dctkit - Discrete Calculus Toolkit

dctkit implements operators from Algebraic Topology, Discrete Exterior Calculus and Discrete Differential Geometry to provide a mathematical language for building discrete physical models.

Features:

• uses jax as a backend for numerical computations
• manipulation of simplicial complexes of any dimension: computation of boundary/coboundary operators, circumcenters, dual/primal volumes
• manipulation of (primal/dual) cochains: addition, multiplication by scalar, inner product, coboundary, Hodge star, codifferential, Laplace-de Rham
• manipulation of vector-valued and tensor-valued cochains: discrete vector and tensor fields, sharp operator
• interface to the pygmo optimization library
• interface to the PETSc non-linear solvers and optimizers
• routines for solving optimal control problems
• implementation of discrete physical models: Dirichlet energy, Poisson, Euler's Elastica, isotropic linear elasticity

Installation

Dependencies should be installed within a conda environment. We recommend using mamba since it is much faster than conda at solving the environment and downloading the dependencies. To create a suitable environment based on the provided .yaml file, use the command

$ mamba env create -f environment.yaml

Otherwise, update an existing environment using the same .yaml file.

After activating the environment, clone the git repository and launch the following command

$ pip install -e .

to install a development version of the dctkit library.

Running the tests:

$ tox

Generating the docs:

$ tox -e docs

Running the benchmarks:

$ sh ./bench/run_bench

The Markdown file bench_results.md will be generated containing the results.

Reference performance (HP Z2 Workstation G9 - 12th Gen Intel i9-12900K (24) @ 5.200GHz - NVIDIA RTX A4000 - 64GB RAM - ArchLinux kernel v6.7.8)

Command	Mean [s]	Min [s]	Max [s]	Relative
python bench_poisson.py scipy cpu	1.739 ± 0.014	1.724	1.754	2.04 ± 0.04
python bench_poisson.py pygmo cpu	0.852 ± 0.017	0.835	0.876	1.00
python bench_poisson.py jaxopt cpu	2.872 ± 0.018	2.851	2.898	3.37 ± 0.07
python bench_poisson.py jaxopt gpu	3.297 ± 0.056	3.210	3.367	3.87 ± 0.10

Usage

Read the full documentation (including API docs).

Example: solving discrete Poisson equation in 1D (variational formulation):

import dctkit as dt
from dctkit import config
from dctkit.mesh import util
from dctkit.dec import cochain as C
from dctkit.math.opt import optctrl as oc
import jax.numpy as jnp
from matplotlib.pyplot import plot

# configure the JAX backend: sets precision and platform (CPU/GPU)
# defaults: float64 precision and CPU platform
# MUST be called before using any function of dctkit
config()

# generate mesh and create SimplicialComplex object
num_nodes = 10
L = 1.
mesh, _ = util.generate_line_mesh(num_nodes, L)
S = util.build_complex_from_mesh(mesh)

# compute Hodge star operator for later use
S.get_hodge_star()

# initial guess for the solution vector (coefficients of a primal 0-chain)
# except for the node 0, where the boundary condition is prescribed
u = jnp.ones(num_nodes-1, dtype=dt.float_dtype)

# source term (primal 0-cochain)
f = C.CochainP0(complex=S, coeffs=jnp.ones(num_nodes))

# discrete Dirichlet energy with source term
def energy(u):
     # zero Dirichlet bc at x=0
     u = jnp.insert(u, 0, 0.)
     # wrap JAX array (when calling minimization routine) into a cochain
     uc = C.CochainP0(complex=S, coeffs=u)
     du = C.coboundary(uc)
     return C.inner_product(du, du)-C.inner_product(uc, f)

# set optimization problem
prb = oc.OptimizationProblem(dim=num_nodes-1, state_dim=num_nodes-1, objfun=energy)
# set additional arguments (empty dictionary) of the objective function,
# other than the state array
prb.set_obj_args({})

x = prb.solve(x0=u)

# add boundary condition
x = jnp.insert(x, 0, 0.)

plot(x)