The Merriman-Bence-Osher threshold dynamics method is an efficient algorithm to simulate the motion by mean curvature, in which two steps of convolution with diffusion kernel and thresholding alternate. It has the advantages of being easy to implement and with high efficiency. In this paper, we propose an efficient threshold dynamics method for dislocation dynamics in a slip plane. We show that this proposed threshold dislocation dynamics method is able to give correct two leading orders in dislocation velocity, including both the $O(\log \varepsilon)$ local curvature force and the $O(1)$ nonlocal force due to the long-range stress field generated by the dislocations, where $\varepsilon$ is the dislocation core size. This is different from the available threshold dynamics methods in the literature which only give the leading order local velocities associated with mean curvature or its anisotropic generalizations of the moving fronts. We also propose a numerical method based on spatial variable stretching to overcome the numerical limitations brought by physical settings in this threshold dislocation dynamics method. Specifically, this variable stretching method is able to correct the mobility and to rescale the velocity, which can be applied generally to any threshold dynamics method. We validate the proposed threshold dislocation dynamics method by numerical simulations of various motions and interaction of dislocations.
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The eigenvalue method, suggested by the developer of the extensively used Analytic Hierarchy Process methodology, exhibits right-left asymmetry: the priorities derived from the right eigenvector do not necessarily coincide with the priorities derived from the reciprocal left eigenvector. This paper offers a comprehensive numerical experiment to compare the two eigenvector-based weighting procedures and their reasonable alternative of the row geometric mean with respect to four measures. The underlying pairwise comparison matrices are constructed randomly with different dimensions and levels of inconsistency. The disagreement between the two eigenvectors turns out to be not always a monotonic function of these important characteristics of the matrix. The ranking contradictions can affect alternatives with relatively distant priorities. The row geometric mean is found to be almost at the midpoint between the right and inverse left eigenvectors, making it a straightforward compromise between them.
Iterative refinement (IR) is a popular scheme for solving a linear system of equations based on gradually improving the accuracy of an initial approximation. Originally developed to improve upon the accuracy of Gaussian elimination, interest in IR has been revived because of its suitability for execution on fast low-precision hardware such as analog devices and graphics processing units. IR generally converges when the error associated with the solution method is small, but is known to diverge when this error is large. We propose and analyze a novel enhancement to the IR algorithm by adding a line search optimization step that guarantees the algorithm will not diverge. Numerical experiments verify our theoretical results and illustrate the effectiveness of our proposed scheme.
The sequential composition of propositional logic programs has been recently introduced. This paper studies the sequential {\em decomposition} of programs by studying Green's relations $\mathcal{L,R,J}$ -- well-known in semigroup theory -- between programs. In a broader sense, this paper is a further step towards an algebraic theory of logic programming.
Quantum computing is a growing field where the information is processed by two-levels quantum states known as qubits. Current physical realizations of qubits require a careful calibration, composed by different experiments, due to noise and decoherence phenomena. Among the different characterization experiments, a crucial step is to develop a model to classify the measured state by discriminating the ground state from the excited state. In this proceedings we benchmark multiple classification techniques applied to real quantum devices.
An underlying mechanism for successful deep learning (DL) with a limited deep architecture and dataset, namely VGG-16 on CIFAR-10, was recently presented based on a quantitative method to measure the quality of a single filter in each layer. In this method, each filter identifies small clusters of possible output labels, with additional noise selected as labels out of the clusters. This feature is progressively sharpened with the layers, resulting in an enhanced signal-to-noise ratio (SNR) and higher accuracy. In this study, the suggested universal mechanism is verified for VGG-16 and EfficientNet-B0 trained on the CIFAR-100 and ImageNet datasets with the following main results. First, the accuracy progressively increases with the layers, whereas the noise per filter typically progressively decreases. Second, for a given deep architecture, the maximal error rate increases approximately linearly with the number of output labels. Third, the average filter cluster size and the number of clusters per filter at the last convolutional layer adjacent to the output layer are almost independent of the number of dataset labels in the range [3, 1,000], while a high SNR is preserved. The presented DL mechanism suggests several techniques, such as applying filter's cluster connections (AFCC), to improve the computational complexity and accuracy of deep architectures and furthermore pinpoints the simplification of pre-existing structures while maintaining their accuracies.
This study presents an importance sampling formulation based on adaptively relaxing parameters from the indicator function and/or the probability density function. The formulation embodies the prevalent mathematical concept of relaxing a complex problem into a sequence of progressively easier sub-problems. Due to the flexibility in constructing relaxation parameters, relaxation-based importance sampling provides a unified framework for various existing variance reduction techniques, such as subset simulation, sequential importance sampling, and annealed importance sampling. More crucially, the framework lays the foundation for creating new importance sampling strategies, tailoring to specific applications. To demonstrate this potential, two importance sampling strategies are proposed. The first strategy couples annealed importance sampling with subset simulation, focusing on low-dimensional problems. The second strategy aims to solve high-dimensional problems by leveraging spherical sampling and scaling techniques. Both methods are desirable for fragility analysis in performance-based engineering, as they can produce the entire fragility surface in a single run of the sampling algorithm. Three numerical examples, including a 1000-dimensional stochastic dynamic problem, are studied to demonstrate the proposed methods.
The recently introduced Genetic Column Generation (GenCol) algorithm has been numerically observed to efficiently and accurately compute high-dimensional optimal transport plans for general multi-marginal problems, but theoretical results on the algorithm have hitherto been lacking. The algorithm solves the OT linear program on a dynamically updated low-dimensional submanifold consisting of sparse plans. The submanifold dimension exceeds the sparse support of optimal plans only by a fixed factor $\beta$. Here we prove that for $\beta \geq 2$ and in the two-marginal case, GenCol always converges to an exact solution, for arbitrary costs and marginals. The proof relies on the concept of c-cyclical monotonicity. As an offshoot, GenCol rigorously reduces the data complexity of numerically solving two-marginal OT problems from $O(\ell^2)$ to $O(\ell)$ without any loss in accuracy, where $\ell$ is the number of discretization points for a single marginal. At the end of the paper we also present some insights into the convergence behavior in the multi-marginal case.
We are interested in numerical algorithms for computing the electrical field generated by a charge distribution localized on scale $l$ in an infinite heterogeneous correlated random medium, in a situation where the medium is only known in a box of diameter $L\gg l$ around the support of the charge. We show that the algorithm of Lu, Otto and Wang, suggesting optimal Dirichlet boundary conditions motivated by the multipole expansion of Bella, Giunti and Otto, still performs well in correlated media. With overwhelming probability, we obtain a convergence rate in terms of $l$, $L$ and the size of the correlations for which optimality is supported with numerical simulations. These estimates are provided for ensembles which satisfy a multi-scale logarithmic Sobolev inequality, where our main tool is an extension of the semi-group estimates established by the first author. As part of our strategy, we construct sub-linear second-order correctors in this correlated setting which is of independent interest.
For distributions over discrete product spaces $\prod_{i=1}^n \Omega_i'$, Glauber dynamics is a Markov chain that at each step, resamples a random coordinate conditioned on the other coordinates. We show that $k$-Glauber dynamics, which resamples a random subset of $k$ coordinates, mixes $k$ times faster in $\chi^2$-divergence, and assuming approximate tensorization of entropy, mixes $k$ times faster in KL-divergence. We apply this to Ising models $\mu_{J,h}(x)\propto \exp(\frac1 2\left\langle x,Jx \right\rangle + \langle h,x\rangle)$ with $\|J\|<1-c$ (the regime where fast mixing is known), where we show that we can implement each step of $\widetilde O(n/\|J\|_F)$-Glauber dynamics efficiently with a parallel algorithm, resulting in a parallel algorithm with running time $\widetilde O(\|J\|_F) = \widetilde O(\sqrt n)$.
The goal of explainable Artificial Intelligence (XAI) is to generate human-interpretable explanations, but there are no computationally precise theories of how humans interpret AI generated explanations. The lack of theory means that validation of XAI must be done empirically, on a case-by-case basis, which prevents systematic theory-building in XAI. We propose a psychological theory of how humans draw conclusions from saliency maps, the most common form of XAI explanation, which for the first time allows for precise prediction of explainee inference conditioned on explanation. Our theory posits that absent explanation humans expect the AI to make similar decisions to themselves, and that they interpret an explanation by comparison to the explanations they themselves would give. Comparison is formalized via Shepard's universal law of generalization in a similarity space, a classic theory from cognitive science. A pre-registered user study on AI image classifications with saliency map explanations demonstrate that our theory quantitatively matches participants' predictions of the AI.
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