Numerical Analysis / Análisis Numérico

Overview
Researchers
International Collaborators
Former Members
Visitors
Meetings
Academic articles
Links

Research directions

We are mainly interested in the development and analysis of numerical algorithms to approximate solutions of partial differential equations. Our group is focused on three main goals:

DEVELOPMENT. We investigate novel numerical methods that, for certain criteria, improve the existing methods and enable an efficient computational implementation.

ANALYSIS. We use mathematical analysis to give theoretical proofs about the properties and performance of numerical methods.

IMPLEMENTATION. We contribute and develope libraries to obtain numerical evidence about the performance of numerical methods. Such libraries may be used to do academic experiments or even high-performance computing.

Team

IGNACIO MUGA. His work has been focused on the analysis of numerical methods (semi-analytical and FEM-based methods) including applications to geophysics, acoustics, and electromagnetism. Right now he is particularly interested in Discontinuous Petrov-Galerkin (DPG) methods and the development of residual minimization algorithms for variational problems in Banach spaces. The working network of professor Muga includes world-class research centers such as the Institute for Computational Engineering and Sciences (ICES) - USA, the Basque Center for Applied Mathematics (BCAM) - Spain, and the University of Nottingham - UK.

SERGIO ROJAS. His research interests are numerical analysis of partial differential equations, finite element methods, discontinuous Galerkin methods.

PAULINA SEPÚLVEDA. Her interests are in developing accurate, stable and efficient numerical methods for the solution of real-world problems based on theoretical foundations. In particular, she is interested in developing space-time methods with the aim of exploiting adaptive space-time mesh-refinement techniques. Among these methods, she has been working in space-time Tent-pitching techniques, space-time Petrov-Galerkin methods, and space-time discontinuous Petrov-Galerkin methods for the Schrodinger and wave equations. She has experience in developing and contributing to finite element and finite volume software libraries, written in different programming languages: C++, Fortran, Python, and Matlab. Her collaboration network includes researchers at Lawrence Livermore National Lab, Portland State University, US. Geological Survey, and the Basque Center for applied mathematics.

LILIANA CAMARGO. Her research interests are numerical analysis of partial differential equations, finite element methods, hybridizable discontinuous Galerkin methods, A posteriori error analysis, fluid mechanics and electromagnetism.

PATRICK VEGA. His research interests are numerical analysis of partial differential equations,finite element methods, discontinuous Galerkin methods, A posteriori error analysis, fluid mechanics and electromagnetism.

Professors

Muga Urquiza, Ignacio

Doctor en Ciencias de la Ingeniería con Mención en Modelamiento Matemático, Universidad de Chile, Chile.

mail:	ignacio.muga@pucv.cl
phone:	(+56) 32 227 4033
website:	Academic Website

Rojas, Sergio

Doctor en Ciencias de la Ingeniería, Pontificia Universidad Católica de Chile, Chile

mail:	sergio.rojas.h@pucv.cl
phone:	(+56) 32 227 4024
website:	Academic Website

Sepúlveda Salas, Paulina

PhD. in Mathematical Sciences, Portland State University, USA.

mail:	paulina.sepulveda@pucv.cl
phone:	(+56) 32 227 4024
website:	Academic Website

Postdoctoral fellows

Camargo, Liliana

mail:	lilianacamargomazuera @ gmail.com
phone:	(+56) —
website:	Academic Website

Vega, Patrick

Doctor en Ciencias Aplicadas c/m Ingeniería Matemática, Universidad de Concepción, Chile.

mail:	patrick.vega@pucv.cl
phone:	(+56) 32 227 4024
website:	Academic Website

PhD. Students

Perera, Jorge

PhD. Student, Pontificia Universidad Católica de Valparaíso, Chile.

mail:	jorgeperera1597@gmail.com
phone:
website:	Academic Website

MSc. Students

Mondaca, Francisco

MS. Student, Pontificia Universidad Católica de Valparaíso, Chile.

mail:	francisco.mondaca.t@mail.ucv.cl
phone:
website:	Academic Website

Ruminot, Ignacio

MSc. Student, Pontificia Universidad Católica de Valparaíso, Chile.

mail:	iruminota@gmail.com
phone:
website:	Academic Website

BSc. students

Herrera, Victoria

BSc. Student, Pontificia Universidad Católica de Valparaíso, Chile.

mail:	victoria.herrera.b@mail.pucv.cl
phone:
website:	Academic Website

Zamorano, Nicolas

BSc. Student, Pontificia Universidad Católica de Valparaíso, Chile.

mail:	nicolas.zamo12@gmail.com
phone:
website:	Academic Website

International Collaborators

Jay Gopalakrishnan, Portland State University, USA. (academic page)
David Pardo, University of the Basque Country, Spain. (academic page)
Judith Muñoz-Matute, Basque Center for Applied Mathematics, Spain. (academic page)
Matteo Croci, Basque Center for Applied Mathematics, Spain. (academic page)
Maciej Paszynski, AGH University of Science and Technology, Poland. (academic page)
Maciej Wozniak, AGH University of Science and Technology, Poland. (academic page)
Marcin Los, AGH University of Science and Technology, Poland. (academic page)
Kris van der Zee, Nottingham University, UK. (academic page)
Ignacio Brevis, Nottingham University, UK. (academic page)
Luis Espath, Nottingham University, UK. (academic page)
Stéphane Lanteri, Inria centre at Université Côte d’Azur, France. (academic page)
Jamie Taylor, CUNEF Universidad, Spain. (academic page)
Théophile Chaumont-Frelet, Inria centre at Université Côte d’Azur, France. (academic page)

Former Members

Brevis, Ignacio

Doctor en Ciencias de la Ingeniería con mención en Modelación Matemática, Universidad de Chile, Chile.

mail:	ignacio.brevis.v@gmail.com
phone:	(+56) —
website:	Academic Website

Former PhD. Students

Millar Vasquez, Felipe

PhD. Student, Pontificia Universidad Católica de Valparaíso, Chile.

mail:	felipe.millar.v@mail.pucv.cl
phone:
website:	Academic Website

Former MSc. Students

Pinochet Soto, Gabriel

MSc. in Mathematics, Pontificia Universidad Católica de Valparaíso, Chile.

mail:	gpinochetsoto@gmail.com
phone:
website:	Academic Website

Former BSc. Students

Gonzalez, Carlos

BSc. in mathematics, Pontificia Universidad Católica de Valparaíso, Chile.

mail:	carlgonzamo98@gmail.com
phone:
website:	Academic Website

Visitors 2023

Witold Dzwinel. Full Professor. AGH University of Science and Technology, Poland.
Maciej Paszyński. Full Professor, AGH University of Science and Technology, Poland.
Paweł Maczuga. PhD. Student. AGH University of Science and Technology, Poland.
Julen Alvarez-Aramberri. Postdoctoral Researcher, UPV/EHU, Spain
Judit Muñoz-Matute. Postdoctoral Researcher Oden Institute for Computational Engineering and Sciences, USA.
Ángel J. Omella. Postdoctoral Researcher UPV/EHU, Spain.
David Pardo. Research Professor. UPV/EHU, Spain.
Leszek Siwik. Assistant Professor. AGH University of Science and Technology, Poland.

Visitors 2022

Carlos Uriarte. PhD. Student. Basque Center for Applied Mathematics, Spain.
Otilio Rojas. Researcher, Barcelona Supercomputing Center, Spain.
Lisa J. Fauci. Full Profesor, Tulane University, USA.
Zhiqiang Cai. Full Professor, Perdue University, USA.
Marcin Łoś. PhD. in Computer Sciences, AGH University of Sciences and Technology, Poland
Maciej Woźniak. PhD. in Computer Sciences, AGH University of Sciences and Technology, Poland.

Visitors 2020

Rodriguez Melendez, Oscar. PhD. Student, Basque Center for Applied Mathematics, Spain.

Permanent Seminar

Seminario Caleta Numérica

Past Events

Valparaíso Numérico IV

Academic articles

Submitted manuscripts

J. Trynda, M. Woźniak, and S. Rojas: A study of concurrent multi-frontal solvers for modern massively parallel architectures (2023). arXiv:2306.08994 [math.NA].
J. M. Taylor, M. Bastidas, D. Pardo, and I. Muga: Deep Fourier Residual method for solving time-harmonic Maxwell’s equations (2023). arXiv:2305.09578 [math.NA].
I. Brevis, I. Muga, D. Pardo, O. Rodríguez, and K. G. van der Zee: Learning quantities of interest from parametric PDEs: An efficient neural-weighted Minimal Residual approach (2023). arXiv:2304.01722 [math.NA].
L. Camargo, S. Rojas, and P. Vega: Minimum-residual a posteriori error estimates for a hybridizable discontinuous Galerkin discretization of the Helmholtz equation (2023). arXiv:2304.00418 [math.NA].
J. G. Hasbani, P. Sepúlveda, I. Muga, V. M. Calo, and S. Rojas: Adaptive stabilized finite elements via residual minimization onto bubble enrichments (2023). arXiv:2303.17982 [math.NA].
M. Skotniczny, A. Paszyńska, S. Rojas, and M. Paszyński: Complexity of direct and iterative solvers on space-time formulations versus time-marching schemes for h-refined grids towards singularities (2022). arXiv:2110.05804 [math.NA].
T. Chaumont-Frelet and P. Vega: Frequency-explicit a posteriori error estimates for discontinuous Galerkin discretizations of Maxwell’s equations (2022). arXiv:2208.01475 [math.NA].

Papers in refereed journals

2023

I. Muga, S. Rojas, and P. Vega: An adaptive superconvergent mixed finite element method based on local residual minimization. SIAM Journal on Numerical Analysis, accepted for publication, 2023.
T. Służalec, M. Dobija, A. Paszyńska, I. Muga, M. Łoś, and M. Paszyński: Automatic stabilization of finite-element simulations using neural networks and hierarchical matrices. Computer Methods in Applied Mechanics and Engineering, 411, Paper No. 116073, 2023.
T. Służalec, R. Grzeszczuk, S. Rojas, W. Dzwinel, and M. Paszyński: Quasi-optimal hp-finite element refinements towards singularities via deep neural network prediction. Computers and Mathematics with Applications, 142, pages 157–174 , 2023.
P. Sepúlveda, I. Muga, N. Sáinz, R. G. Rojas, S. Ossandón: Nanoionics from a quantum mechanics point of view: Mathematical modeling and numerical simulation. Computer Methods in Applied Mechanics and Engineering, 407, Paper No. 115926, 2023.
C. Uriarte, D. Pardo, I. Muga, and J. Muñoz-Matute: A deep double Ritz method (D²RM) for solving partial differential equations. Computer Methods in Applied Mechanics and Engineering, 405, Paper No. 115892, 2023.
J. M. Taylor, D. Pardo, and I. Muga: A deep Fourier residual method for solving PDEs using neural networks. Computer Methods in Applied Mechanics and Engineering, 405, Paper No. 115850, 2023.

2022

I. Brevis, I. Muga, and K. G. van der Zee. Neural control of discrete weak formulations: Galerkin, least-squares and minimal-residual methods with quasi-optimal weights. Computer Methods in Applied Mechanics and Engineering, 402, Paper No. 115716, 2022.
A. Erdozain, I. Muga, V. Péron, G. Pinochet. Semi-analytical solutions for the problem of the electric potential set in a borehole with a highly conductive casing. GEM-International Journal on Geomathematics, 13, 1-20, 2022.
M. Woźniak, A. Szyszka, and S. Rojas. A study of efficient concurrent integration methods of B-Spline basis functions in IGA-FEM. Journal of Computational Science, 64, Paper No. 101857, 2022.
T. Chaumont-Frelet and P. Vega. Frequency-explicit a posteriori error estimates for finite element discretizations of Maxwell’s equations. SIAM Journal on Numerical Analysis, 60, 1774-1798, 2022.
T. Chaumont-Frelet and P. Vega. Frequency-explicit approximability estimates for finite element discretizations of Maxwell’s equations. Calcolo, 59, Paper No. 22, 2022.
F. Kyburg, S. Rojas, and V. M. Calo. Incompressible flow modeling using an adaptive stabilized finite element method based on residual minimization. International Journal for Numerical Methods in Engineering, 123, 1717-1735, 2022.
F. Millar, I. Muga, S. Rojas, and K. G. van der Zee. Projection in negative norms and the regularization of rough linear functionals. Numerische Mathematik, 150, 1087-1121, 2022.

2021

R. J. Cier, T. Poulet, S. Rojas, M. Veveakis, and V. M. Calo. Automatically adaptive stabilized finite elements and continuation analysis for compaction banding in geomaterials. International Journal for Numerical Methods in Engineering, 122, 6234-6252, 2021.
R. J. Cier, S. Rojas, and V. M. Calo. Automatically adaptive, stabilized finite element method via residual minimization for heterogeneous, anisotropic advection–diffusion–reaction problems. Computer Methods in Applied Mechanics and Engineering, 385, Paper No. 114027, 2021.
T. Chaumont-Frelet, S. Lanteri, and P. Vega. A posteriori error estimates for finite element discretizations of Maxwell’s equations coupled with a non-local hydrodynamic Drude model. Computer Methods in Applied Mechanics and Engineering, 385, Paper No. 114002, 2021.
M. Łoś, J. Muñoz-Matute, I. Muga, and M. Paszyński. Isogeometric residual minimization (iGRM) for non-stationary Stokes and Navier–Stokes problems. Computers and Mathematics with Applications, 95, 200–214, 2021.
I. Brevis, I. Muga, and K. G. van der Zee. A machine-learning minimal-residual (ML-MRes) framework for goal-oriented finite element discretizations. Computers and Mathematics with Applications, 95, 186-199, 2021.
S. Rojas, D. Pardo, P. Behnoudfar, and V. M. Calo. Goal-oriented adaptivity for a conforming residual minimization method in a dual discontinuous Galerkin norm. Computer Methods in Applied Mechanics and Engineering, 377, Paper No. 113686, 2021.
M. Łoś, S. Rojas, M. Paszyński, I. Muga, and V. M. Calo. DGIRM: Discontinuous Galerkin based Isogeometric Residual Minimization for the Stokes problem. Journal of Computational Science, 50, Paper No. 101306, 2021.
R. J. Cier, S. Rojas, and V. M. Calo. A nonlinear weak constraint enforcement method for advection-dominated diffusion problems. Mechanics Research Communications, 112, Paper No. 103602, 2021.
V. M. Calo, M. Łoś, Q. Deng, I. Muga, and M. Paszyński. Isogeometric Residual Minimization Method (iGRM) with direction splitting preconditioner for stationary advection-dominated diffusion problems. Computers Methods in Applied Mechanics and Engineering, 373, Paper No. 113214 , 2021.

2020

V. M. Calo, A. Ern, I. Muga, and S. Rojas. An Adaptive Stabilized Conforming Finite Element Method via Residual Minimization on Dual Discontinuous Galerkin Norms. Computer Methods in Applied Mechanics and Engineering, 363, Paper No. 112891, 2020.
M. Łoś, J. Muñoz-Matute, I. Muga, and M. Paszyński. Isogeometric Residual Minimization Method (iGRM) with direction splitting for non-stationary advection–diffusion problems. Computers and Mathematics with Applications, 79(2), pages 213-229 , 2020.
I. Muga, and K. G. van der Zee. Discretization of Linear Problems in Banach Spaces: Residual Minimization, Nonlinear Petrov-Galerkin, and Monotone Mixed Methods. SIAM J. Numer. Anal., 58(6), pages 3406-3426, 2020.

2019

I. Brevis, Á. Rodríguez-Rozas, J. H. Ortega, and D. Pardo. Source Time Reversal (STR) Method for Linear Elasticity, Computers and Mathematics with Applications, 77(5), 1358-1375, 2019.
J. Gopalakrishnan and P. Sepúlveda. A space-time DPG method for the wave equation in multiple dimensions. In collection “Space-Time Methods: Applications to Partial Differential Equations” , 117-140, 2019.
P. Houston, I. Muga, S. Roggendorf, and K. G. van der Zee. The Convection-Diffusion-Reaction Equation in Non-Hilbert Sobolev Spaces: A Direct Proof of the Inf-Sup Condition and Stability of Galerkin’s Method. Comput. Methods Appl. Math., 19(3), 503–522, 2019.
I. Muga, M. J. W. Tyler, and K. G. van der Zee. The Discrete-Dual Minimal-Residual Method (DDMRes) for Weak Advection-Reaction Problems in Banach spaces. Comput. Methods Appl. Math., 19(3), 557-579.
R. Rebolledo, S. A. Navarrete, S. Kéfi, S. Rojas, and P. A. Marquet. An Open-System Approach to Complex Biological Networks. SIAM Journal on Applied Mathematics, 79(2), 619–640. 2019.
S. Rojas, I. Muga, and C. Jerez-Hanckes. The outgoing time-harmonic electromagnetic wave in a half-space with non-absorbing impedance boundary condition. ESAIM: M2AN, 53(1), 325-350, 2019.

2018

V. Darrigrand, Á. Rodríguez‐Rozas, I. Muga, D. Pardo, A. Romkes, and S. Prudhomme. Goal‐oriented adaptivity using unconventional error representations for the multidimensional Helmholtz equation. Int. J. Numer. Meth. Engng., 113(1), 22–42, 2018.
S. Rojas, R. Hein, and M. Durán. On an equivalent representation of the Green’s function for the Helmholtz problem in a non-absorbing impedance half-plane. Computers & Mathematics with Applications, 75(11), 3903–3917, 2018.
M. Shahriari, S. Rojas, D. Pardo, Á. Rodríguez-Rozas, A. B. Shaaban, V. M. Calo, and I. Muga. A Numerical 1.5D Method for the Rapid Simulation of Geophysical Resistivity Measurements. Geosciences, 8(6), pages 225, 2018.
K. Voronin, C. S. Lee, M. Neumüller, P. Sepulveda, and P. Vassilevski. Space-time discretizations using constrained first-order system least-squares (CFOSLS). Journal of Computational Physics, 373, pages 863-876, 2018.

2017

I. Brevis, J. H. Ortega, and D. Pardo. A Source Time Reversal Method for Seismicity Induced by Mining, Inverse Problems and Imaging, Vol. 10(1), 25-45, 2017.
L. Demkowicz, J. Gopalakrishnan, S. Nagaraj, and P. Sepúlveda. A Spacetime DPG Method for the Schrödinger Equation. SIAM J. Numer. Anal., 55(4), pages 1740–1759, 2017.

2016

S. Rojas, I. Muga, and D. Pardo. A quadrature-free method for simulation and inversion of 1.5D direct current (DC) borehole measurements. Computational Geosciences, 20(6), pages 1301-1318, 2016.

2015

V. Darrigrand, D. Pardo, and I. Muga. Goal-oriented adaptivity using unconventional error representations for the 1D Helmholtz equation. Computers and Mathematics with Applications, 69(9), pages 964-979, 2015.
J. Gopalakrishnan, P. Monk, and P. Sepúlveda. A tent pitching scheme motivated by Friedrichs theory. Computers and Mathematics with Applications, 70(5), pages 1114-1135, 2015.
F. Millar and D. Mora. A finite element method for the buckling problem of simply supported Kirchhoff plates. Journal of Computational and Applied Mathematics, 286(1), pages 68-78, 2015.
I. Muga, D. Pardo, P. J. Matuszyk, C. Torres-Verdín. Semi-analytical response of acoustic logging measurements in frequency domain. Computers and Mathematics with Applications, 70(4), pages 314-329, 2015.