To address the challenges of interoperability in computational science, we present the latest updates to the software package MaRDI Open Interfaces. This software package aims to decrease the time and coding/testing efforts spent by computational scientists on tasks such as writing bindings to numerical solvers and adapting experiment codes to the varying interfaces of solvers for the same problem type (e.g., for benchmarking, which solver is better). By streamlining these tasks, this software package helps researchers focus on the actual essence of their computational projects. Here, we demonstrate a recently developed interface for nonlinear optimization and illustrate how it can be applied for computational experiments with optimization problems. As an example of such problem, we consider training of physics-informed neural networks to predict the solutions of viscous Burgers' equation.

MaRDI Open Interfaces is a software package that aims to improve interoperability in scientific computing, particularly, for nonlinear optimization. To this end, this package holds two main characteristics. First, it provides unified interfaces for typical numerical problems to help switching between solvers for the same problem type. Second, it automates data marshalling between programming languages. Hence, computational scientists can conduct experiments faster by using the package, with fewer code-modification and testing efforts. In this work we describe the general structure of the software package and show examples with the interface for nonlinear optimization.

WepresentatwolevelSchwarzmethodasanalternativetoAlgebraicMultigridmethod(AMG) used as the last level (coarse) solver of the p-multigrid pMG preconditioner for pressure Poission equation resulting from Spectral/Finite element descretization of incompressible Navier-Stokes eqaution. Proposed Schwarz method consits of a local problem in the original pMG coarse space and a global coarse problem. Main contribution of the paper is a novel, structured and a non-nested coarse space for the global coarse problem. Structured nature of the proposed global coarse space enable communication-free interpolation between the original p-multgrid coarse space and the global coarse problem. We demonstrate the effectiveness of the proposed method compared to the state of the art AMG solver BoomerAMG by a series of experiments performed using Nek5000/RS, a suite of highly scalable incompressible Navier-Stokes solvers, on Summit/Frontier supercomputers at Oak Ridge Leadership Computing Facility.

The extended complex plane is a fundamental object in complex analysis, hyperbolic geometry, and mathematical physics. Its geometry is governed by Möbius transformations, with the cross ratio serving as a central invariant. We present a formalization of these concepts in the Lean4 theorem prover. The extended complex plane is represented using Mathlib's Option type over $\mathbb{C}$, where the additional element represents the point at infinity. On this foundation, we define Möbius transformations, their action on the extended complex plane, and the cross ratio. We formalize several basic properties of Möbius transformations, including their group structure, and identify them with a projective general linear group. We also prove the uniqueness of a Möbius transformation mapping any three distinct points to any other three distinct points, and the invariance of the cross ratio. All proofs are machine-checked in Lean 4. The complete development comprises approximately 6,000 lines of Lean code, including about 40 definitions and 150 lemmas and theorems. This work provides a verified foundation for future formalizations of conformal geometry, hyperbolic models, modular forms, and applications in mathematical physics.

We present a hardware-oriented description of GoldenFloat (GF), a static-split floating-point family generated by a single closed rule, and three concrete artefacts: (i) an open multi-width RTL generator covering GF4-GF256 with a continuous-integration differential sweep against a correctly-rounded reference; (ii) an integer-backed Lucas-exact accumulator path verified at 500-digit precision for n = 1, ..., 256; and (iii) a GF16 FPGA codec passing a 35-of-35 testbench at 323 MHz on Artix-7 (Xilinx XC7A35T). A format-conformance oracle (Corona) ships in the same repository and is used as the blackbox check in our continuous-integration audit. The rule and its scope. For each total width N >= 4, the exponent width is e = round((N-1)/phi^2) with fraction f = N-1-e and phi = (1+sqrt(5))/2. The rule reproduces the realised exponent widths of nine formats GF4, GF8, GF12, GF16, GF20, GF24, GF32, GF64, GF256 (9/9) and extends consistently to GF128, GF512, GF1024. The rule is positioned alongside posit (2022 Posit Standard), takum (Hunhold 2024, 2025), OCP-MX (Rouhani et al. 2023), and the IEEE P3109 multi-width float draft, all of which are width-spanning families under a parameterised rule. We make no per-rung accuracy or superiority claim against any of them. What is open. The breadth/toolchain-coherence framing is recorded as an open conjecture with a pre-registered falsification path: a matched-substrate FPGA experiment and a matched-budget software ablation. A falsification ledger (FL-002) records the open questions and the experiments that would settle them. An RTL-correctness erratum dated 2026-05-31 is reported in Section 5.5; the fabricated TTSKY26b dies carry the defective multiplier portfolio, and the corrected generator is the regeneration baseline.

Recently, graphics processors (GPUs) have been increasingly leveraged in a variety of scientific computing applications. However, architectural differences between CPUs and GPUs necessitate the development of algorithms that take advantage of GPU hardware. As sparse matrix vector multiplication (SPMV) operations are commonly used in finite element analysis, a new SPMV algorithm and several variations are developed for unstructured finite element meshes on GPUs. The effective bandwidth of current GPU algorithms and the newly proposed algorithms are measured and analyzed for 15 sparse matrices of varying sizes and varying sparsity structures. The effects of optimization and differences between the new GPU algorithm and its variants are then subsequently studied. Lastly, both new and current SPMV GPU algorithms are utilized in the GPU CG Solver in GPU finite element simulations of the heart. These results are then compared against parallel PETSc finite element implementation results. The effective bandwidth tests indicate that the new algorithms compare very favorably with current algorithms for a wide variety of sparse matrices and can yield very notable benefits. GPU finite element simulation results demonstrate the benefit of using GPUs for finite element analysis, and also show that the proposed algorithms can yield speedup factors up to 12-fold for real finite element applications.