Advancing Teaching and Research with Python
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1 Advancing Teaching and Research with Python Hans Petter Langtangen Center for Biomedical Computing (CBC) at Simula Research Laboratory Dept. of Informatics, University of Oslo EuroSciPy 2010
2 Teaching Research Discussion
3 1 Teaching 2 Research 3 Discussion
4 How modern is science education, really? Observations: Strong focus on IT in education: mostly for communication but minor impact on the scientific content of courses Mismatch between computers in research/industry and education! Our goals: Use numerics, programming and simulation from day 1 Create the future science courses
5 How modern is science education, really? Observations: Strong focus on IT in education: mostly for communication but minor impact on the scientific content of courses Mismatch between computers in research/industry and education! Our goals: Use numerics, programming and simulation from day 1 Create the future science courses
6 How modern is science education, really? Observations: Strong focus on IT in education: mostly for communication but minor impact on the scientific content of courses Mismatch between computers in research/industry and education! Our goals: Use numerics, programming and simulation from day 1 Create the future science courses
7 We are implementing a major reform in science education Where: University of Oslo First semester: Classical calculus Numerical calculus Scientific computer programming Second semester: Vector calculus w/numerics Linear algebra w/numerics Mechanics w/simulation Third semester: More calculus algebra w/numerics Science courses w/simulation
8 We are implementing a major reform in science education Where: University of Oslo First semester: Classical calculus Numerical calculus Scientific computer programming Second semester: Vector calculus w/numerics Linear algebra w/numerics Mechanics w/simulation Third semester: More calculus algebra w/numerics Science courses w/simulation
9 We are implementing a major reform in science education Where: University of Oslo First semester: Classical calculus Numerical calculus Scientific computer programming Second semester: Vector calculus w/numerics Linear algebra w/numerics Mechanics w/simulation Third semester: More calculus algebra w/numerics Science courses w/simulation
10 We are implementing a major reform in science education Where: University of Oslo First semester: Classical calculus Numerical calculus Scientific computer programming Second semester: Vector calculus w/numerics Linear algebra w/numerics Mechanics w/simulation Third semester: More calculus algebra w/numerics Science courses w/simulation
11 We are implementing a major reform in science education Where: University of Oslo First semester: Classical calculus Numerical calculus Scientific computer programming Second semester: Vector calculus w/numerics Linear algebra w/numerics Mechanics w/simulation Third semester: More calculus algebra w/numerics Science courses w/simulation
12 We are implementing a major reform in science education Where: University of Oslo First semester: Classical calculus Numerical calculus Scientific computer programming Second semester: Vector calculus w/numerics Linear algebra w/numerics Mechanics w/simulation Third semester: More calculus algebra w/numerics Science courses w/simulation
13 Why Python and not Matlab, C++ or Java?
14 Why Python and not Matlab, C++ or Java?
15 Popularity of major programming languages 20 Ratings in percent (TIOBE Index, April 2010) C Java C++ PHP VB C# Python Perl JavaScript
16 Python is a convenient programming environment
17 Python is a convenient programming environment
18 Python is a convenient programming environment
19 Python is a convenient programming environment
20 Python is a convenient programming environment
21 Python is a convenient programming environment
22 Python is a convenient programming environment
23 Python is a convenient programming environment
24 A new introductory programming course for science What we teach: Python programming: Matlab-style numerical programming Java/C++-style programming with classes Examples on basic numerical methods (differentiation, integration, ODEs) Examples on applications to physics, finance, biology,...
25 A new introductory programming course for science Course material: Book written for the course Slides for the lectures A large collection of exercises (most with solutions) Multiple choice tests A Python-based computing platform
26 How far do we go in this course? The final compulsory project: mü +f( u)+s(u) = F(t), u(0) = U 0, u(0) = V 0 For example, pendulum with air resistance: mü C D A u u +mg sin(u) = F(t), u(0) = U 0, u(0) = 0 Tasks: Matlab-style implementation Visualization: u(t), u(t), u vs. u Class-style implementation (class Problem, Solver, Viz) Class hierarchy for choices of f, s and F e.g., f( u) = 0, or β u, or 1 2 C D A u u
27 Experience with Python for teaching Ideal choice as first computing language Natural transition from Matlab-style to Java/C++-style Compared to Matlab: improved program structure, abstraction, code reuse, mathematical understanding Excellent student critiques
28 Programming is understanding Problem: make a new data type for polynomials >>> p1 = Polynomial({0: 1, 1: -1}) # 1 - x >>> p2 = Polynomial({1: 1, 4: -6, 5: -1}) # x - 6x^4 - x^5 >>> p3 = p1 + p2 >>> p4 = p1*p2 >>> print p4 x - x^2-6x^4 + 5x^5 + x^6 The add function is given Task: implement the multiply function ( ) M N c i x i d j x j i=0 j=0 Recall: everybody is drilled in polynomial multiplication, (1 x)(1 6x 4 x 5 ) =...
29 Programming is understanding Problem: make a new data type for polynomials >>> p1 = Polynomial({0: 1, 1: -1}) # 1 - x >>> p2 = Polynomial({1: 1, 4: -6, 5: -1}) # x - 6x^4 - x^5 >>> p3 = p1 + p2 >>> p4 = p1*p2 >>> print p4 x - x^2-6x^4 + 5x^5 + x^6 The add function is given Task: implement the multiply function ( ) M N c i x i d j x j i=0 j=0 Recall: everybody is drilled in polynomial multiplication, (1 x)(1 6x 4 x 5 ) =...
30 Programming is understanding Problem: make a new data type for polynomials >>> p1 = Polynomial({0: 1, 1: -1}) # 1 - x >>> p2 = Polynomial({1: 1, 4: -6, 5: -1}) # x - 6x^4 - x^5 >>> p3 = p1 + p2 >>> p4 = p1*p2 >>> print p4 x - x^2-6x^4 + 5x^5 + x^6 The add function is given Task: implement the multiply function ( ) M N c i x i d j x j i=0 j=0 Recall: everybody is drilled in polynomial multiplication, (1 x)(1 6x 4 x 5 ) =...
31 1 Teaching 2 Research 3 Discussion
32 Python + compiled loops = high-performance computing 2 u = (k(x,y) u) with finite differences t2 Implementation arith. harm. Pure Fortran scipy.weave Instant f2py Cython Vectorized code (slices) Plain Python
33 Some scientific software projects for large-scale simulation General FE PDE solution: FEniCS, General FE PDE solution: PyADH, General FV PDE solution: FiPy, Discontinuous Galerkin FE PDE solver: hedge, mathema.tician.de/software/hedge Quantum mechanics: GPAW, wiki.fysik.dtu.dk/gpaw N-body dynamics: pnbody, obswww.unige.ch/ revaz/pnbody
34 FEniCS: write simulator in Python, autogenerate C++ Specify equation/model as a(u,v) = L(v) in Python Running this Python code generates specialized C++ code, linked to general C++ libraries Result: specialized application Generality + efficiency!
35 FEniCS example: The Poisson equation 2 u = f Variational forms: Implementation: a(u,v) = L(v) a(v,u) = v udx Ω L(v) = vf dx V = FunctionSpace(mesh, Lagrange, 1) v = TestFunction(V) u = TrialFunction(V) f = Expression( sin(x[0])*cos(x[1]) ) a = inner(grad(v), grad(u))*dx L = v*f*dx u = VariationalProblem(a, L).solve() Ω
36 FEniCS example: The Poisson equation 2 u = f Variational forms: Implementation: a(u,v) = L(v) a(v,u) = v udx Ω L(v) = vf dx V = FunctionSpace(mesh, Lagrange, 1) v = TestFunction(V) u = TrialFunction(V) f = Expression( sin(x[0])*cos(x[1]) ) a = inner(grad(v), grad(u))*dx L = v*f*dx u = VariationalProblem(a, L).solve() Ω
37 FEniCS example: The Poisson equation 2 u = f Variational forms: Implementation: a(u,v) = L(v) a(v,u) = v udx Ω L(v) = vf dx V = FunctionSpace(mesh, Lagrange, 1) v = TestFunction(V) u = TrialFunction(V) f = Expression( sin(x[0])*cos(x[1]) ) a = inner(grad(v), grad(u))*dx L = v*f*dx u = VariationalProblem(a, L).solve() Ω
38 Example of an autogenerated element matrix routine
39 FEniCS example: Stokes problem (1) Differential equations: 2 u + p = f u = 0 Variational forms: a((v,q),(u,p)) = L((v,q)) where a((v,q),(u,p)) = v u v p +q udx Ω L((v,q)) = v f dx Ω
40 FEniCS example: Stokes problem (2) Implementation: V = VectorFunctionSpace(mesh, "Lagrange", 2) Q = FunctionSpace(mesh, "Lagrange", 1) W = V * Q # Taylor-Hood mixed finite element v, q = TestFunctions(W) u, p = TrialFunctions(W) f = Constant((0, 0)) a = (inner(grad(v), grad(u)) - div(v)*p + q*div(u))*dx L = inner(v, f)*dx
41 Comparing math and code for the Stokes problem Key mathematical formulas: a((v,q),(u,p)) = L((v,q)) = Corresponding code: Ω Ω v u v p +q udx v f dx a = (inner(grad(v), grad(u)) - div(v)*p + q*div(u))*dx L = inner(v, f)*dx
42 RANS turbulence modeling Models: Navier-Stokes LES One-equation k-ǫ, k-ω, v 2 -f Reynolds stress models Structure-based turbulence Flexibility: Compare models Hand-tuned linearization Coupled vs. segregated solution Picard vs. Newton iteration
43 Examples on RANS equations u t +u u = 1 p +ν 2 u+f u u u = 0 u u = 2 k2 1 ǫ 2 ( u+ ut )+ 2 3 ki k t +u k = (ν k k)+p k ǫ D, ǫ t +u ǫ = (ν ǫ ǫ)+(c ǫ1 P k C ǫ2 f 2 ǫ) ǫ k +E ǫ = 2νs : s, s = 1 2 ( u +( u ) T ) ν k = ν + ν T σ k...
44 Solving PDEs: equation and code Variational form: Code: F k = (u k,v k )+((ν k ) k, v k )+((P k ),v k ) (ǫ k k,v k ) (D,v k ) F_k = - inner(dot(u, grad(k)), v_k)*dx \ - nu_k_*inner(grad(k), grad(v_k))*dx \ + P_k_*v_k*dx \ - e_*k/k_*v_k*dx - D_*v_k*dx
45 The Python part of FEniCS is growing
46 mpi4py can parallelize Python codes High-level interface to MPI: Arbitrary Python objects can be sent via MPI Efficient treatment of arrays Can parallelize black-box legacy codes via domain decomposition Older alternatives: pypar, PyMPI
47 Parallelizing legacy codes (1) We had an old F77 code for solving the Boussinesq wave equations for tsunami simulation: η t + q = 0 ( 1 η q = (H +αη) φ+ǫh 6 t 1 3 φ t + α 2 φ φ+η ǫ ( 2 H H φ t ) H φ H ) + ǫ 6 H2 2 φ t = 0 The code applies sophisticated numerics in tricky code that would be hard to parallelize
48 Parallelizing legacy codes (2) Ideas: Apply overlapping domain decomposition for parallelizing the mathematical problem - code that algorithm in Python Let Python call the old F77 code as subdomain solver, fed with right boundary conditions No modifications of the old F77 code All parallelization outside, in Python Speedup results: # cores wall time speedup time steps, cells python 2.6, numpy 1.3, pypar 2.1.4
49 Parallelizing legacy codes (2) Ideas: Apply overlapping domain decomposition for parallelizing the mathematical problem - code that algorithm in Python Let Python call the old F77 code as subdomain solver, fed with right boundary conditions No modifications of the old F77 code All parallelization outside, in Python Speedup results: # cores wall time speedup time steps, cells python 2.6, numpy 1.3, pypar 2.1.4
50 Parallelizing legacy codes (2) Ideas: Apply overlapping domain decomposition for parallelizing the mathematical problem - code that algorithm in Python Let Python call the old F77 code as subdomain solver, fed with right boundary conditions No modifications of the old F77 code All parallelization outside, in Python Speedup results: # cores wall time speedup time steps, cells python 2.6, numpy 1.3, pypar 2.1.4
51 1 Teaching 2 Research 3 Discussion
52 Installing Python and all its modules can be a nightmare Manual install: jungle of packages, platforms, OS versions... Enthought distribution pythonxy.com (Win) Sage Ubuntu/Debian: apt-get install... Run Ubuntu dual boot or VirtualBox
53 Installing Python and all its modules can be a nightmare Manual install: jungle of packages, platforms, OS versions... Enthought distribution pythonxy.com (Win) Sage Ubuntu/Debian: apt-get install... Run Ubuntu dual boot or VirtualBox
54 Installing Python and all its modules can be a nightmare Manual install: jungle of packages, platforms, OS versions... Enthought distribution pythonxy.com (Win) Sage Ubuntu/Debian: apt-get install... Run Ubuntu dual boot or VirtualBox
55 Python has a potentially very clean syntax for math r S k = νexp ( C 2 ) 2 σ 3/2 sin(kx) S_k = self.nu*np.exp(-self.parameters[ C_2 ] \ *(pow(self.radius,2)/self.parameters[ sigma ]**(3.0/2.0))) \ *np.sin(self.parameters[ wavenumber ]*x) S_k = nu*exp(-c_2*r**2/sigma**(3./2))*sin(k*x) Same variable names in code as in mathematics Introduce local variables, before formulas, to strip off self, dicts, etc. for name in self.parameters: exec(name + = + str(self.parameters[name])) The from module import * import is not that evil?
56 Python has a potentially very clean syntax for math r S k = νexp ( C 2 ) 2 σ 3/2 sin(kx) S_k = self.nu*np.exp(-self.parameters[ C_2 ] \ *(pow(self.radius,2)/self.parameters[ sigma ]**(3.0/2.0))) \ *np.sin(self.parameters[ wavenumber ]*x) S_k = nu*exp(-c_2*r**2/sigma**(3./2))*sin(k*x) Same variable names in code as in mathematics Introduce local variables, before formulas, to strip off self, dicts, etc. for name in self.parameters: exec(name + = + str(self.parameters[name])) The from module import * import is not that evil?
57 Python has a potentially very clean syntax for math r S k = νexp ( C 2 ) 2 σ 3/2 sin(kx) S_k = self.nu*np.exp(-self.parameters[ C_2 ] \ *(pow(self.radius,2)/self.parameters[ sigma ]**(3.0/2.0))) \ *np.sin(self.parameters[ wavenumber ]*x) S_k = nu*exp(-c_2*r**2/sigma**(3./2))*sin(k*x) Same variable names in code as in mathematics Introduce local variables, before formulas, to strip off self, dicts, etc. for name in self.parameters: exec(name + = + str(self.parameters[name])) The from module import * import is not that evil?
58 Python has a potentially very clean syntax for math r S k = νexp ( C 2 ) 2 σ 3/2 sin(kx) S_k = self.nu*np.exp(-self.parameters[ C_2 ] \ *(pow(self.radius,2)/self.parameters[ sigma ]**(3.0/2.0))) \ *np.sin(self.parameters[ wavenumber ]*x) S_k = nu*exp(-c_2*r**2/sigma**(3./2))*sin(k*x) Same variable names in code as in mathematics Introduce local variables, before formulas, to strip off self, dicts, etc. for name in self.parameters: exec(name + = + str(self.parameters[name])) The from module import * import is not that evil?
59 Python has a potentially very clean syntax for math r S k = νexp ( C 2 ) 2 σ 3/2 sin(kx) S_k = self.nu*np.exp(-self.parameters[ C_2 ] \ *(pow(self.radius,2)/self.parameters[ sigma ]**(3.0/2.0))) \ *np.sin(self.parameters[ wavenumber ]*x) S_k = nu*exp(-c_2*r**2/sigma**(3./2))*sin(k*x) Same variable names in code as in mathematics Introduce local variables, before formulas, to strip off self, dicts, etc. for name in self.parameters: exec(name + = + str(self.parameters[name])) The from module import * import is not that evil?
60 How can we make Python take off in science? Make better documentation and many more examples (numpy, scipy, f2py, Cython,...) Make all Matlab commands available Support unified interfaces to ODE solvers, linear algebra, optimization, random numbers, visualization, etc. Point to real-world high-performance applications Help people to get started
61 Quick summary of what I have covered... Python for teaching is excellent! Try it! Python for PDE solution provides a new generation of simulation software Still some non-resolved issues...
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