Current research

My work concerns geometric PDE and fluid dynamics, numerical and geometric discretization, approximation theory and neural realization of refinement schemes, and computational approaches to quantum field theory. Newer research information and publications are available on my new research page and publication list.

Preprints (my arXiv)

  1. Projection geometry and relaxed quasi-orthogonality for inf-sup stable Galerkin methods [arXiv] 2026.
  2. with Boldsaikhan Bolorkhuu. Exact ReLU realization of binary affine refinement iterates via reflection folding and cone switching [arXiv] 2026.
  3. with Boldsaikhan Bolorkhuu. Exact ReLU realization of affine one-dimensional refinement iterates via residual memory and offset frames [arXiv] 2026.
  4. Conditional Euclidean-Hamiltonian reductions for sign-problem toy models [arXiv] 2026.
  5. Exact ReLU realization of tensor-product refinement iterates [arXiv] 2026.
  6. with Boldsaikhan Bolorkhuu. Exact Loop Controllers for ReLU Realization of Homogeneous Curve Refinements [arXiv] 2026.
  7. Possible Reforms of the Tibetan Lunisolar Calendar [arXiv] 2026.
  8. An expository review of the Chebyshev-Sylvester method in prime number theory [arXiv] 2025.
  9. A direct proof of the equivalence between Dirichlet's principle and Perron's method [arXiv] 2025.
  10. with Michael Holst and David Maxwell. A scaling approach to elliptic theory for geometrically-natural differential operators with Sobolev-type coefficients [arXiv] 2023.
  11. with Vladmir Sicca Gonçalves. A prescribed scalar and boundary mean curvature problem on compact manifolds with boundary [arXiv] 2021.
  12. with Michael Holst and Yunrong Zhu. Local convergence of adaptive methods for nonlinear partial differential equations [arXiv] 2010.

Journal articles

  1. with Luubaatar Badarch, Dorj Byambaa, Hansaem Park, Fady Alnajjar, and Munkhjargal Gochoo. TCubes: Non-privacy invasive dataset for activities of daily living as thermal cubes [journal, DOI]. Expert Systems with Applications, 333, 133943 (2027).
  2. Rotationally invariant dynamical lattice regulators for Euclidean quantum field theories [arXiv, journal, DOI]. Journal of High Energy Physics, 07 (2026) 036.
  3. Projectional continuous data assimilation on the torus: Resonant and kernel-free regimes [arXiv]. Journal of Mathematical Fluid Mechanics, accepted for publication (2026).
  4. with Nella Rotundo. Regularity of solutions of the Navier-Stokes-alpha-beta equations with wall-eddy boundary conditions [arXiv]. Zeitschrift für angewandte Mathematik und Physik, accepted for publication (2026).
  5. with Gerard Awanou. Uniqueness of Weak Solutions to Second Boundary Value Problems for Generated Prescribed Jacobian Equations [arXiv]. Electronic Journal of Differential Equations, accepted for publication, Special Issue 03 (2026).
  6. Convergence Theory of Adaptive Mixed Finite Element Methods for the Stokes Problem [arXiv, journal, DOI]. Journal of Mathematical Sciences, 279, 794–813 (2024).
  7. with Yue-Hong Feng, Haifeng Hu, Ming Mei, and Guojing Zhang. Relaxation Time Limits of Subsonic Steady States for Multidimensional Hydrodynamic Model of Semiconductors [journal, DOI]. SIAM Journal on Mathematical Analysis, 56(5), 6933–6962 (2024).
  8. with Vladmir Sicca Gonçalves. A prescribed scalar and boundary mean curvature problem and the Yamabe classification on asymptotically Euclidean manifolds with inner boundary [arXiv, journal, DOI, correction]. Journal of Geometric Analysis, 33, 342 (2023).
  9. with Kalani Rubasinghe, Guangming Yao, and Jing Niu. Polyharmonic splines interpolation on scattered data in 2D and 3D with applications [journal, DOI, corrigendum]. Engineering Analysis with Boundary Elements, 156, 240–250 (2023). Corrigendum (2024).
  10. with Kalani Rubasinghe, Guangming Yao, and Wen Li. Solving Nonlinear Elliptic PDEs in 2D and 3D Using Polyharmonic Splines and Low-Degree of Polynomials [journal, DOI]. International Journal of Computational Methods, 20(8), 2250051 (2023).
  11. with Ibrahim Al Balushi, Wen Jiang and Tae-Yeon Kim. A posteriori analysis of a B-spline based finite-element method for the stationary quasi-geostrophic equations of the ocean [link]. Computer Methods in Applied Mechanics and Engineering, 371:113317 (2020).
  12. with Erick Schulz. Convergence of discrete exterior calculus approximations for Poisson problems [arxiv]. Discrete & Computational Geometry, 63(2):346–376 (2020).
  13. with Michael Holst and Caleb Meier. Non-CMC solutions of the Einstein constraint equations on compact manifolds with apparent horizon boundaries [arXiv]. Communications in Mathematical Physics, 357(2):467–517 (2018).
  14. with Ibrahim Al Balushi, Wen Jiang and Tae-Yeon Kim. Adaptivity of a B-spline based finite element method for modeling wind-driven ocean circulation [link]. Computer Methods in Applied Mechanics and Engineering, 332:1–24 (2018).
  15. Convergence rates of adaptive methods, Besov spaces, and multilevel approximation [arxiv]. Foundations of Computational Mathematics, 17(4), 917-956 (2017).
  16. with Rustum Choksi and Ihsan Topaloglu. Axisymmetric critical points of a nonlocal isoperimetric problem on the two-sphere [arxiv]. ESAIM: Control, Optimisation and Calculus of Variations, 21(1), 247-270 (2015).
  17. with Douglas Arnold, Richard Falk and Johnny Guzmán. On the consistency of the combinatorial codifferential [arXiv]. Transactions of the American Mathematical Society, 366, 5487-5502 (2014).
  18. with Michael Holst. The Lichnerowicz equation on compact manifolds with boundary. [arXiv]. Classical and Quantum Gravity, 30, 205011 (2013). Selected as a highlight of the journal in 2013-2014.
  19. Adaptive boundary element methods with convergence rates [arXiv]. Numerische Mathematik, 124(3), 471-516 (2013).
  20. with Michael Holst and Evelyn Lunasin. Analysis of a general family of regularized Navier-Stokes and magnetohydrodynamics models [arXiv]. Journal of Nonlinear Science, 20(5), 523-567 (2010).
  21. with Michael Holst and Gabriel Nagy. Rough solutions of the Einstein constraints on closed manifolds without near-CMC conditions [arXiv]. Communications in Mathematical Physics, 288, 547-613 (2009).
  22. with Michael Holst and Gabriel Nagy. Far-from-constant mean curvature solutions of Einstein's constraint equations with positive Yamabe metrics [arXiv]. Physical Review Letters, 100, 161101 (2008).
  23. An optimal adaptive wavelet method for nonsymmetric and indefinite elliptic problems [pdf]. Journal of Computational and Applied Mathematics, 211(1), 90-102 (2008).
  24. with Helmut Harbrecht and Rob Stevenson. An optimal adaptive wavelet method without coarsening of the iterands [pdf]. Mathematics of Computation, 76, 615-629 (2007).
  25. with Rob Stevenson. Computation of singular integral operators in wavelet coordinates [pdf]. Computing, 76, 77-107 (2006).
  26. with Rob Stevenson. Computation of differential operators in wavelet coordinates [pdf]. Mathematics of Computation, 75, 697-709 (2006).

Earlier projects

  1. with Hani Ali. On critical and supercritical regularized Navier-Stokes equations.
  2. with Young Ju Lee. On a modified Oldroyd-B model.
  3. with Michael Holst and Yongcheng Zhou. Adaptive coupling of finite element and boundary element methods for a class of nonlinear interface problems.

Thesis

Adaptive wavelet algorithms for solving operator equations [summary | samenvatting | main part (1.6M)]. PhD thesis. Department of Mathematics, Utrecht University. August 2006. ISBN 90-393-4365-9. [errata for the printed version]

Talks

2025
  1. PDE seminar (invited lecture). Jiangxi Normal University, Nanchang.
  2. PDE seminar (invited lecture). Nanchang University, Nanchang.
2024
  1. PDE seminar (invited lecture). Inner Mongolia University, Hohhot.
2023
  1. Canadian Mathematical Society Winter Meeting (invited meeting). Montreal.
  2. International Conference on Computational and Applied Mathematics (invited meeting). Ulan Bator.
  3. International Workshop on Mathematical Analysis and Applications (invited meeting). Ulan Bator.
  4. International Conference on Applied Sciences and Engineering (invited meeting). Ulan Bator.
2022
  1. International Conference on Optimization, Simulation and Control (invited meeting). Ulan Bator.
2021
  1. International Conference on Applied Sciences and Engineering (invited meeting). Ulan Bator.
2019
  1. On convergence of discrete exterior calculus. Oberwolfach workshop on Innovative Algorithms. Oberwolfach. September 6, 2019.
  2. Einstein constraint equations. Mongolian Academy of Sciences. Ulan Bator. July 5, 2019.
  3. On analysis of discrete exterior calculus. AMS Sectional Meeting. Honolulu. March 24, 2019.
2018
  1. Elliptic estimates for operators with nonsmooth coefficients. Capital Normal University. Bejing. China. July 23, 2018.
  2. A prescribed scalar-mean curvature problem. Chinese Academy of Sciences. Bejing. China. July 23, 2018.
  3. Elliptic estimates for operators with nonsmooth coefficients. International Conference on Partial Differential Equations. Changchun. China. July 18-21, 2018.
  4. A prescribed scalar-mean curvature problem. National University of Mongolia. Ulan Bator. Mongolia. July 5, 2018.
  5. On analysis of discrete exterior calculus. Huazhong University of Science and Technology. Wuhan. China. June 21, 2018.
  6. Elliptic estimates for operators with nonsmooth coefficients. Wuhan University. Wuhan. China. June 21, 2018.
  7. Some scaling estimates in Besov and Triebel-Lizorkin spaces. McGill Geometric Analysis Seminar. February 7, 2018.
  8. Elliptic estimates for operators with nonsmooth coefficients. Workshop on General Relativity and Finite Element Exterior Calculus. University of California, San Digeo. January 13-16, 2018.
2017
  1. Elliptic estimates for operators with rough coefficients. McGill Geometric Analysis Seminar. September 27, 2017.
  2. On analysis of discrete exterior calculus. BIRS Workshop on Geometric Numerical Integration and Structure-Preserving Discretization. Banff. June 11-16, 2017.
  3. On analysis of discrete exterior calculus. Trimester Seminar. Hausdroff Research Instritute for Mathematics. Bonn. April 20, 2017.
2016
  1. A prescribed scalar-mean curvature problem. Session on Geometric PDEs. Canadian Mathematical Society Winter Meeting. Niagara Falls. December 2-5, 2016.
  2. Approximation properties of finite element exterior calculus. Session on Structure-Preserving Discretizations. Canadian Mathematical Society Winter Meeting. Niagara Falls. December 2-5, 2016.
  3. On approximation classes of adaptive methods. Oberwolfach workshop on Adaptive Algorithms. Oberwolfach. September 22, 2016.
  4. Well posedness theory for some regularized models of turbulence. PDE Seminar. National University of Mongolia. Ulan Bator. July 29, 2016.
2015
  1. Well posedness theory for some regularized models of turbulence. Session on Nonlinear Evolutionary Equations. Canadian Mathematical Society Winter Meeting. Montreal. December 6, 2015.
  2. On analysis of discrete exterior calculus. Computational Mathematics Seminar. University of Pittsburgh. Pittsburgh. November 10, 2015.
  3. On analysis of discrete exterior calculus. Applied Mathematics Seminar. University of Leicester. Leicester. October 29, 2015.
  4. On analysis of discrete exterior calculus. Scientific Computation Seminar. University of Nottingham. Nottingham. October 28, 2015.
  5. On approximation classes of adaptive finite element methods. International Congress on Industrial and Applied Mathematics. Beijing. August 14, 2015.
  6. The Navier-Stokes equations with random initial data. Stochastic Analysis and Applications Mongolia. National University of Mongolia. Ulan Bator. August 6, 2015.
  7. The Lichnerowicz equation and the prescribed scalar-mean curvature problem in the compact-with-boundary setting. Focus Program on 100 Years of General Relativity. Fields Institute. Toronto. May 13, 2015.
2014
  1. On approximation classes of adaptive finite element methods [slides]. LMS-EPSRC Durham Symposium. Durham. July 9, 2014.
  2. Adaptive mixed finite element methods for the Stokes problem. McGill Applied Math Working Seminar. February 25, 2014.
2013
  1. The Einstein constraint equations on compact manifolds with or without boundary. PIMS-AMI seminar. Edmonton. October 25, 2013.
  2. Convergence rates of adaptive boundary element methods [slides]. Numerical Analysis & PDE seminar. University of Delaware. May 2, 2013.
  3. On the consistency of the combinatorial codifferential [slides]. Joint Mathematics Meetings. San Diego. January 11, 2013.
2012
  1. Adaptive boundary element methods with convergence rates. ESI Workshop on Wavelet Methods in Scientific Computing. Vienna. November 12, 2012.
  2. On well-posedness of the Navier-Stokes-αβ equations with the wall-eddy boundary conditions [slides]. BIRS Workshop on Regularized and LES Methods for Turbulence. Banff. May 18, 2012.
  3. Nonconstant mean curvature solutions of the Einstein constraint equations [slides]. CRM Workshop on Geometric PDE. Montreal. April 27, 2012.
  4. Einstein constraint equations. McGill Applied Math Working Seminar. March 14, 2012.
  5. Einstein field equations. McGill Applied Math Working Seminar. Montreal. February 8, 2012.
2011
  1. Adaptive boundary element methods with convergence rates. SIAM Conference on Analysis of PDEs. San Diego. November 14, 2011.
  2. Partial regularity results for generalized alpha models of turbulence [slides]. SIAM Conference on Analysis of PDEs. San Diego. November 14, 2011.
  3. Adaptive boundary element methods with convergence rates [slides]. CRM-McGill Applied Mathematics Seminar. Montreal. September 19, 2011.
  4. Generalized Navier-Stokes type models. McGill Applied Math Working Seminar. Montreal. September 14, 2011.
2010
  1. Wavelet Galerkin methods. Applied Mathematics Seminar. National University of Mongolia. Ulan Bator. June 9, 2010.
  2. Einstein's constraint equations. Analysis Seminar. National University of Mongolia. Ulan Bator. June 7, 2010.
  3. Partial regularity for generalized Navier-Stokes equations. McGill Analysis Seminar. McGill University. Montreal. Febuary 12, 2010.
  4. Mathematical general relativity [slides]. Winter school in pure and applied math. McGill University. Montreal. January 8-11, 2010.
2009
  1. Local convergence of adaptive finite element methods for nonlinear problems [slides]. CRM-McGill Applied Mathematics Seminar. Montreal. November 23, 2009.
  2. Solutions of the Einstein constraint equations on asymptotically Euclidean manifolds. 25th Pacific Coast Gravity Meeting. University of Oregon, Eugene. March 27-28, 2009.
  3. On analysis and numerical treatment of Einstein's constraint equations in general relativity [slides]. CRM-McGill Applied Mathematics Seminar. McGill University. Montreal. Canada. February 13, 2009.
2008
  1. Finite elements for the Lichnerowicz equation. Mathematical and Computational Physics Group Meeting. University of California, San Diego. April 21, 2008.
  2. Non-constant mean curvature solutions to Einstein's constraint equations: A fixed point argument [slides]. 24th Pacific Coast Gravity Meeting. University of California, Santa Barbara. March 21-22, 2008.
  3. Analysis and convergent adaptive solution of the Einstein constraint equations [slides]. Workshop on Adaptive Numerical Methods for PDE's. Wolfgang Pauli Institute. Vienna University. Austria. January 21-25, 2008.
  4. A short tour of Sobolev spaces. Mathematical and Computational Physics Group Meeting. University of California, San Diego. February 27, 2008.
2007
  1. Einstein constraints on closed manifolds. Mathematical and Computational Physics Group Meeting. University of California, San Diego. October 24, 2007.
  2. A priori estimates for Petrov-Galerkin approximation on Riemannian manifolds. Mathematical and Computational Physics Group Meeting. University of California, San Diego. June 7, 2007.
  3. Wavelet methods for operator equations. Mathematical and Computational Physics Group Meeting. University of California, San Diego. February 6, 2007.
2006
  1. Adaptive wavelet algorithms for solving operator equations [slides]. General Mathematical Colloquium. Mathematisch Instituut. Universiteit Utrecht. September 21, 2006.
  2. Adaptive wavelet algorithms with truncated residuals [slides]. EC-IHP Breaking Complexity Final Meeting. Wolfgang Pauli Institute. Vienna University. Austria. September 14-18, 2006.
  3. Adaptive wavelet methods with truncated residuals. Postgraduate seminar IGPM. RWTH Aachen. Germany. August 31, 2006.
  4. Adaptive wavelet algorithms for solving operator equations. International Conference on Multilevel Iterative Methods. Peking University. Beijing. China. August 14-18, 2006.
  5. Adaptive wavelet methods with truncated residuals. EC-IHP Breaking Complexity Spring Meeting. Politecnico di Torino. Turin. Italy. April 20-22, 2006.
  6. An optimal adaptive wavelet method without coarsening of the iterands [slides]. Joint HASSIP/DFG-SPP1114 Workshop "Recent Progress in Wavelet Analysis and Frame Theory". Universität Bremen. Bremen. Germany. January 23-26, 2006.
2005
  1. An optimal adaptive wavelet method for strongly elliptic operator equations [slides]. Workshop on Fast Numerical Solution of PDE's. Mathematisch Instituut. Universiteit Utrecht. December 20-22, 2005.
  2. Optimal adaptive wavelet methods for linear operator equations [slides for screen, for printer]. Biweekly numerical mathematics colloquium. Mathematisch Instituut. Universiteit Utrecht. June 17, 2005.
2004
  1. Computation of operators in wavelet coordinates [slides]. EC-IHP Breaking Complexity Mid-Term Meeting. Istituto di Matematica Applicata alle Tecnologie Industriali. Pavia. Italy. December 9 - 10, 2004.
  2. Computation of operators in wavelet coordinates [slides]. 6th Minisimposium "Topics In and Around Numerical Analysis". KdV Institute for Mathematics. Amsterdam. October 27, 2004.