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The main topic of this paper are algorithms for computing Nash equilibria. We cast our particular methods as instances of a general algorithmic abstraction, namely, a method we call {\em algorithmic boosting}, which is also relevant to other fixed-point computation problems. Algorithmic boosting is the principle of computing fixed points by taking (long-run) averages of iterated maps and it is a generalization of exponentiation. We first define our method in the setting of nonlinear maps. Secondly, we restrict attention to convergent linear maps (for computing dominant eigenvectors, for example, in the PageRank algorithm) and show that our algorithmic boosting method can set in motion {\em exponential speedups in the convergence rate}. Thirdly, we show that algorithmic boosting can convert a (weak) non-convergent iterator to a (strong) convergent one. We also consider a {\em variational approach} to algorithmic boosting providing tools to convert a non-convergent continuous flow to a convergent one. Then, by embedding the construction of averages in the design of the iterated map, we constructively prove the existence of Nash equilibria (and, therefore, Brouwer fixed points). We then discuss implementations of averaging and exponentiation, an important matter even for the scalar case. We finally discuss a relationship between dominant (PageRank) eigenvectors and Nash equilibria.

The numerical solution of continuum damage mechanics (CDM) problems suffers from convergence-related challenges during the material softening stage, and consequently existing iterative solvers are subject to a trade-off between computational expense and solution accuracy. In this work, we present a novel unified arc-length (UAL) method, and we derive the formulation of the analytical tangent matrix and governing system of equations for both local and non-local gradient damage problems. Unlike existing versions of arc-length solvers that monolithically scale the external force vector, the proposed method treats the latter as an independent variable and determines the position of the system on the equilibrium path based on all the nodal variations of the external force vector. This approach renders the proposed solver substantially more efficient and robust than existing solvers used in CDM problems. We demonstrate the considerable advantages of the proposed algorithm through several benchmark 1D problems with sharp snap-backs and 2D examples under various boundary conditions and loading scenarios. The proposed UAL approach exhibits a superior ability of overcoming critical increments along the equilibrium path. Moreover, the proposed UAL method is 1-2 orders of magnitude faster than force-controlled arc-length and monolithic Newton-Raphson solvers.

In this paper we revisit the classical Cauchy problem for Laplace's equation as well as two further related problems in the light of regularisation of this highly ill-conditioned problem by replacing integer derivatives with fractional ones. We do so in the spirit of quasi reversibility, replacing a classically severely ill-posed PDE problem by a nearby well-posed or only mildly ill-posed one. In order to be able to make use of the known stabilising effect of one-dimensional fractional derivatives of Abel type we work in a particular rectangular (in higher space dimensions cylindrical) geometry. We start with the plain Cauchy problem of reconstructing the values of a harmonic function inside this domain from its Dirichlet and Neumann trace on part of the boundary (the cylinder base) and explore three options for doing this with fractional operators. The two other related problems are the recovery of a free boundary and then this together with simultaneous recovery of the impedance function in the boundary condition. Our main technique here will be Newton's method. The paper contains numerical reconstructions and convergence results for the devised methods.

Many classical constructions, such as Plotkin's and Turyn's, were generalized by matrix product (MP) codes. Quasi-twisted (QT) codes, on the other hand, form an algebraically rich structure class that contains many codes with best-known parameters. We significantly extend the definition of MP codes to establish a broader class of generalized matrix product (GMP) codes that contains QT codes as well. We propose a generator matrix formula for any linear GMP code and provide a condition for determining the code size. We prove that any QT code has a GMP structure. Then we show how to build a generator polynomial matrix for a QT code from its GMP structure, and vice versa. Despite that the class of QT codes contains many codes with best-known parameters, we present different examples of GMP codes with best-known parameters that are neither MP nor QT. Two different lower bounds on the minimum distance of GMP codes are presented; they generalize their counterparts in the MP codes literature. The second proposed lower bound replaces the non-singular by columns matrix with a less restrictive condition. Some examples are provided for comparing the two proposed bounds, as well as showing that these bounds are tight.

This manuscript is a research resource description and presents a large and novel Electronic Health Records (EHR) data resource, American Family Cohort (AFC). The AFC data is derived from Centers for Medicare and Medicaid Services (CMS) certified American Board of Family Medicine (ABFM) PRIME registry. The PRIME registry is the largest national Qualified Clinical Data Registry (QCDR) for Primary Care. The data is converted to a popular common data model, the Observational Health Data Sciences and Informatics (OHDSI) Observational Medical Outcomes Partnership (OMOP) Common Data Model (CDM). The resource presents approximately 90 million encounters for 7.5 million patients. All 100% of the patients present age, gender, and address information, and 73% report race. Nealy 93% of patients have lab data in LOINC, 86% have medication data in RxNorm, 93% have diagnosis in SNOWMED and ICD, 81% have procedures in HCPCS or CPT, and 61% have insurance information. The richness, breadth, and diversity of this research accessible and research ready data is expected to accelerate observational studies in many diverse areas. We expect this resource to facilitate research in many years to come.

We consider a one-dimensional singularly perturbed 4th order problem with the additional feature of a shift term. An expansion into a smooth term, boundary layers and an inner layer yields a formal solution decomposition, and together with a stability result we have estimates for the subsequent numerical analysis. With classical layer adapted meshes we present a numerical method, that achieves supercloseness and optimal convergence orders in the associated energy norm. We also consider coarser meshes in view of the weak layers. Some numerical examples conclude the paper and support the theory.

This paper presents a novel spatial discretisation method for the reliable and efficient simulation of Bose-Einstein condensates modelled by the Gross-Pitaevskii equation and the corresponding nonlinear eigenvector problem. The method combines the high-accuracy properties of numerical homogenisation methods with a novel super-localisation approach for the calculation of the basis functions. A rigorous numerical analysis demonstrates superconvergence of the approach compared to classical polynomial and multiscale finite element methods, even in low regularity regimes. Numerical tests reveal the method's competitiveness with spectral methods, particularly in capturing critical physical effects in extreme conditions, such as vortex lattice formation in fast-rotating potential traps. The method's potential is further highlighted through a dynamic simulation of a phase transition from Mott insulator to Bose-Einstein condensate, emphasising its capability for reliable exploration of physical phenomena.

We consider a sharp interface formulation for the multi-phase Mullins-Sekerka flow. The flow is characterized by a network of curves evolving such that the total surface energy of the curves is reduced, while the areas of the enclosed phases are conserved. Making use of a variational formulation, we introduce a fully discrete finite element method. Our discretization features a parametric approximation of the moving interfaces that is independent of the discretization used for the equations in the bulk. The scheme can be shown to be unconditionally stable and to satisfy an exact volume conservation property. Moreover, an inherent tangential velocity for the vertices on the discrete curves leads to asymptotically equidistributed vertices, meaning no remeshing is necessary in practice. Several numerical examples, including a convergence experiment for the three-phase Mullins-Sekerka flow, demonstrate the capabilities of the introduced method.

We propose a parallel (distributed) version of the spectral proper orthogonal decomposition (SPOD) technique. The parallel SPOD algorithm distributes the spatial dimension of the dataset preserving time. This approach is adopted to preserve the non-distributed fast Fourier transform of the data in time, thereby avoiding the associated bottlenecks. The parallel SPOD algorithm is implemented in the PySPOD (//github.com/MathEXLab/PySPOD) library and makes use of the standard message passing interface (MPI) library, implemented in Python via mpi4py (//mpi4py.readthedocs.io/en/stable/). An extensive performance evaluation of the parallel package is provided, including strong and weak scalability analyses. The open-source library allows the analysis of large datasets of interest across the scientific community. Here, we present applications in fluid dynamics and geophysics, that are extremely difficult (if not impossible) to achieve without a parallel algorithm. This work opens the path toward modal analyses of big quasi-stationary data, helping to uncover new unexplored spatio-temporal patterns.

We propose an approach to compute inner and outer-approximations of the sets of values satisfying constraints expressed as arbitrarily quantified formulas. Such formulas arise for instance when specifying important problems in control such as robustness, motion planning or controllers comparison. We propose an interval-based method which allows for tractable but tight approximations. We demonstrate its applicability through a series of examples and benchmarks using a prototype implementation.