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An invertible function is bi-Lipschitz if both the function and its inverse have bounded Lipschitz constants. Nowadays, most Normalizing Flows are bi-Lipschitz by design or by training to limit numerical errors (among other things). In this paper, we discuss the expressivity of bi-Lipschitz Normalizing Flows and identify several target distributions that are difficult to approximate using such models. Then, we characterize the expressivity of bi-Lipschitz Normalizing Flows by giving several lower bounds on the Total Variation distance between these particularly unfavorable distributions and their best possible approximation. Finally, we discuss potential remedies which include using more complex latent distributions.

Working from a Poisson-Gaussian noise model, a multi-sample extension of the Photon Counting Histogram Expectation Maximization (PCH-EM) algorithm is derived as a general-purpose alternative to the Photon Transfer (PT) method. This algorithm is derived from the same model, requires the same experimental data, and estimates the same sensor performance parameters as the time-tested PT method, all while obtaining lower uncertainty estimates. It is shown that as read noise becomes large, multiple data samples are necessary to capture enough information about the parameters of a device under test, justifying the need for a multi-sample extension. An estimation procedure is devised consisting of initial PT characterization followed by repeated iteration of PCH-EM to demonstrate the improvement in estimate uncertainty achievable with PCH-EM; particularly in the regime of Deep Sub-Electron Read Noise (DSERN). A statistical argument based on the information theoretic concept of sufficiency is formulated to explain how PT data reduction procedures discard information contained in raw sensor data, thus explaining why the proposed algorithm is able to obtain lower uncertainty estimates of key sensor performance parameters such as read noise and conversion gain. Experimental data captured from a CMOS quanta image sensor with DSERN is then used to demonstrate the algorithm's usage and validate the underlying theory and statistical model. In support of the reproducible research effort, the code associated with this work can be obtained on the MathWorks File Exchange (Hendrickson et al., 2024).

Robust Markov Decision Processes (RMDPs) are a widely used framework for sequential decision-making under parameter uncertainty. RMDPs have been extensively studied when the objective is to maximize the discounted return, but little is known for average optimality (optimizing the long-run average of the rewards obtained over time) and Blackwell optimality (remaining discount optimal for all discount factors sufficiently close to 1). In this paper, we prove several foundational results for RMDPs beyond the discounted return. We show that average optimal policies can be chosen stationary and deterministic for sa-rectangular RMDPs but, perhaps surprisingly, that history-dependent (Markovian) policies strictly outperform stationary policies for average optimality in s-rectangular RMDPs. We also study Blackwell optimality for sa-rectangular RMDPs, where we show that {\em approximate} Blackwell optimal policies always exist, although Blackwell optimal policies may not exist. We also provide a sufficient condition for their existence, which encompasses virtually any examples from the literature. We then discuss the connection between average and Blackwell optimality, and we describe several algorithms to compute the optimal average return. Interestingly, our approach leverages the connections between RMDPs and stochastic games.

libEnsemble is a Python-based toolkit for running dynamic ensembles, developed as part of the DOE Exascale Computing Project. The toolkit utilizes a unique generator--simulator--allocator paradigm, where generators produce input for simulators, simulators evaluate those inputs, and allocators decide whether and when a simulator or generator should be called. The generator steers the ensemble based on simulation results. libEnsemble communicates between a manager and workers. Flexibility is provided through multiple manager--worker communication substrates each of which has different benefits. These include Python's multiprocessing, mpi4py, and TCP. Multisite ensembles are supported using Balsam or Globus Compute. We overview the unique characteristics of libEnsemble as well as current and potential interoperability with other packages in the workflow ecosystem. We highlight libEnsemble's dynamic resource features: libEnsemble can detect system resources (nodes, cores, and GPUs) and assign these in a portable way. These features allow users to specify resources required for each simulation automatically on a range of systems, including Frontier, Aurora, and Perlmutter. Such ensembles can include multiple simulation types, some using GPUs and others using only CPUs, sharing nodes for maximum efficiency. We demonstrate libEnsemble's capabilities, scalability, and scientific impact via a Gaussian process surrogate training problem for the longitudinal density profile at the exit of a plasma accelerator stage using Wake-T and WarpX simulations. We also describe the benefits of libEnsemble's generator--simulator coupling, which easily exposes to the user the ability to cancel, and portably kill, running simulations. Such control can be directed from the generator or allocator based on models that are updated with intermediate simulation output.

We show that the known list-decoding algorithms for univariate multiplicity and folded Reed-Solomon (FRS) codes can be made to run in nearly-linear time. This yields, to our knowledge, the first known family of codes that can be decoded in nearly linear time, even as they approach the list decoding capacity. Univariate multiplicity codes and FRS codes are natural variants of Reed-Solomon codes that were discovered and studied for their applications to list-decoding. It is known that for every $\epsilon >0$, and rate $R \in (0,1)$, there exist explicit families of these codes that have rate $R$ and can be list-decoded from a $(1-R-\epsilon)$ fraction of errors with constant list size in polynomial time (Guruswami & Wang (IEEE Trans. Inform. Theory, 2013) and Kopparty, Ron-Zewi, Saraf & Wootters (SIAM J. Comput. 2023)). In this work, we present randomized algorithms that perform the above tasks in nearly linear time. Our algorithms have two main components. The first builds upon the lattice-based approach of Alekhnovich (IEEE Trans. Inf. Theory 2005), who designed a nearly linear time list-decoding algorithm for Reed-Solomon codes approaching the Johnson radius. As part of the second component, we design nearly-linear time algorithms for two natural algebraic problems. The first algorithm solves linear differential equations of the form $Q\left(x, f(x), \frac{df}{dx}, \dots,\frac{d^m f}{dx^m}\right) \equiv 0$ where $Q$ has the form $Q(x,y_0,\dots,y_m) = \tilde{Q}(x) + \sum_{i = 0}^m Q_i(x)\cdot y_i$. The second solves functional equations of the form $Q\left(x, f(x), f(\gamma x), \dots,f(\gamma^m x)\right) \equiv 0$ where $\gamma$ is a high-order field element. These algorithms can be viewed as generalizations of classical algorithms of Sieveking (Computing 1972) and Kung (Numer. Math. 1974) for computing the modular inverse of a power series, and might be of independent interest.

This paper focuses on discussing Newton's method and its hybrid with machine learning for the steady state Navier-Stokes Darcy model discretized by mixed element methods. First, a Newton iterative method is introduced for solving the relative discretized problem. It is proved technically that this method converges quadratically with the convergence rate independent of the finite element mesh size, under certain standard conditions. Later on, a deep learning algorithm is proposed for solving this nonlinear coupled problem. Following the ideas of an earlier work by Huang, Wang and Yang (2020), an Int-Deep algorithm is constructed by combining the previous two methods so as to further improve the computational efficiency and robustness. A series of numerical examples are reported to show the numerical performance of the proposed methods.

We study the lattice Green's function (LGF) of the screened Poisson equation on a two-dimensional rectangular lattice. This LGF arises in numerical analysis, random walks, solid-state physics, and other fields. Its defining characteristic is the screening term, which defines different regimes. When its coefficient is large, we can accurately approximate the LGF with an exponentially converging asymptotic expansion, and its convergence rate monotonically increases with the coefficient of the screening term. To tabulate the LGF when the coefficient is not large, we derive a one-dimensional integral representation of the LGF. We show that the trapezoidal rule can approximate this integral with exponential convergence, and we propose an efficient algorithm for its evaluation via the Fast Fourier Transform. We discuss applications including computing the LGF of the three-dimensional Poisson equation with one periodic direction and the return probability of a two-dimensional random walk with killing.

We introduce PennyLane's Lightning suite, a collection of high-performance state-vector simulators targeting CPU, GPU, and HPC-native architectures and workloads. Quantum applications such as QAOA, VQE, and synthetic workloads are implemented to demonstrate the supported classical computing architectures and showcase the scale of problems that can be simulated using our tooling. We benchmark the performance of Lightning with backends supporting CPUs, as well as NVidia and AMD GPUs, and compare the results to other commonly used high-performance simulator packages, demonstrating where Lightning's implementations give performance leads. We show improved CPU performance by employing explicit SIMD intrinsics and multi-threading, batched task-based execution across multiple GPUs, and distributed forward and gradient-based quantum circuit executions across multiple nodes. Our data shows we can comfortably simulate a variety of circuits, giving examples with up to 30 qubits on a single device or node, and up to 41 qubits using multiple nodes.

We extend the fourth order, two stage Multi-Derivative Runge Kutta (MDRK) scheme of Li and Du to the Flux Reconstruction (FR) framework by writing both of the stages in terms of a time averaged flux and then use the approximate Lax-Wendroff procedure. Numerical flux is computed in each stage using D2 dissipation and EA flux, enhancing Fourier CFL stability and accuracy respectively. A subcell based blending limiter is developed for the MDRK scheme, which ensures that the limited scheme is provably admissibility preserving. Along with being admissibility preserving, the blending scheme is constructed to minimize dissipation errors by using Gauss-Legendre solution points and performing MUSCL-Hancock reconstruction on subcells. The accuracy enhancement of the blending scheme is numerically verified on compressible Euler's equations, with test cases involving shocks and small-scale structures.

Classical tests are available for the two-sample test of correspondence of distribution functions. From these, the Kolmogorov-Smirnov test provides also the graphical interpretation of the test results, in different forms. Here, we propose modifications of the Kolmogorov-Smirnov test with higher power. The proposed tests are based on the so-called global envelope test which allows for graphical interpretation, similarly as the Kolmogorov-Smirnov test. The tests are based on rank statistics and are suitable also for the comparison of $n$ samples, with $n \geq 2$. We compare the alternatives for the two-sample case through an extensive simulation study and discuss their interpretation. Finally, we apply the tests to real data. Specifically, we compare the height distributions between boys and girls at different ages, as well as sepal length distributions of different flower species using the proposed methodologies.