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Linear complexity for FE forward modeling: Transfer matrices, fast FE solver methods and a first sensitivity study Carsten Wolters Tensor weiter erlaeutern.

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Presentation on theme: "Linear complexity for FE forward modeling: Transfer matrices, fast FE solver methods and a first sensitivity study Carsten Wolters Tensor weiter erlaeutern."— Presentation transcript:

1 Linear complexity for FE forward modeling: Transfer matrices, fast FE solver methods and a first sensitivity study Carsten Wolters Tensor weiter erlaeutern Zeitableitung Programmlaenge, 1Mill. WM-Rendering->5000 nodes 1:10 raus und anfangs erwaehnen Institut für Biomagnetismus und Biosignalanalyse, Westfälische Wilhelms-Universität Münster Münster, January 24, 2017

2 Outline Linear complexity for FEM forward modeling: Transfer matrices
Fast FE solver methods Modeling and influence of white matter anisotropy

3 Outline Linear complexity for FEM forward modeling: Transfer matrices
Fast FE solver methods Modeling and influence of white matter anisotropy

4 Link to own work for transfer matrices:
[Wolters, Grasedyck & Hackbusch, Inverse Problems, 2004] Link to own work for transfer matrices: PEBBLES: Parallel and Element Based grey Box Linear Equation Solver

5 [Weinstein, Zhukov & Johnson, Annals Biomed. Eng.,2000]
[Wolters, Grasedyck & Hackbusch, Inverse Problems, 2004] EEG transfer matrix D: electric displacement rho: electric free charge density E: electric field B: magnetic induction mu: magnetic permeability (vacuum) j: electric current density Phi: electric scalar potential Psi: magnetic flux A: magnetic vector potential divA=0: Coulomb’s gauge conductor loop

6 [Wolters, Grasedyck & Hackbusch, Inverse Problems, 2004]
MEG transfer matrix D: electric displacement rho: electric free charge density E: electric field B: magnetic induction mu: magnetic permeability (vacuum) j: electric current density Phi: electric scalar potential Psi: magnetic flux A: magnetic vector potential divA=0: Coulomb’s gauge conductor loop

7 [Wolters, Grasedyck & Hackbusch, Inverse Problems, 2004]
MEG transfer matrix D: electric displacement rho: electric free charge density E: electric field B: magnetic induction mu: magnetic permeability (vacuum) j: electric current density Phi: electric scalar potential Psi: magnetic flux A: magnetic vector potential divA=0: Coulomb’s gauge conductor loop

8 [Wolters, Grasedyck & Hackbusch, Inverse Problems, 2004]
MEG transfer matrix D: electric displacement rho: electric free charge density E: electric field B: magnetic induction mu: magnetic permeability (vacuum) j: electric current density Phi: electric scalar potential Psi: magnetic flux A: magnetic vector potential divA=0: Coulomb’s gauge conductor loop

9 Computing the transfer matrices
[Wolters, Grasedyck & Hackbusch, Inverse Problems, 2004] Computing the transfer matrices der Tensor muss erklärt werden!!

10 Outline Linear complexity for FEM forward modeling: Transfer matrices
Fast FE solver methods Modeling and influence of white matter anisotropy

11 Link to this work: [Wolters, Vorlesungsskriptum, Chapter 7.2]
[Wolters, Kuhn, Anwander & Reitzinger, Comp. Vis. Sci.., 2002] [Lew, Wolters, Dierkes, Roer & MacLeod, Appl.Num.Math., 2009] Link to this work: PEBBLES: Parallel and Element Based grey Box Linear Equation Solver

12 FEM solver aspects [Wolters, Vorlesungsskriptum, Chapter 7.2]
[Wolters, Kuhn, Anwander & Reitzinger, Comp. Vis. Sci.., 2002] [Lew, Wolters, Dierkes, Roer & MacLeod, Appl.Num.Math., 2009] FEM solver aspects D: electric displacement rho: electric free charge density E: electric field B: magnetic induction mu: magnetic permeability (vacuum) j: electric current density Phi: electric scalar potential Psi: magnetic flux A: magnetic vector potential divA=0: Coulomb’s gauge conductor loop

13 Preconditioned Conjugate Gradient Method
[Hackbusch, Springer, 1994] Preconditioned Conjugate Gradient Method PCG accuracy

14 Iterative solver for FE-equation system
[Wolters, Vorlesungsskriptum, Chapter 7.2] [Wolters, Kuhn, Anwander & Reitzinger, Comp. Vis. Sci.., 2002] [Lew, Wolters, Dierkes, Roer & MacLeod, Appl.Num.Math., 2009] Iterative solver for FE-equation system FE discretization, meshsize: with condition number der Tensor muss erklärt werden!! Example:

15 [Wolters, Reitzinger, Basermann, Burkhardt, Hartmann, Kruggel & Anwander, Proceedings of BIOMAG Helsinki, 2000] Link to this work: PEBBLES: Parallel and Element Based grey Box Linear Equation Solver

16 Iterative solver for FE-equation system
[Wolters, Reitzinger, Basermann, Burkhardt, Hartmann, Kruggel & Anwander, Proceedings of BIOMAG Helsinki, 2000] Iterative solver for FE-equation system der Tensor muss erklärt werden!!

17 Iterative solver for FE-equation system
[Wolters, Reitzinger, Basermann, Burkhardt, Hartmann, Kruggel & Anwander, Proceedings of BIOMAG Helsinki, 2000] Iterative solver for FE-equation system FE discretization, meshsize: with condition number Convergence of the CG solver: c wird asymptotische Konvergenzrate genannt, denn lim_{i->infty}(c^i\frac{2}{1+c^{2i}})^(1/i)=c Formel wird mit Hilfe der Chebyshev-Polynome bewiesen und ist nur eine obere Schranke, denn es wird ja \sigma_M=[a,b] gewaehlt, obwohl womoeglich \sigma_M=[a,a´]\cup[b´,b] moeglich waere. Example: Strategy: Preconditioning!

18 Jacobi preconditioning
[Wolters, Vorlesungsskriptum, Chapter 7.2] [Wolters, Kuhn, Anwander & Reitzinger, Comp. Vis. Sci.., 2002] [Lew, Wolters, Dierkes, Roer & MacLeod, Appl.Num.Math., 2009] Jacobi preconditioning

19 Incomplete Cholesky preconditioning
[Wolters, Vorlesungsskriptum, Chapter 7.2] [Wolters, Kuhn, Anwander & Reitzinger, Comp. Vis. Sci.., 2002] [Lew, Wolters, Dierkes, Roer & MacLeod, Appl.Num.Math., 2009] Incomplete Cholesky preconditioning

20 Incomplete Cholesky preconditioning
[Wolters, Vorlesungsskriptum, Chapter 7.2] [Wolters, Kuhn, Anwander & Reitzinger, Comp. Vis. Sci.., 2002] [Lew, Wolters, Dierkes, Roer & MacLeod, Appl.Num.Math., 2009] Incomplete Cholesky preconditioning

21 MultiGrid (MG) preconditioning
[Hackbusch, Springer, 1994] [Wolters, Vorlesungsskriptum, Chapter 7.2] [Wolters, Kuhn, Anwander & Reitzinger, Comp. Vis. Sci.., 2002] [Lew, Wolters, Dierkes, Roer & MacLeod, Appl.Num.Math., 2009] MultiGrid (MG) preconditioning Principle of MG Smoother for high-frequency error components is effective

22 MultiGrid (MG) preconditioning
[Hackbusch, Springer, 1994] [Wolters, Vorlesungsskriptum, Chapter 7.2] [Wolters, Kuhn, Anwander & Reitzinger, Comp. Vis. Sci.., 2002] [Lew, Wolters, Dierkes, Roer & MacLeod, Appl.Num.Math., 2009] MultiGrid (MG) preconditioning Principle of MG Smoother for low-frequency error components not effective

23 MultiGrid (MG) preconditioning
[Hackbusch, Springer, 1994] [Wolters, Vorlesungsskriptum, Chapter 7.2] [Wolters, Kuhn, Anwander & Reitzinger, Comp. Vis. Sci.., 2002] [Lew, Wolters, Dierkes, Roer & MacLeod, Appl.Num.Math., 2009] MultiGrid (MG) preconditioning Principle of the MG Smoother for high-frequency error components Coarse grid correction for low-frequency error components

24 MultiGrid (MG) preconditioning
[Hackbusch, Springer, 1994] [Wolters, Vorlesungsskriptum, Chapter 7.2] [Wolters, Kuhn, Anwander & Reitzinger, Comp. Vis. Sci.., 2002] [Lew, Wolters, Dierkes, Roer & MacLeod, Appl.Num.Math., 2009] MultiGrid (MG) preconditioning

25 Geometric Multigrid Geometric MG (GMG) Complexity: O(N)
[Hackbusch, Springer, 1994] [Wolters, Vorlesungsskriptum, Chapter 7.2] [Wolters, Kuhn, Anwander & Reitzinger, Comp. Vis. Sci.., 2002] [Lew, Wolters, Dierkes, Roer & MacLeod, Appl.Num.Math., 2009] Geometric Multigrid Complexity: O(N) Convergence rate: h-independent high Geometric MG (GMG) Problem-specific smoother for high-frequency error components Coarse grid correction for low-frequency error components Problems in our application Choice of the smoother is difficult (Inhomogeneities/Anisotropies) Generation of the coarse grids difficult Strategy: Algebraic MG (AMG)!

26 Algebraic Multigrid Principle of the AMG:
[Ruge and Stüben, SIAM, 1986; Stüben, GMD, Tech.Report, 1999] [Wolters, Vorlesungsskriptum, Chapter 7.2] [Wolters, Kuhn, Anwander & Reitzinger, Comp. Vis. Sci.., 2002] [Lew, Wolters, Dierkes, Roer & MacLeod, Appl.Num.Math., 2009] Algebraic Multigrid Principle of the AMG: Smoother is given (Gauss-Seidel) Coarse grids and interpolation matrices are constructed from the entries of K Diagonal entries <-> nodes Nondiagonal entries <-> edges Only one high resolution FE mesh! Problems: AMG-interpolation non-optimal: Some few error components are not well reduced, i.e., some few eigenvalues of the AMG iteration matrix are close to 1 Strategy: AMG-CG!

27 Preconditioned Conjugate Gradient Method
[Hackbusch, Springer, 1994] [Wolters, Vorlesungsskriptum, Chapter 7.2] [Wolters, Kuhn, Anwander & Reitzinger, Comp. Vis. Sci.., 2002] [Lew, Wolters, Dierkes, Roer & MacLeod, Appl.Num.Math., 2009] Preconditioned Conjugate Gradient Method PCG accuracy

28 [Lew, Wolters, Dierkes, Roer & MacLeod, Appl.Num.Math., 2009]
Solver: “Maximal relative error over all eccentricities” versus “solution time” highlighted Nodes: 360,056 Elements: 2,165,281

29 [Lew, Wolters, Dierkes, Roer & MacLeod, Appl.Num.Math., 2009]
FE models tuned for subtraction (group 1) and for the direct potential methods (group 2)

30 Comparison of different preconditioners
[Lew, Wolters, Dierkes, Roer & MacLeod, Appl.Num.Math., 2009] Comparison of different preconditioners

31 [Wolters, Grasedyck, Anwander & Hackbusch, Biomag, 2004]
Link to this work: PEBBLES: Parallel and Element Based grey Box Linear Equation Solver

32 Efficiency of the fast FE transfer matrix approaches
[Wolters, Grasedyck, Anwander & Hackbusch, Biomag, 2004] Efficiency of the fast FE transfer matrix approaches Head model: Tetrahedral FE, 147,287 nodes, 892,119 elements der Tensor muss erklärt werden!!

33 Efficiency of the fast FE transfer matrix approaches
[Wolters, Grasedyck, Anwander & Hackbusch, Biomag, 2004] Efficiency of the fast FE transfer matrix approaches Head model: Tetrahedral FE, 147,287 nodes, 892,119 elements Influence space: brain surface mesh, 2mm resolution, 9555 nodes der Tensor muss erklärt werden!!

34 Efficiency of the fast FE transfer matrix approaches
[Wolters, Grasedyck, Anwander & Hackbusch, Biomag, 2004] Efficiency of the fast FE transfer matrix approaches Head model: Tetrahedral FE, 147,287 nodes, 892,119 elements Influence space: brain surface mesh, 2mm resolution, 9555 nodes Number of FE forward solutions: EEG, 71 electrodes: * 3 = 28665 MEG, 147 channels: 9555 * 2 = (tangential constraint) FE approach and dipole model: Venant der Tensor muss erklärt werden!!

35 Parallel AMG-CG on distributed memory computers
[Wolters, Kuhn, Anwander & Reitzinger, Comp.Vis.Sci, 2002] Parallel AMG-CG on distributed memory computers ``Element-wise’’ partitioning into subdomains Irregular k way partitioning scheme for irregular graphs Parallel MultiRHS-AMG-CG: Communication is only necessary for: Smoother (-Jacobi within interface nodes, Gauss-Seidel between blocks and for inner nodes) distribution of coarse grid solution inner products within PCG method

36 Results for parallel solver methods
[Wolters, Kuhn, Anwander & Reitzinger, Comp.Vis.Sci, 2002] Results for parallel solver methods SGI Origin2000, each processor 195MHz, MIPS 10000 Distribution of memory About a linear speedup for moderate processor number Jacobi-CG on 1 proc <-> AMG-CG on 8 procs: Speedup factor of about 80 (10 MG, 8 parallel.) Anisotropy and inhomogeneity does not change performance results Stable preconditioning for moderate processor numbers Solver time comparison: Unknowns:

37 Outline Linear complexity for FEM forward modeling: Transfer matrices
Fast FE solver methods Modeling and influence of white matter anisotropy

38 [Wolters, Vorlesungsskriptum, Chapter 5.4.2]
[Sen et al., Geophysics, 1981] Differential effective medium approach (EMA) for electrical conductivity : Conductivity tensor eigenvalue : Conductivity of the intracellular medium : Conductivity of the extracellular medium : Electromagnetic depolarization factor : Porosity of the medium

39 Differential effective medium approach (EMA) for water diffusion
[Wolters, Vorlesungsskriptum, Chapter 5.4.2] [Latour et al., PNAS, 1994] Differential effective medium approach (EMA) for water diffusion D: electric displacement rho: electric free charge density E: electric field B: magnetic induction mu: magnetic permeability (vacuum) j: electric current density Phi: electric scalar potential Psi: magnetic flux A: magnetic vector potential divA=0: Coulomb’s gauge conductor loop

40 Differential effective medium approach (EMA) for water diffusion
[Wolters, Vorlesungsskriptum, Chapter 5.4.2] [Tuch et al., Ann NYAS, 1999] Differential effective medium approach (EMA) for water diffusion D: electric displacement rho: electric free charge density E: electric field B: magnetic induction mu: magnetic permeability (vacuum) j: electric current density Phi: electric scalar potential Psi: magnetic flux A: magnetic vector potential divA=0: Coulomb’s gauge conductor loop

41 Differential effective medium approach (EMA) for water diffusion
[Wolters, Vorlesungsskriptum, Chapter 5.4.2] [Tuch et al., PNAS, 2001] Differential effective medium approach (EMA) for water diffusion D: electric displacement rho: electric free charge density E: electric field B: magnetic induction mu: magnetic permeability (vacuum) j: electric current density Phi: electric scalar potential Psi: magnetic flux A: magnetic vector potential divA=0: Coulomb’s gauge conductor loop

42 [Wolters, Anwander, Tricoche, Weinstein, Koch & MacLeod, NeuroImage, 2006]
Link to this work: PEBBLES: Parallel and Element Based grey Box Linear Equation Solver

43 5 compartment anisotropic head modeling
[Wolters, Anwander, Tricoche, Weinstein, Koch & MacLeod, NeuroImage, 2006] 5 compartment anisotropic head modeling Nodes: ,287 Elements: 892,119 5 compartment tetrahedral FE model D: electric displacement rho: electric free charge density E: electric field B: magnetic induction mu: magnetic permeability (vacuum) j: electric current density Phi: electric scalar potential Psi: magnetic flux A: magnetic vector potential divA=0: Coulomb’s gauge conductor loop

44 5 compartment anisotropic head modeling
[Wolters, Anwander, Tricoche, Weinstein, Koch & MacLeod, NeuroImage, 2006] Material labeling: Normed colored tensor ellipsoids Tensor ellipsoids in the barycenter of tetrahedra elements

45 5 compartment tetrahedral FE head model
[Wolters, Anwander, Tricoche, Weinstein, Koch & MacLeod, NeuroImage, 2006] 5 compartment tetrahedral FE head model Mainly radially oriented thalamic dipole D: electric displacement rho: electric free charge density E: electric field B: magnetic induction mu: magnetic permeability (vacuum) j: electric current density Phi: electric scalar potential Psi: magnetic flux A: magnetic vector potential divA=0: Coulomb’s gauge conductor loop

46 How volume currents are affected by tissue compartments
[Wolters, Anwander, Tricoche, Weinstein, Koch & MacLeod, NeuroImage, 2006] How volume currents are affected by tissue compartments highlighted 46 46

47 Impact of white matter anisotropy on volume currents
[Wolters, Anwander, Tricoche, Weinstein, Koch & MacLeod, NeuroImage, 2006] Impact of white matter anisotropy on volume currents highlighted Isotropic white matter Anisotropic white matter 47 47

48 Parallelity of return current and main conductivity tensor eigenvector
[Wolters, Anwander, Tricoche, Weinstein, Koch & MacLeod, NeuroImage, 2006] Parallelity of return current and main conductivity tensor eigenvector highlighted Isotropic WM Anisotropic WM

49 Thank you for your attention!

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