Partitioning graphs is a common and useful operation in many areas, from parallel computing to VLSI design to sparse matrix algorithms. In this paper, we introduce Mongoose, a multilevel hybrid graph par- titioning algorithm and library. Building on previous work in multilevel partitioning frameworks and com- binatoric approaches, we introduce novel stall-reducing and stall-free coarsening strategies, as well as an efficient hybrid algorithm leveraging 1) traditional combinatoric methods and 2) continuous quadratic pro- gramming formulations. We demonstrate how this new hybrid algorithm outperforms either strategy in isolation, and we also compare Mongoose to METIS and demonstrate its effectiveness on large and social networking (power law) graphs.

SuiteSparse:GraphBLAS is a full implementation of the GraphBLAS standard, which defines a set of sparse matrix operations on an extended algebra of semirings using an almost unlimited variety of operators and types. When applied to sparse adjacency matrices, these algebraic operations are equivalent to computations on graphs. GraphBLAS provides a powerful and expressive framework for creating graph algorithms based on the elegant mathematics of sparse matrix operations on a semiring. An overview of the GraphBLAS specification is given, followed by a description of the key features of its implementation in the SuiteSparse:GraphBLAS package.

There are several AD tools available, which all implement different strategies for the reverse mode of AD. The major strategies are primal value taping (implemented e.g. by ADOL-c) and Jacobi taping (implemented e.g. by adept and dco/c++). Especially for Jacobi taping, recent advances by using expression templates make this approach very attractive for large scale software. The current implementations are either closed source or miss essential features and flexibility. Therefore, we present the new AD tool CoDiPack (Code Differentiation Package) in this paper. It is specifically designed for a minimal memory consumption and optimal runtime, such that it can be used for the differentiation of large scale software. An essential part of the design of CoDiPack is the modular layout and the recursive data structures, which do not only allow the efficient implementation of the Jacobi taping approach, but will also enable other approaches like the primal value taping or new research ideas. We will also present the performance value of CoDiPack on a generic PDE example and on the SU2 code.

IPscatt is a free toolbox in MATLAB that solves the inverse medium problem in scattering in two and three dimensions. It uses the Helmholtz equation as a physical model of scattering and provides point sources or plane waves as incident fields as well as simulated near or far field data with noise. Further, it can deal with real-world data in two dimensions from Institute Fresnel. A rapid variational regularization scheme minimizes the defect, i.e. the difference between the predicted data from the model and the given data, as well as penalty terms promoting sparsity, total variation and physical bounds. These regularization scheme relies on a primal-dual algorithm and was introduced
in [F. Bürgel, K. S. Kazimierski, and A. Lechleiter 2017 *Journal of Computational Physics*, **339**:130]. This paper provides a survey on the mathematical concepts, connects them with the implementation and shows the practical usage of the toolbox via guides.

The maximum flow problem is one of the most common network flow problems. This problem involves finding the maximum possible amount of flow between two designated nodes on a network with arcs having flow capacities. The push-relabel algorithm is one of the fastest algorithms to solve this problem. We present a shared memory parallel push-relabel algorithm. Graph coloring is used to avoid collisions between threads for concurrent push and relabel operations. In addition, excess values of target nodes are updated using atomic instructions to prevent race conditions. The experiments show that our algorithm is competitive for wide graphs with low diameters. Results from two different data sets are included, computer vision problems and DIMACS challenge problems. These are compared with existing push-relabel and pseudoflow implementations. We show that high speedup rates are possible using our coloring based parallelization technique on sparse networks. However, we also observe that the pseudoflow algorithm runs faster than the push-relabel algorithm on dense and long networks.

In a recent paper, Lucambio Pérez and Prudente extended the Wolfe conditions for the vector-valued optimization. Here, we propose a line search algorithm for finding a step size satisfying the strong Wolfe conditions in the vector optimization setting. Well definedness and finite termination results are provided. We discuss practical aspects related to the algorithm and present some numerical experiments illustrating its applicability. Codes supporting this paper are written in Fortran 90 and are freely available for download.

Multi-degree splines are smooth piecewise-polynomial functions where the pieces can have different degrees. We describe a simple algorithmic construction of a set of basis functions for the space of multi-degree splines, with similar properties to standard B-splines. These basis functions are called multi-degree B-splines (or MDB-splines). The construction relies on an extraction operator that represents all MDB-splines as linear combinations of local B-splines of different degrees. This enables the use of existing efficient algorithms for B-spline evaluations and refinements in the context of multi-degree splines. A Matlab implementation is provided to illustrate the computation and use of MDB-splines.

A scientific framework for simulations of large-scale networks, such as is required for the analysis of critical infrastructure interaction and interdependencies, is needed for applications on exascale computers. Such a framework must be able to manage heterogeneous physics and unstructured topology, and must be reusable. To this end we have developed DMNetwork, a class in PETSc that provides data and topology management and migration for network problems, along with multiphysics solvers to exploit the problem structure. It eases the application development cycle by providing the necessary infrastructure through simple abstrac- tions to define and query the network. This paper presents the design of the DMNetwork, illustrates its user interface, and demonstrates its ability to solve large network problems through the numerical simulation of a water pipe network with more than 2 billion variables on extreme-scale computers using up to 30,000 processor cores.