ABSTRACT. The PETSc 3.24 release introduced PetscRegressor, a new top-level component that provides basic infrastructure and a general API for machine learning tasks at a higher level of abstraction than the purely algebraic “solvers” view required when implementing machine learning algorithms entirely from scratch using TAO optimizers. This presentation will describe the varied motivations for using PETSc for machine learning; provide a short overview of the general interface and usage model of PetscRegressor; present some experiences and examples of using it in different science applications; and discuss future directions for further development of PetscRegressor and other components of a planned machine learning toolkit in PETSc -- one that can scale across distributed-memory parallel machines and facilitate machine learning in an online fashion inside PETSc-based simulations.

ABSTRACT. In the current implementation of Newton-Krylov-Schwarz in PETSc for solving linear and/or nonlinear algebraic systems, the preconditioner is usually constructed using the information of the "system operator", but not the residual or solution vectors. This is reasonable since the solution is not available when the preconditioner is initially constructed. However, once the iteration starts, more and more information about the solution becomes available, and some of the information can be used to enhance the preconditioner so that the rest of the computation can be made faster. Some examples will be discussed in the talk.

ABSTRACT. Although multigrid is asymptotically optimal for solving many important partial differential equations, its efficiency relies heavily on the careful selection of its individual algorithmic components. In contrast to recent approaches that can optimize certain multigrid components using deep learning techniques, we adopt a complementary strategy, employing evolutionary algorithms to construct efficient multigrid cycles from proven algorithmic building blocks. We generate efficient algebraic multigrid (AMG) methods with flexible cycling, that is, non-recursive cycles with level-dependent smoothing sequences and arbitrary cycling patterns. The search space with such non-standard cycles is intractable to navigate manually, and is generated using genetic programming (GP) guided by context-free grammars. Thus, the AMG methods are generated not only by tuning parameters but also by altering the algorithmic scheme within the limits imposed by the grammar. The resulting methodology discovers novel flexible AMG cycles that outperform standard AMG methods. The flexible cycling capabilities have been implemented in hypre BoomerAMG and also exposed via a PETSc interface, allowing PETSc-based applications to leverage GP-designed AMG solvers and preconditioners in a non-intrusive manner. We demonstrate the effectiveness of the approach in simulation codes using PETSc, where the generated non-standard AMG methods exhibit superior performance compared to standard BoomerAMG settings.

ABSTRACT. We present a first foray into formal reasoning for the transformation of computational meshes. The formal definition is based on the DMPlex abstraction in PETSc. We should a few elementary proofs, and then demonstrate that the output complexity of mesh transformation is linear, so that pieces of meshes can be efficiently generated on the fly. All examples use the Rocq language for proof verification.

ABSTRACT. Time-stepping libraries like PETScTS typically accept callbacks to functions defining a system of ordinary differential (or differential algebraic equations) and provide a broad suite of time-stepping schemes. Also included is expert knowledge for things such as adaptive time-step selection and adjoints.

Frequently missing from these libraries are robust support for full support fully implicit, collocation-type Runge-Kutta methods such as Gauss-Legendre and RadauIIA. Perhaps this is a chicken-and-egg problem: people don't use them because libraries don't support them, but libraries don't support them because people don't use them! However, there is a growing literature that, if the implementation challenges of such methods are overcome, then they can actually be more efficient than more widely-applied methods in terms of accuracy obtained per work expended.

Many such empirical results are based on Irksome, a Firedrake-based package that can generate the complex, stage-coupled systems required by these methods and deploy powerful, PETSc-based solution algorithms. In this talk, I hope to: 1) Demonstrate the power of fully implicit methods 2) Show why they are quite difficult to implement generally in PETScTS 3) Suggest some open opportunities to enhance Irksome via hooks for methods that PETScTS supplies

ABSTRACT. The transmission eigenvalue problem arises from the inverse scattering theory for inhomogeneous media. Transmission eigenvalues can be reconstructed from the far-field data of the scattered wave, and they can be used to estimate the material properties of the scatterer, such as the refractive index. We consider acoustic waves via a Helmholtz-like PDE, discretized with FreeFEM, and discuss different discretization schemes proposed in the literature. The resulting algebraic eigenvalue problem is large, generalized, and non-symmetric, where the wanted eigenvalues are interior. They are computed with SLEPc using either shift-and-invert Krylov-Schur or a contour integral eigensolver. We analyze various aspects that are relevant to performance, such as using MUMPS' mixed-precision capabilities when solving the associated linear systems.

ABSTRACT. Frequency-domain, Finite Element based 3D imaging with Full Waveform Inversion [1] requires solving large sparse, complex and highly indefinite linear systems. These systems must be solved for multiple right-hand sides, which classically motivates the use of sparse direct solvers. Unfortunately, these become prohibitive when the system size grows (hundreds of millions of unknowns), and iterative solvers become necessary.

We present our FWI code based on Gmsh [2], which integrates various solvers: overlapping, one-level Optimized Restricted Additive Schwarz, non-overlapping Optimized Schwarz Methods (which we showed to be more affordable [3]), and recent two-level methods based on harmonic coarse spaces [4] and MS-GFEM [5], implemented in PETSc through HPDDM [6].

We explore the combinations between these wave solvers and the FWI gradient-based nonlinear optimizers to find synergies. In particular, we study the tradeoffs between setup time and solve time, as the number of RHS and the number of systems to solve depend on the choice of optimizer as well as the imaging setup.

[1] A. Tarantola, "Inversion of seismic reflection data in the acoustic approximation," Geophysics, vol. 49, no. 8, pp. 1259–1266, 1984. [2] C. Geuzaine and J.-F. Remacle, "Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities," Int. J. Numer. Meth. Engng., vol. 79, no. 11, pp. 1309–1331, 2009. [3] B. Martin, P. Jolivet, and C. Geuzaine, "Comparison of substructured non-overlapping domain decomposition and overlapping additive Schwarz methods for large-scale Helmholtz problems with multiple sources," J. Comput. Phys., vol. 548, 114557, 2026. [4] Q. Hu and Z. Li, "A novel coarse space applying to the weighted Schwarz method for Helmholtz equations," arXiv:2402.06905, 2024. [5] C. Ma, C. Alber, R. Scheichl, and Y. Zhang, "Two-level Restricted Additive Schwarz Preconditioner based on Multiscale Spectral Generalized FEM for Heterogeneous Helmholtz Problems," J. Sci. Comput., vol. 105, 2025. [6] P. Jolivet, J. E. Roman, and S. Zampini, "KSPHPDDM and PCHPDDM: Extending PETSc with advanced Krylov methods and robust multilevel overlapping Schwarz preconditioners," Comput. Math. Appl., vol. 84, pp. 277–295, 2021.

ABSTRACT. We consider an effective new method for solving trust-region and norm-regularization problems that arise as subproblems in many optimization applications. We show that the solutions to such subproblems lie on a manifold of approximately very low rank as a function of their controlling parameters (trust-region radius or regularization weight). Based on this, we build a basis for this manifold using an efficient extended-Krylov-subspace iteration that involves a single matrix factorization. The problems within the subspace using such a basis may be solved at very low cost using effective high-order root-finding methods. This then provides an alternative to common methods using multiple factorizations or standard Krylov subspaces. We provide numerical results to illustrate the effectiveness of our {\tt TREK}/{\tt NREK} approach.