We propose an adaptive iteratively linearized finite element method (AILFEM) in the context of strongly monotone nonlinear operators in Hilbert spaces. The approach combines adaptive mesh-refinement with an energy-contractive linearization scheme (e.g., the Ka\v{c}anov method) and a norm-contractive algebraic solver (e.g., an optimal geometric multigrid method). Crucially, a novel parameter-free algebraic stopping criterion is designed and we prove that it leads to a uniformly bounded number of algebraic solver steps. Unlike available results requiring sufficiently small adaptivity parameters to ensure even plain convergence, the new AILFEM algorithm guarantees full R-linear convergence for arbitrary adaptivity parameters. Thus, parameter-robust convergence is guaranteed. Moreover, for sufficiently small adaptivity parameters, the new adaptive algorithm guarantees optimal complexity, i.e., optimal convergence rates with respect to the overall computational cost and, hence, time.
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We study when low coordinate degree functions (LCDF) -- linear combinations of functions depending on small subsets of entries of a vector -- can hypothesis test between high-dimensional probability measures. These functions are a generalization, proposed in Hopkins' 2018 thesis but seldom studied since, of low degree polynomials (LDP), a class widely used in recent literature as a proxy for all efficient algorithms for tasks in statistics and optimization. Instead of the orthogonal polynomial decompositions used in LDP calculations, our analysis of LCDF is based on the Efron-Stein or ANOVA decomposition, making it much more broadly applicable. By way of illustration, we prove channel universality for the success of LCDF in testing for the presence of sufficiently "dilute" random signals through noisy channels: the efficacy of LCDF depends on the channel only through the scalar Fisher information for a class of channels including nearly arbitrary additive i.i.d. noise and nearly arbitrary exponential families. As applications, we extend lower bounds against LDP for spiked matrix and tensor models under additive Gaussian noise to lower bounds against LCDF under general noisy channels. We also give a simple and unified treatment of the effect of censoring models by erasing observations at random and of quantizing models by taking the sign of the observations. These results are the first computational lower bounds against any large class of algorithms for all of these models when the channel is not one of a few special cases, and thereby give the first substantial evidence for the universality of several statistical-to-computational gaps.
A preconditioning strategy is proposed for the iterative solve of large numbers of linear systems with variable matrix and right-hand side which arise during the computation of solution statistics of stochastic elliptic partial differential equations with random variable coefficients sampled by Monte Carlo. Building on the assumption that a truncated Karhunen-Lo\`{e}ve expansion of a known transform of the random variable coefficient is known, we introduce a compact representation of the random coefficient in the form of a Voronoi quantizer. The number of Voronoi cells, each of which is represented by a centroidal variable coefficient, is set to the prescribed number $P$ of preconditioners. Upon sampling the random variable coefficient, the linear system assembled with a given realization of the coefficient is solved with the preconditioner whose centroidal variable coefficient is the closest to the realization. We consider different ways to define and obtain the centroidal variable coefficients, and we investigate the properties of the induced preconditioning strategies in terms of average number of solver iterations for sequential simulations, and of load balancing for parallel simulations. Another approach, which is based on deterministic grids on the system of stochastic coordinates of the truncated representation of the random variable coefficient, is proposed with a stochastic dimension which increases with the number $P$ of preconditioners. This approach allows to bypass the need for preliminary computations in order to determine the optimal stochastic dimension of the truncated approximation of the random variable coefficient for a given number of preconditioners.
Algorithms for initializing particle distribution in SPH simulations of complex geometries have been proven essential for improving the accuracy of SPH simulations. However, no such algorithms exist for boundary integral SPH models, which can model complex geometries without needing virtual particle layers. This study introduces a Boundary Integral based Particle Initialization (BIPI) algorithm. It consists of a particle-shifting technique carefully designed to redistribute particles to fit the boundary by using the boundary integral formulation for particles adjacent to the boundary. The proposed BIPI algorithm gives special consideration to particles adjacent to the boundary to prevent artificial volume compression. It can automatically produce a "uniform" particle distribution with reduced and stabilized concentration gradient for domains with complex geometrical shapes. Finally, a number of examples are presented to demonstrate the effectiveness of the proposed algorithm.
We establish a coding theorem and a matching converse theorem for separate encodings and joint decoding of individual sequences using finite-state machines. The achievable rate region is characterized in terms of the Lempel-Ziv (LZ) complexities, the conditional LZ complexities and the joint LZ complexity of the two source sequences. An important feature that is needed to this end, which may be interesting on its own right, is a certain asymptotic form of a chain rule for LZ complexities, which we establish in this work. The main emphasis in the achievability scheme is on the universal decoder and its properties. We then show that the achievable rate region is universally attainable by a modified version of Draper's universal incremental Slepian-Wolf (SW) coding scheme, provided that there exists a low-rate reliable feedback link.
We consider a statistical model for symmetric matrix factorization with additive Gaussian noise in the high-dimensional regime where the rank $M$ of the signal matrix to infer scales with its size $N$ as $M = o(N^{1/10})$. Allowing for a $N$-dependent rank offers new challenges and requires new methods. Working in the Bayesian-optimal setting, we show that whenever the signal has i.i.d. entries the limiting mutual information between signal and data is given by a variational formula involving a rank-one replica symmetric potential. In other words, from the information-theoretic perspective, the case of a (slowly) growing rank is the same as when $M = 1$ (namely, the standard spiked Wigner model). The proof is primarily based on a novel multiscale cavity method allowing for growing rank along with some information-theoretic identities on worst noise for the Gaussian vector channel. We believe that the cavity method developed here will play a role in the analysis of a broader class of inference and spin models where the degrees of freedom are large arrays instead of vectors.
Maximum entropy (Maxent) models are a class of statistical models that use the maximum entropy principle to estimate probability distributions from data. Due to the size of modern data sets, Maxent models need efficient optimization algorithms to scale well for big data applications. State-of-the-art algorithms for Maxent models, however, were not originally designed to handle big data sets; these algorithms either rely on technical devices that may yield unreliable numerical results, scale poorly, or require smoothness assumptions that many practical Maxent models lack. In this paper, we present novel optimization algorithms that overcome the shortcomings of state-of-the-art algorithms for training large-scale, non-smooth Maxent models. Our proposed first-order algorithms leverage the Kullback-Leibler divergence to train large-scale and non-smooth Maxent models efficiently. For Maxent models with discrete probability distribution of $n$ elements built from samples, each containing $m$ features, the stepsize parameters estimation and iterations in our algorithms scale on the order of $O(mn)$ operations and can be trivially parallelized. Moreover, the strong $\ell_{1}$ convexity of the Kullback--Leibler divergence allows for larger stepsize parameters, thereby speeding up the convergence rate of our algorithms. To illustrate the efficiency of our novel algorithms, we consider the problem of estimating probabilities of fire occurrences as a function of ecological features in the Western US MTBS-Interagency wildfire data set. Our numerical results show that our algorithms outperform the state of the arts by one order of magnitude and yield results that agree with physical models of wildfire occurrence and previous statistical analyses of wildfire drivers.
We give a short survey of recent results on sparse-grid linear algorithms of approximate recovery and integration of functions possessing a unweighted or weighted Sobolev mixed smoothness based on their sampled values at a certain finite set. Some of them are extended to more general cases.
The exponential trapezoidal rule is proposed and analyzed for the numerical integration of semilinear integro-differential equations. Although the method is implicit, the numerical solution is easily obtained by standard fixed-point iteration, making its implementation straightforward. Second-order convergence in time is shown in an abstract Hilbert space framework under reasonable assumptions on the problem. Numerical experiments illustrate the proven order of convergence.
The purpose of this work is to introduce a strategy for determining the nodes and weights of a low-cardinality positive cubature formula nearly exact for polynomials of a given degree over spherical polygons. In the numerical section we report the results about numerical cubature over a spherical polygon $\cal P$ approximating Australia and reconstruction of functions over such $\cal P$, also affected by perturbations, via hyperinterpolation and some of its variants. The open-source Matlab software used in the numerical tests is available at the author's homepage.
Mendelian randomization is an instrumental variable method that utilizes genetic information to investigate the causal effect of a modifiable exposure on an outcome. In most cases, the exposure changes over time. Understanding the time-varying causal effect of the exposure can yield detailed insights into mechanistic effects and the potential impact of public health interventions. Recently, a growing number of Mendelian randomization studies have attempted to explore time-varying causal effects. However, the proposed approaches oversimplify temporal information and rely on overly restrictive structural assumptions, limiting their reliability in addressing time-varying causal problems. This paper considers a novel approach to estimate time-varying effects through continuous-time modelling by combining functional principal component analysis and weak-instrument-robust techniques. Our method effectively utilizes available data without making strong structural assumptions and can be applied in general settings where the exposure measurements occur at different timepoints for different individuals. We demonstrate through simulations that our proposed method performs well in estimating time-varying effects and provides reliable inference results when the time-varying effect form is correctly specified. The method could theoretically be used to estimate arbitrarily complex time-varying effects. However, there is a trade-off between model complexity and instrument strength. Estimating complex time-varying effects requires instruments that are unrealistically strong. We illustrate the application of this method in a case study examining the time-varying effects of systolic blood pressure on urea levels.
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