We propose a well-posed Maxwell-type boundary condition for the linear moment system in half-space. As a reduction of the Boltzmann equation, the moment equations are available to model Knudsen layers near a solid wall, where proper boundary conditions should be prescribed. Utilizing an orthogonal decomposition, we separate the part with a damping term from the system and then impose a new class of Maxwell-type boundary conditions on it. Due to the new boundary condition, we show that the half-space boundary value problem admits a unique solution with explicit expressions. Instantly, the well-posedness of the linear moment system is achieved. We apply the procedure to classical flow problems with the Shakhov collision term, such as the velocity slip and temperature jump problems. The model can capture Knudsen layers with very high accuracy using only a few moments.
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Compact finite-difference (FD) schemes specify derivative approximations implicitly, thus to achieve parallelism with domain-decomposition suitable partitioning of linear systems is required. Consistent order of accuracy, dispersion, and dissipation is crucial to maintain in wave propagation problems such that deformation of the associated spectra of the discretized problems is not too severe. In this work we consider numerically tuning spectral error, at fixed formal order of accuracy to automatically devise new compact FD schemes. Grid convergence tests indicate error reduction of at least an order of magnitude over standard FD. A proposed hybrid matching-communication strategy maintains the aforementioned properties under domain-decomposition. Under evolution of linear wave-propagation problems utilizing exponential integration or explicit Runge-Kutta methods improvement is found to remain robust. A first demonstration that compact FD methods may be applied to the Z4c formulation of numerical relativity is provided where we couple our header-only, templated C++ implementation to the highly performant GR-Athena++ code. Evolving Z4c on test-bed problems shows at least an order in magnitude reduction in phase error compared to FD for propagated metric components. Stable binary-black-hole evolution utilizing compact FD together with improved convergence is also demonstrated.
Bandits with Knapsacks (BwK), the generalization of the Multi-Armed Bandits under budget constraints, has received a lot of attention in recent years. It has numerous applications, including dynamic pricing, repeated auctions, etc. Previous work has focused on one of the two extremes: Stochastic BwK where the rewards and consumptions of the resources each round are sampled from an i.i.d. distribution, and Adversarial BwK where these values are picked by an adversary. Achievable guarantees in the two cases exhibit a massive gap: No-regret learning is achievable in Stochastic BwK, but in Adversarial BwK, only competitive ratio style guarantees are achievable, where the competitive ratio depends on the budget. What makes this gap so vast is that in Adversarial BwK the guarantees get worse in the typical case when the budget is more binding. While ``best-of-both-worlds'' type algorithms are known (algorithms that provide the best achievable guarantee in both extreme cases), their guarantees degrade to the adversarial case as soon as the environment is not fully stochastic. Our work aims to bridge this gap, offering guarantees for a workload that is not exactly stochastic but is also not worst-case. We define a condition, Approximately Stationary BwK, that parameterizes how close to stochastic or adversarial an instance is. Based on these parameters, we explore what is the best competitive ratio attainable in BwK. We explore two algorithms that are oblivious to the values of the parameters but guarantee competitive ratios that smoothly transition between the best possible guarantees in the two extreme cases, depending on the values of the parameters. Our guarantees offer great improvement over the adversarial guarantee, especially when the available budget is small. We also prove bounds on the achievable guarantee, showing that our results are approximately tight when the budget is small.
We develop a new interior point method for solving linear programs. Our algorithm is universal in the sense that it matches the number of iterations of any interior point method that uses a self-concordant barrier function up to a factor $O(n^{1.5}\log n)$ for an $n$-variable linear program in standard form. The running time bounds of interior point methods depend on bit-complexity or condition measures that can be unbounded in the problem dimension. This is in contrast with the simplex method that always admits an exponential bound. Our algorithm also admits a combinatorial upper bound, terminating with an exact solution in $O(2^{n} n^{1.5}\log n)$ iterations. This complements previous work by Allamigeon, Benchimol, Gaubert, and Joswig (SIAGA 2018) that exhibited a family of instances where any path-following method must take exponentially many iterations.
We introduce the Weak-form Estimation of Nonlinear Dynamics (WENDy) method for estimating model parameters for non-linear systems of ODEs. The core mathematical idea involves an efficient conversion of the strong form representation of a model to its weak form, and then solving a regression problem to perform parameter inference. The core statistical idea rests on the Errors-In-Variables framework, which necessitates the use of the iteratively reweighted least squares algorithm. Further improvements are obtained by using orthonormal test functions, created from a set of $C^{\infty}$ bump functions of varying support sizes. We demonstrate that WENDy is a highly robust and efficient method for parameter inference in differential equations. Without relying on any numerical differential equation solvers, WENDy computes accurate estimates and is robust to large (biologically relevant) levels of measurement noise. For low dimensional systems with modest amounts of data, WENDy is competitive with conventional forward solver-based nonlinear least squares methods in terms of speed and accuracy. For both higher dimensional systems and stiff systems, WENDy is typically both faster (often by orders of magnitude) and more accurate than forward solver-based approaches. We illustrate the method and its performance in some common population and neuroscience models, including logistic growth, Lotka-Volterra, FitzHugh-Nagumo, Hindmarsh-Rose, and a Protein Transduction Benchmark model. Software and code for reproducing the examples is available at (//github.com/MathBioCU/WENDy).
In this paper we deal with the problem of sequential testing of multiple hypotheses. The main goal is minimising the expected sample size (ESS) under restrictions on the error probabilities. We use a variant of the method of Lagrange multipliers which is based on the minimisation of an auxiliary objective function (called Lagrangian). This function is defined as a weighted sum of all the test characteristics we are interested in: the error probabilities and the ESSs evaluated at some points of interest. In this paper, we use a definition of the Lagrangian function involving the ESS evaluated at any finite number of fixed parameter points (not necessarily those representing the hypotheses). Then we develop a computer-oriented method of minimisation of the Lagrangian function, that provides, depending on the specific choice of the parameter points, optimal tests in different concrete settings, like in Bayesian, Kiefer-Weiss and other settings. To exemplify the proposed methods for the particular case of sampling from a Bernoulli population we develop a set of computer algorithms for designing sequential tests that minimise the Lagrangian function and for the numerical evaluation of test characteristics like the error probabilities and the ESS, and other related. For the Bernoulli model, we made a series of computer evaluations related to the optimality of sequential multi-hypothesis tests, in a particular case of three hypotheses. A numerical comparison with the matrix sequential probability ratio test is carried out.
Elliptic interface problems whose solutions are $C^0$ continuous have been well studied over the past two decades. The well-known numerical methods include the strongly stable generalized finite element method (SGFEM) and immersed FEM (IFEM). In this paper, we study numerically a larger class of elliptic interface problems where their solutions are discontinuous. A direct application of these existing methods fails immediately as the approximate solution is in a larger space that covers discontinuous functions. We propose a class of high-order enriched unfitted FEMs to solve these problems with implicit or Robin-type interface jump conditions. We design new enrichment functions that capture the imposed discontinuity of the solution while keeping the condition number from fast growth. A linear enriched method in 1D was recently developed using one enrichment function and we generalized it to an arbitrary degree using two simple discontinuous one-sided enrichment functions. The natural tensor product extension to the 2D case is demonstrated. Optimal order convergence in the $L^2$ and broken $H^1$-norms are established. We also establish superconvergence at all discretization nodes (including exact nodal values in special cases). Numerical examples are provided to confirm the theory. Finally, to prove the efficiency of the method for practical problems, the enriched linear, quadratic, and cubic elements are applied to a multi-layer wall model for drug-eluting stents in which zero-flux jump conditions and implicit concentration interface conditions are both present.
Consider the problem of solving systems of linear algebraic equations $Ax=b$ with a real symmetric positive definite matrix $A$ using the conjugate gradient (CG) method. To stop the algorithm at the appropriate moment, it is important to monitor the quality of the approximate solution. One of the most relevant quantities for measuring the quality of the approximate solution is the $A$-norm of the error. This quantity cannot be easily computed, however, it can be estimated. In this paper we discuss and analyze the behaviour of the Gauss-Radau upper bound on the $A$-norm of the error, based on viewing CG as a procedure for approximating a certain Riemann-Stieltjes integral. This upper bound depends on a prescribed underestimate $\mu$ to the smallest eigenvalue of $A$. We concentrate on explaining a phenomenon observed during computations showing that, in later CG iterations, the upper bound loses its accuracy, and is almost independent of $\mu$. We construct a model problem that is used to demonstrate and study the behaviour of the upper bound in dependence of $\mu$, and developed formulas that are helpful in understanding this behavior. We show that the above mentioned phenomenon is closely related to the convergence of the smallest Ritz value to the smallest eigenvalue of $A$. It occurs when the smallest Ritz value is a better approximation to the smallest eigenvalue than the prescribed underestimate $\mu$. We also suggest an adaptive strategy for improving the accuracy of the upper bounds in the previous iterations.
Deep neural networks (DNNs) are sensitive to adversarial examples, resulting in fragile and unreliable performance in the real world. Although adversarial training (AT) is currently one of the most effective methodologies to robustify DNNs, it is computationally very expensive (e.g., 5-10X costlier than standard training). To address this challenge, existing approaches focus on single-step AT, referred to as Fast AT, reducing the overhead of adversarial example generation. Unfortunately, these approaches are known to fail against stronger adversaries. To make AT computationally efficient without compromising robustness, this paper takes a different view of the efficient AT problem. Specifically, we propose to minimize redundancies at the data level by leveraging data pruning. Extensive experiments demonstrate that the data pruning based AT can achieve similar or superior robust (and clean) accuracy as its unpruned counterparts while being significantly faster. For instance, proposed strategies accelerate CIFAR-10 training up to 3.44X and CIFAR-100 training to 2.02X. Additionally, the data pruning methods can readily be reconciled with existing adversarial acceleration tricks to obtain the striking speed-ups of 5.66X and 5.12X on CIFAR-10, 3.67X and 3.07X on CIFAR-100 with TRADES and MART, respectively.
Pre-trained deep neural network language models such as ELMo, GPT, BERT and XLNet have recently achieved state-of-the-art performance on a variety of language understanding tasks. However, their size makes them impractical for a number of scenarios, especially on mobile and edge devices. In particular, the input word embedding matrix accounts for a significant proportion of the model's memory footprint, due to the large input vocabulary and embedding dimensions. Knowledge distillation techniques have had success at compressing large neural network models, but they are ineffective at yielding student models with vocabularies different from the original teacher models. We introduce a novel knowledge distillation technique for training a student model with a significantly smaller vocabulary as well as lower embedding and hidden state dimensions. Specifically, we employ a dual-training mechanism that trains the teacher and student models simultaneously to obtain optimal word embeddings for the student vocabulary. We combine this approach with learning shared projection matrices that transfer layer-wise knowledge from the teacher model to the student model. Our method is able to compress the BERT_BASE model by more than 60x, with only a minor drop in downstream task metrics, resulting in a language model with a footprint of under 7MB. Experimental results also demonstrate higher compression efficiency and accuracy when compared with other state-of-the-art compression techniques.
The field of few-shot learning has recently seen substantial advancements. Most of these advancements came from casting few-shot learning as a meta-learning problem. Model Agnostic Meta Learning or MAML is currently one of the best approaches for few-shot learning via meta-learning. MAML is simple, elegant and very powerful, however, it has a variety of issues, such as being very sensitive to neural network architectures, often leading to instability during training, requiring arduous hyperparameter searches to stabilize training and achieve high generalization and being very computationally expensive at both training and inference times. In this paper, we propose various modifications to MAML that not only stabilize the system, but also substantially improve the generalization performance, convergence speed and computational overhead of MAML, which we call MAML++.
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