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This paper addresses the deconvolution problem of estimating a square-integrable probability density from observations contaminated with additive measurement errors having a known density. The estimator begins with a density estimate of the contaminated observations and minimizes a reconstruction error penalized by an integrated squared $m$-th derivative. Theory for deconvolution has mainly focused on kernel- or wavelet-based techniques, but other methods including spline-based techniques and this smoothness-penalized estimator have been found to outperform kernel methods in simulation studies. This paper fills in some of these gaps by establishing asymptotic guarantees for the smoothness-penalized approach. Consistency is established in mean integrated squared error, and rates of convergence are derived for Gaussian, Cauchy, and Laplace error densities, attaining some lower bounds already in the literature. The assumptions are weak for most results; the estimator can be used with a broader class of error densities than the deconvoluting kernel. Our application example estimates the density of the mean cytotoxicity of certain bacterial isolates under random sampling; this mean cytotoxicity can only be measured experimentally with additive error, leading to the deconvolution problem. We also describe a method for approximating the solution by a cubic spline, which reduces to a quadratic program.

In building practical applications of evolutionary computation (EC), two optimizations are essential. First, the parameters of the search method need to be tuned to the domain in order to balance exploration and exploitation effectively. Second, the search method needs to be distributed to take advantage of parallel computing resources. This paper presents BLADE (BLAnket Distributed Evolution) as an approach to achieving both goals simultaneously. BLADE uses blankets (i.e., masks on the genetic representation) to tune the evolutionary operators during the search, and implements the search through hub-and-spoke distribution. In the paper, (1) the blanket method is formalized for the (1 + 1)EA case as a Markov chain process. Its effectiveness is then demonstrated by analyzing dominant and subdominant eigenvalues of stochastic matrices, suggesting a generalizable theory; (2) the fitness-level theory is used to analyze the distribution method; and (3) these insights are verified experimentally on three benchmark problems, showing that both blankets and distribution lead to accelerated evolution. Moreover, a surprising synergy emerges between them: When combined with distribution, the blanket approach achieves more than $n$-fold speedup with $n$ clients in some cases. The work thus highlights the importance and potential of optimizing evolutionary computation in practical applications.

PDE-constrained inverse problems are some of the most challenging and computationally demanding problems in computational science today. Fine meshes that are required to accurately compute the PDE solution introduce an enormous number of parameters and require large scale computing resources such as more processors and more memory to solve such systems in a reasonable time. For inverse problems constrained by time dependent PDEs, the adjoint method that is often employed to efficiently compute gradients and higher order derivatives requires solving a time-reversed, so-called adjoint PDE that depends on the forward PDE solution at each timestep. This necessitates the storage of a high dimensional forward solution vector at every timestep. Such a procedure quickly exhausts the available memory resources. Several approaches that trade additional computation for reduced memory footprint have been proposed to mitigate the memory bottleneck, including checkpointing and compression strategies. In this work, we propose a close-to-ideal scalable compression approach using autoencoders to eliminate the need for checkpointing and substantial memory storage, thereby reducing both the time-to-solution and memory requirements. We compare our approach with checkpointing and an off-the-shelf compression approach on an earth-scale ill-posed seismic inverse problem. The results verify the expected close-to-ideal speedup for both the gradient and Hessian-vector product using the proposed autoencoder compression approach. To highlight the usefulness of the proposed approach, we combine the autoencoder compression with the data-informed active subspace (DIAS) prior to show how the DIAS method can be affordably extended to large scale problems without the need of checkpointing and large memory.

We present an alternating least squares type numerical optimization scheme to estimate conditionally-independent mixture models in $\mathbb{R}^n$, without parameterizing the distributions. Following the method of moments, we tackle an incomplete tensor decomposition problem to learn the mixing weights and componentwise means. Then we compute the cumulative distribution functions, higher moments and other statistics of the component distributions through linear solves. Crucially for computations in high dimensions, the steep costs associated with high-order tensors are evaded, via the development of efficient tensor-free operations. Numerical experiments demonstrate the competitive performance of the algorithm, and its applicability to many models and applications. Furthermore we provide theoretical analyses, establishing identifiability from low-order moments of the mixture and guaranteeing local linear convergence of the ALS algorithm.

Cochran's $Q$ statistic is routinely used for testing heterogeneity in meta-analysis. Its expected value (under an incorrect null distribution) is part of several popular estimators of the between-study variance, $\tau^2$. Those applications generally do not account for the studies' use of estimated variances in the inverse-variance weights that define $Q$ (more explicitly, $Q_{IV}$). Importantly, those weights make approximating the distribution of $Q_{IV}$ rather complicated. As an alternative, we are investigating a $Q$ statistic, $Q_F$, whose constant weights use only the studies' arm-level sample sizes. For log-odds-ratio, log-relative-risk, and risk difference as the measure of effect, these simulations study approximations to the distributions of $Q_F$ and $Q_{IV}$, as the basis for tests of heterogeneity. We present the results in 132 Figures, 153 pages in total.

Sampling from high-dimensional distributions is a fundamental problem in statistical research and practice. However, great challenges emerge when the target density function is unnormalized and contains isolated modes. We tackle this difficulty by fitting an invertible transformation mapping, called a transport map, between a reference probability measure and the target distribution, so that sampling from the target distribution can be achieved by pushing forward a reference sample through the transport map. We theoretically analyze the limitations of existing transport-based sampling methods using the Wasserstein gradient flow theory, and propose a new method called TemperFlow that addresses the multimodality issue. TemperFlow adaptively learns a sequence of tempered distributions to progressively approach the target distribution, and we prove that it overcomes the limitations of existing methods. Various experiments demonstrate the superior performance of this novel sampler compared to traditional methods, and we show its applications in modern deep learning tasks such as image generation. The programming code for the numerical experiments is available at //github.com/yixuan/temperflow.

Gaussian processes are widely used as priors for unknown functions in statistics and machine learning. To achieve computationally feasible inference for large datasets, a popular approach is the Vecchia approximation, which is an ordered conditional approximation of the data vector that implies a sparse Cholesky factor of the precision matrix. The ordering and sparsity pattern are typically determined based on Euclidean distance of the inputs or locations corresponding to the data points. Here, we propose instead to use a correlation-based distance metric, which implicitly applies the Vecchia approximation in a suitable transformed input space. The correlation-based algorithm can be carried out in quasilinear time in the size of the dataset, and so it can be applied even for iterative inference on unknown parameters in the correlation structure. The correlation-based approach has two advantages for complex settings: It can result in more accurate approximations, and it offers a simple, automatic strategy that can be applied to any covariance, even when Euclidean distance is not applicable. We demonstrate these advantages in several settings, including anisotropic, nonstationary, multivariate, and spatio-temporal processes. We also illustrate our method on multivariate spatio-temporal temperature fields produced by a regional climate model.

With the recent study of deep learning in scientific computation, the Physics-Informed Neural Networks (PINNs) method has drawn widespread attention for solving Partial Differential Equations (PDEs). Compared to traditional methods, PINNs can efficiently handle high-dimensional problems, but the accuracy is relatively low, especially for highly irregular problems. Inspired by the idea of adaptive finite element methods and incremental learning, we propose GAS, a Gaussian mixture distribution-based adaptive sampling method for PINNs. During the training procedure, GAS uses the current residual information to generate a Gaussian mixture distribution for the sampling of additional points, which are then trained together with historical data to speed up the convergence of the loss and achieve higher accuracy. Several numerical simulations on 2D and 10D problems show that GAS is a promising method that achieves state-of-the-art accuracy among deep solvers, while being comparable with traditional numerical solvers.

Coupled cluster theory is considered to be the ``gold standard'' ansatz of molecular quantum chemistry. The finite-size error of the correlation energy in periodic coupled cluster calculations for three-dimensional insulating systems has been observed to satisfy the inverse volume scaling, even in the absence of any correction schemes. This is surprising, as simpler theories that utilize only a subset of the coupled cluster diagrams exhibit much slower decay of the finite-size error, which scales inversely with the length of the system. In this study, we present a rigorous numerical analysis that explains the underlying mechanisms behind this phenomenon in the context of coupled cluster doubles (CCD) calculations, and reconciles a few seemingly paradoxical statements with respect to the finite-size scaling. Our findings also have implications on how to effectively address finite-size errors in practical quantum chemistry calculations for periodic systems.

With the rapid increase of large-scale, real-world datasets, it becomes critical to address the problem of long-tailed data distribution (i.e., a few classes account for most of the data, while most classes are under-represented). Existing solutions typically adopt class re-balancing strategies such as re-sampling and re-weighting based on the number of observations for each class. In this work, we argue that as the number of samples increases, the additional benefit of a newly added data point will diminish. We introduce a novel theoretical framework to measure data overlap by associating with each sample a small neighboring region rather than a single point. The effective number of samples is defined as the volume of samples and can be calculated by a simple formula $(1-\beta^{n})/(1-\beta)$, where $n$ is the number of samples and $\beta \in [0,1)$ is a hyperparameter. We design a re-weighting scheme that uses the effective number of samples for each class to re-balance the loss, thereby yielding a class-balanced loss. Comprehensive experiments are conducted on artificially induced long-tailed CIFAR datasets and large-scale datasets including ImageNet and iNaturalist. Our results show that when trained with the proposed class-balanced loss, the network is able to achieve significant performance gains on long-tailed datasets.