineering Enhancement Fellowship,
Ken Kennedy Institute for Information Technology, August 2016 - May 2021

### Research Interests

• Nonlinear approximation theory
• Functional analysis for neural networks
• Numerical methods for PDE's
• Image segmentation
• Biomedical informatics
• Matrix factorization and numerical linear algebra

### Current and Previous Research Topics

#### Kolmogorov Superposition Theorem

The Kolmogorov Superposition Theorem (KST) states that any multivariate continuous function can be represented using a small number of superpositions of continuous univariate functions. Specifically, for every continuous function $\,f: [0,1]^n \rightarrow \mathbb{R}$, there exists univariate functions $\,\chi: \mathbb{R} \rightarrow \mathbb{R}$ and $\,\psi_{p,q}: [0,1] \rightarrow \mathbb{R}$ such that $$f(x_1,\dots,x_n) = \sum_{q=0}^{2n} \chi \left( \sum_{p=1}^n \psi_{p,q}(x_p) \right).$$ In this representation, the functions $\,\psi_{p,q}$ do not depend on the function $\,f$ in question. Therefore, we can associate $\,f$ with its corresponding univariate function $\,\chi$, i.e. the function that enables representation via KST. Using this theorem, we effectively trade the number of variables for smoothness: the functions $\,\psi$ and $\,\chi$ are inherently nonsmooth. My research explores how to computationally represent multivariate functions using KST while maintaining the best possible smoothness of these functions.

With Dr. Matthew Knepley, University at Buffalo.

#### Inertia for Hierarchical Semiseparable (HSS) Matrices

HSS matrix factorization exploits a hierarchical low-rank off-diagonal structure to make the solution of large linear algebra problems feasible. The software package STRUMPACK computes such factorizations for both dense and sparse matrices, making use of multithreading and distributed memory to enable peak performance. However, recent versions of STRUMPACK do not have the capability of finding the inertia of a matrix. This limitation prevents STRUMPACK from being used as a direct solver for optimization problems, where the inertia is needed to evaluate the optimality of a solution. My contributions to STRUMPACK involve adding routines to find the inertia for both dense and sparse HSS matrices. Future work will extend my routines for the distributed memory case.

With Dr. Xiaoye (Sherry) Li and Dr. Pieter Ghysels, Lawrence Berkeley National Laboratory.

#### Unrolled NN Methods + PDE Functionals for Image Segmentation

More coming soon!

With Dr. Beatrice Riviere, Rice University, and Dr. David Fuentes, MD Anderson Cancer Center.

### Publications and Presentations

Here are materials from recent (and not-so-recent) research I have done. Please see my CV for a full list of publications and presentations.

#### Publications

Construction of Lipschitz Inner Functions via Reparameterization for the Kolmogorov Superposition Theorem.
Actor, J. A. and M. G. Knepley.
In preparation (2018).
Computation for the Kolmogorov Superposition Theorem.
Actor, J. A.
Rice University, Thesis, Master of Arts (2018).
An Algorithm for Computing Lipschitz Inner Functions in Kolmogorov's Superposition Theorem.
Actor, J. A. and M. G. Knepley.
Submitted (2018).
Break-off Model for $CaCO_3$ Fouling in Heat Exchangers.
Babuska, I., Silva, R. S., and Actor, J. A.
International Journal of Heat and Mass Transfer 116 (2018), 104–114.

#### Presentations and Posters

Finding the Inertia of HSS Matrices
Presentation, CAAM Graduate Seminar, September 2018.
Inertia of HSS matrices using STRUMPACK
Poster, CSSSP Poster Session, Lawrence Berkeley National Laboratory, August 2018.
Exploiting Lipschitz Continuity for the Kolmogorov Superposition Theorem
Talk, Sparse Grids and Applications 2018, TU Munich, July 2018.
The Kolmogorov Superposition Theorem: A Framework for Multivariate Computation
Thesis Defense, Master of Arts, Rice University, May 2018.
Physics-Based Machine Learning for Image Segmentation
Presentation and Interview, Gulf Coast Consortium, February 2018.
A Primer on Image Segmentation
Presentation, CAAM Graduate Seminar, February 2018.
Kolmogorov Superposition Theorem: Univariate Encodings for Multivariate Functions
Poster, Rice Data Science Conference, Ken Kennedy Institute for Information Technology, October 2017.
Lipschitz Inner Functions in Kolmogorov's Superposition Theorem
Presentation, CAAM Graduate Seminar, September 2017.
Serial Block Face SEM Visualization of Tuberculosis Infected Macrophages
Poster, Fall Meeting of the American Society for Microbiology, Texas Branch, November 2014.

### Mathematical Writing and Academic Presentation

Communicating research is difficult yet necessary. Here are a few of my favorite resources; I reference these almost whenever I have to communicate my research: papers, posters, conference talks, research group meetings, etc.

##### Style Guides and General Resources
• Strunk and White, The Elements of Style
• Steenrod, Halmos, Schiffer, and Dieudonne, How to Write Mathematics

Duncan Hall 2107