Laboratory for Parallel Numerical Algorithms

Research Topics

Tensor Algebra: Multidimensional data (ubiquitous in scientific computing and machine learning) can be effectively treated via tensor abstractions. Tensors and their decompositions provide numerical tools for hypergraph processing, high-order methods, and neural networks.

Matrix Computations: Numerical linear algebra underlies most computational approaches in data sciences. Fast matrix algorithms provide solutions for nonlinear optimization, low-rank approximation, and eigenvalue problems.

Quantum Systems: Tensor representations provide the most natural way to computationally model entanglement (correlation between electrons). We investigate numerical parallel algorithms for tensor computations arising in quantum chemistry (e.g. high-accuracy electronic structure calculations) and quantum information (e.g. quantum circuit simulation).

Communication Complexity: Performance and scalability of algorithms and libraries is constrained by data movement in the memory hierarchy and network. We aim to design parallel algorithms that minimize the amount of communication and number of messages.

High Performance Numerical Libraries: Parallel numerical libraries are the glue between fast algorithms and real-world applications. We pursue application-driven research on algorithms by way of developing general and scalable library routines.


(May 2019) See Edward Hutter's presentation on Communication-avoiding Cholesky-QR2, presented as regular conference paper at IPDPS 2019 in Rio de Janeiro, Brazil.

(May 2019) Welcome (back) to Yuchen Pang and Linjian Ma, who plan to start their PhD work this fall at UIUC as part of LPNA!

(May 2019) LPNA undergraduate researchers are becoming graduates! Zecheng Zhang and Qile Zhi will be pursuing MS programs at Stanford, Naijing Zhang will pursue an MSE at UC Berkeley, Xiaoxiao Wu will pursue an MSCF at CMU, Siyuan Zhang will pursue MS at UIUC, David Zhang will pursue an MS at Cornell, and Hongru Yang will pursue a PhD at UT Austin!

(May 2019) Congratulations to Caleb Ju for winning the Franz Hohn and J.P. Nash Scholarship.

(Feb 2019) Edward Hutter presented on Communication-avoiding Cholesky-QR2 at SIAM CSE 2019 in Spokane. This work will also be presented as a regular conference paper at IPDPS 2019.

(Dec 2018) Edward Hutter is one of two UIUC PhD students in the DOE CSGF class of 2018 (CS @ Illinois news article).

(Nov 2018) Edgar Solomonik is one of three winners of the 2018 IEEE TCHPC Award for Excellence in Early Career Research, also see CS @ Illinois news article.

(May 2018) Congratulations to Pavle Simonovic (completed BS thesis) and Peter Tatkowski (joining ETH Zurich MS program)!

(May 2018) Congratulations to Qile Zhi for winning the Franz Hohn and J.P. Nash Scholarship.

(March 2018) Edward Hutter presented on Communication-avoiding Cholesky-QR2 at SIAM PP 2018 in Tokyo.

(July 2017) Congratulations to Raul Platero for being a winner of the Outstanding Oral Presentation prize at the 2017 Illinois Summer Research Symposium (ISRS)!

(June 2017) Congratulations to Tobias Wicky for finishing his MS thesis and to Edward Hutter for finishing his BS thesis!

(June 2017) Group webpage is up, welcome!


Cyclops Tensor Framework
a distributed-memory library for graph, matrix, and tensor computations

a suite of parallel algorithms for matrix factorization and eigendecomposition



We are always looking for new collaborators and participants. If you are a UIUC student interested in doing research in the area, email Edgar Solomonik (


We are part of the scientific computing group at the University of Illinois at Urbana-Champaign. Below LPNA group photo is from a May 2019 get together to celebrate several graduating students.


Edgar Solomonik
Assistant Professor

Graduate Students

Edward Hutter
PhD Student
Raul Platero
PhD Student
Samah Karim
PhD Student (co-advised with William Gropp)

Undergraduate Students

Zecheng Zhang (CS)
Naijing Zhang (CS)
Hung Woei Neoh (Math & CS)
Wentao Yang (CS)
Xiaoxiao Wu (Stats & CS)
Siyuan Zhang (CS)
Caleb Ju (CS)
Zhaoyu Wu (CS)
Yiqing Zhou (Physics & CS)
Yunxin (David) Zhang (CS)
Tianyi Hao (CS)
Karthik Murthy (CS)
Yifan Zhang (Math, Eng Phys, and Stats)

Master Theses

Tobias Wicky (2017): A communication-avoiding algorithm for solving linear systems of equations with selective inversion

Bachelor Theses

Pavle Simonovic (2018): Shared-memory parallel algorithms for sparse matrix operations
Edward Hutter (2017): QR factorization over tunable processor grids

Past Participants / Independent Study Projects

Eric Song (2018): High-level interface abstractions for parallel tensor decompositions
Ruiqian Yao (2018): Performance modeling, prediction, and training for parallel tensor computation kernels
Linjian Ma (2018): Pairwise perturbation and multigrid in alternating least squares for CP and Tucker decomposition
Eduardo Yap (2018): Generalized tensor contractions by batched matrix multiplication
Peter Tatkowski (2018): Algebraic tensor representaiton of finite element methods; Tensor completion with Cyclops
Qile Zhi (2017): Stability of triangular matrix inversion
Thomas Warther (2017): Parallel neural networks with Cyclops

Video Lectures

web-course Numerical analysis Spring 2018, Fall 2018; CS 450

web-course Parallel numerical algorithms Fall 2017; CS 554

video June 2017; LPNA Lecture; Basics of tensors (Edgar)

video June 2017; LPNA Lecture; Basics of communication complexity (Edgar)

web-course Communication cost analysis of algorithms Fall 2016; CS 598-ES


slides May 2019; IPDPS19; Rio de Janeiro, Brazil; Communication-avoiding Cholesky-QR2 for rectangular matrices (Edward)

February 2019; SIAM CSE19; Spokane, WA; Communication-avoiding Cholesky-QR2 for rectangular matrices (Edward)

March 2018; SIAM PP18; Tokyo, Japan; Communication-avoiding Cholesky-QR2 for rectangular matrices (Edward)

September 2017; BLIS Retreat 2017; Austin TX, USA; Parallel 3D Cholesky-QR2 for rectangular matrices (Edward)

slides July 2017; LPNA Group presentation; Urbana IL, USA; 2D Finite Element Methods and Gather/Scatter (Peter)

slides July 2017; LPNA Group presentation; Urbana IL, USA; Least Squares Updating for Kronecker Products (Raul)

July 2017; LPNA Group presentation; Urbana IL, USA; A new tunable QR factorization algorithm (Edward)

slides May 2017; Illinois Data Science Fundamentals Summit; Urbana IL, USA; Scalable numerical linear algebra for data science (Edgar)

slides May 2017; MolSSI Workshop on Core Software Blocks in Quantum Chemistry: Tensors and Integrals; Monterey Bay CA, USA; An overview of Cyclops Tensor Framework (Edgar)


article Edward Hutter and Edgar Solomonik Communication-avoiding Cholesky-QR2 for rectangular matrices IEEE International Parallel and Distributed Processing Symposium (IPDPS), Rio de Jianero, Brazil, May, 2019, to appear.
report Linjian Ma and Edgar Solomonik Accelerating Alternating Least Squares for Tensor Decomposition by Pairwise Perturbation> arXiv:1811.10573 [math.NA], November 2018.
article Tobias Wicky, Edgar Solomonik, and Torsten Hoefler Communication-avoiding parallel algorithms for solving triangular systems of linear equations IEEE International Parallel and Distributed Processing Symposium (IPDPS), Orlando, FL, June 2017, pp. 678-687. report