Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino Symmetry Preserving The Scope of Problems Discretization Schemes through Function Theoretical Methods in Hypercomplex Variables Numerical Analysis Motivation behind this talk Lie-algebraic discretizations Umbral Calculus Nelson Faustino Revisited Radial algebra approach Appell Sets Center of Mathematics, Computation and Cognition, UFABC su ( 1 , 1 ) symmetries Discretization of nelson.faustino@ufabc.edu.br Operators of Sturm-Liouville type ICNAAM 2017, Thessaloniki, Greece, 25–30 September 2017 Discrete Electromagnetic Schr¨ odinger operators Interplay with Bayesian Statistics 1 / 34
Symmetry Pre- serving Dis- cretiza- tion Schemes The Scope of Problems 1 Nelson Faustino Function Theoretical Methods in Numerical Analysis The Scope of Motivation behind this talk Problems Function Theoretical Methods in Numerical Analysis Lie-algebraic discretizations 2 Motivation behind this talk Umbral Calculus Revisited Lie-algebraic Radial algebra approach discretizations Umbral Calculus Appell Sets Revisited Radial algebra su ( 1 , 1 ) symmetries approach Appell Sets su ( 1 , 1 ) symmetries Discretization of Operators of Sturm-Liouville type 3 Discretization of Operators of Discrete Electromagnetic Schr¨ odinger operators Sturm-Liouville type Interplay with Bayesian Statistics Discrete Electromagnetic Schr¨ odinger operators Interplay with Bayesian Statistics 2 / 34
Symmetry Pre- ’(. . . ) When Columbus set sail, he was like an applied serving Dis- cretiza- mathematician, paid for the search of the solution of a concrete problem: tion Schemes find a way to India. His discovery of the New World was similar to the Nelson Faustino work of a pure mathematician (. . . ) ” The Scope of Vladimir Arnol’d , Notices of AMS, Volume 44, Number 4 (1997) Problems Function Theoretical Methods in Numerical Analysis Motivation behind this talk Lie-algebraic discretizations Umbral Calculus Revisited Radial algebra approach Appell Sets su ( 1 , 1 ) symmetries Discretization of Operators of Sturm-Liouville type Discrete Electromagnetic Figure: From left to right: Discrete Dirac operators on graphs/dual Schr¨ odinger operators graphs vs. 7 − point representation of the ’discrete’ Laplacian ∆ h . Interplay with Bayesian Statistics 3 / 34
Overview Why should we use Finite Difference Dirac Operators? Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino Hypercomplex analysis approach: The Scope of Problems Useful to rewrite our main problem in a more compact form 1 Function Theoretical Methods in (e.g. Lam´ e/Navier-Stokes/Schr¨ odinger equations ); Numerical Analysis Motivation behind this talk Get exact representation formulae to solve vector-field 2 Lie-algebraic problems numerically (discrete counterparts); discretizations Umbral Calculus The regularity conditions that we need to impose on the Revisited 3 Radial algebra approach design of convergence schemes are quite lower in Appell Sets comparison with the usual convergence conditions su ( 1 , 1 ) symmetries associated to standard finite difference schemes. Discretization of Operators of Sturm-Liouville type Discrete Electromagnetic Schr¨ odinger operators Interplay with Bayesian Statistics 4 / 34
Overview Why should we use Finite Difference Dirac Operators? Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino Hypercomplex analysis approach: The Scope of Problems Useful to rewrite our main problem in a more compact form 1 Function Theoretical Methods in (e.g. Lam´ e/Navier-Stokes/Schr¨ odinger equations ); Numerical Analysis Motivation behind this talk Get exact representation formulae to solve vector-field 2 Lie-algebraic problems numerically (discrete counterparts); discretizations Umbral Calculus The regularity conditions that we need to impose on the Revisited 3 Radial algebra approach design of convergence schemes are quite lower in Appell Sets comparison with the usual convergence conditions su ( 1 , 1 ) symmetries associated to standard finite difference schemes. Discretization of Operators of Sturm-Liouville type Discrete Electromagnetic Schr¨ odinger operators Interplay with Bayesian Statistics 4 / 34
Overview Why should we use Finite Difference Dirac Operators? Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino Hypercomplex analysis approach: The Scope of Problems Useful to rewrite our main problem in a more compact form 1 Function Theoretical Methods in (e.g. Lam´ e/Navier-Stokes/Schr¨ odinger equations ); Numerical Analysis Motivation behind this talk Get exact representation formulae to solve vector-field 2 Lie-algebraic problems numerically (discrete counterparts); discretizations Umbral Calculus The regularity conditions that we need to impose on the Revisited 3 Radial algebra approach design of convergence schemes are quite lower in Appell Sets comparison with the usual convergence conditions su ( 1 , 1 ) symmetries associated to standard finite difference schemes. Discretization of Operators of Sturm-Liouville type Discrete Electromagnetic Schr¨ odinger operators Interplay with Bayesian Statistics 4 / 34
Overview Some references Symmetry Pre- Boundary value problems: G¨ urlebeck and Spr¨ oßig - serving Dis- 1 cretiza- Quaternionic and Clifford calculus for Engineers and tion Schemes Nelson Faustino Physicists (1997). The Scope of Discrete Fundamental solutions for Difference Dirac 2 Problems Function Theoretical operators: G¨ urlebeck and Hommel , On finite difference Methods in Numerical Analysis Dirac operators and their fundamental solutions, Adv. Appl. Motivation behind this talk Clifford Algebras, 11 , 89 – 106 (2003). Lie-algebraic discretizations Numerical implementation using discrete counterparts: 3 Umbral Calculus Revisited Faustino, G¨ urlebeck, Hommel, and K¨ ahler - Difference Radial algebra approach Potentials for the Navier-Stokes equations in unbounded Appell Sets domains , J. Diff. Eq. & Appl., Journal of Difference Equations su ( 1 , 1 ) symmetries and Applications, 12(6), 577-595. Discretization of Operators of To take a look for further progresses on this direction, attend Sturm-Liouville 4 type tomorrow ( September 26 ) the morning talks of the Discrete Electromagnetic Schr¨ odinger 13th Symposium on Clifford Analysis and Applications operators Interplay with (ROOM 3) . Bayesian Statistics 5 / 34
Overview Some references Symmetry Pre- Boundary value problems: G¨ urlebeck and Spr¨ oßig - serving Dis- 1 cretiza- Quaternionic and Clifford calculus for Engineers and tion Schemes Nelson Faustino Physicists (1997). The Scope of Discrete Fundamental solutions for Difference Dirac 2 Problems Function Theoretical operators: G¨ urlebeck and Hommel , On finite difference Methods in Numerical Analysis Dirac operators and their fundamental solutions, Adv. Appl. Motivation behind this talk Clifford Algebras, 11 , 89 – 106 (2003). Lie-algebraic discretizations Numerical implementation using discrete counterparts: 3 Umbral Calculus Revisited Faustino, G¨ urlebeck, Hommel, and K¨ ahler - Difference Radial algebra approach Potentials for the Navier-Stokes equations in unbounded Appell Sets domains , J. Diff. Eq. & Appl., Journal of Difference Equations su ( 1 , 1 ) symmetries and Applications, 12(6), 577-595. Discretization of Operators of To take a look for further progresses on this direction, attend Sturm-Liouville 4 type tomorrow ( September 26 ) the morning talks of the Discrete Electromagnetic Schr¨ odinger 13th Symposium on Clifford Analysis and Applications operators Interplay with (ROOM 3) . Bayesian Statistics 5 / 34
Overview Some references Symmetry Pre- Boundary value problems: G¨ urlebeck and Spr¨ oßig - serving Dis- 1 cretiza- Quaternionic and Clifford calculus for Engineers and tion Schemes Nelson Faustino Physicists (1997). The Scope of Discrete Fundamental solutions for Difference Dirac 2 Problems Function Theoretical operators: G¨ urlebeck and Hommel , On finite difference Methods in Numerical Analysis Dirac operators and their fundamental solutions, Adv. Appl. Motivation behind this talk Clifford Algebras, 11 , 89 – 106 (2003). Lie-algebraic discretizations Numerical implementation using discrete counterparts: 3 Umbral Calculus Revisited Faustino, G¨ urlebeck, Hommel, and K¨ ahler - Difference Radial algebra approach Potentials for the Navier-Stokes equations in unbounded Appell Sets domains , J. Diff. Eq. & Appl., Journal of Difference Equations su ( 1 , 1 ) symmetries and Applications, 12(6), 577-595. Discretization of Operators of To take a look for further progresses on this direction, attend Sturm-Liouville 4 type tomorrow ( September 26 ) the morning talks of the Discrete Electromagnetic Schr¨ odinger 13th Symposium on Clifford Analysis and Applications operators Interplay with (ROOM 3) . Bayesian Statistics 5 / 34
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