Coming from hIPPYlib

The inverse-problem classes keep the names and signatures of hIPPYlib, from which that layer is adapted, so a driver script written for it usually ports by changing the imports and the space construction. This page is the map.

hIPPYlib / hIPPYlibx	hIPPyMFEM
pde_varf(u, m, p) in UFL	pde_varf(u, m, p, x) as a JAX density, fields carrying .val and .grad
a ds term in the form	bdr_varf(u, m, p, x, n), passed as PDEVariationalProblem(..., bdr_varf=...)
a dS term in the form	facet_varf(u, m, p, x, n, h), written in jump and avg
dolfinx.fem.functionspace	hm.FunctionSpace.H1 / .L2 / .ND / .RT
fem.dirichletbc	hm.DirichletBC(space, value, bdr_attributes=...)
PDEVariationalProblem	same name, same role
BiLaplacianPrior	same name
PointwiseStateObservation	DiscreteStateObservation (and a PointwiseStateObservation wrapper)
ContinuousStateObservation	same name
Model, ReducedHessian	same names
modelVerify	same name; returns (eps, err_grad, err_H)
ReducedSpaceNewtonCG	same name
BFGS, SteepestDescent	same names
doublePass, doublePassG	same names
GaussianLRPosterior	same name
MCMC, pCN, gpCN, MALA	same names
PETScKrylovSolver	hm.KrylovSolver (MFEM Krylov + hypre), or hm.PETScKrylovSolver
PETScLUSolver	hm.LUSolver, exact on any rank count, or hm.PETScLUSolver
petsc4py.PETSc.Vec	hm.ParVector
amg_method()	accepted and ignored; hypre BoomerAMG is the only option

Differences that change results

The quadrature rule. hIPPYlib’s rule is chosen by FFCx from the form’s estimated degree; here it is 2*max(order) + 2 by default. For a non-polynomial density neither is exact, so the two libraries agree only when given the same rule. The cross-validation harness (Verification and validation) sets them equal, which is why it reaches 1e-13 rather than 1e-6.

The gradient norm. Both default to the discrete \(L^2\) norm through the mass matrix. Selecting the prior-weighted norm changes the iteration history.

Randomized eigensolver draws. The random matrices differ, so the computed eigenvalues agree to the accuracy of the randomized method rather than to round-off. For an exact comparison use the dense spectrum on a coarse mesh, which is what the validation harness does.

Negative curvature inside a trust region. CGSolverSteihaug with a trust region follows a direction of nonpositive curvature to the boundary, as Steihaug’s method prescribes. hIPPYlib’s takes the whole first direction wherever it lands, outside the region when the radius is small, and stops inside the region when the direction appears later. Without a trust region (Newton-CG with a line search) the two agree.

BFGS damping. Powell damping needs \(H y\). hIPPYlib’s BFGS_operator computes it with the two-loop recursion working in its own output vector, so the initial inverse Hessian is applied in place and, for the default rescaled identity, returns zero: a pair that needs damping is not damped and fails the curvature check. Here the recursion has a work vector of its own.

Things hIPPyMFEM has that hIPPYlib does not

• a residual density that may contain anything JAX can differentiate, such as a network, a table or an inner Newton solve, with exact derivatives;

• element kernels that run on a GPU without the residual being rewritten, and, with a CUDA or HIP build of PyMFEM, the matrices and the solves on the card as well, from the same script;

• direct-CSR assembly with a reused sparsity pattern, which is faster than assembling through MFEM’s own integrator interface;

• MFEM’s meshes, element families and I/O.

Things hIPPYlib has that hIPPyMFEM does not

• a form language, and with it automatic quadrature degree estimation, and mixed or block spaces of different families in the problem class itself; here such a system is assembled from its blocks, which come from one density all the same (see tutorial 11);

• compiled C element kernels, which are faster per element on a CPU;

• years of use by many people.