In recent research, the parallel performances of sweeping-type algorithms for high-frequency time-harmonic wave problems have been improved by departing from standard layer-type domain decomposition and introducing a new sweeping strategy on a checkerboard-type domain decomposition, where sweeps can be performed more flexibly. These sweeps can be done by a certain number of steps, each of which provides the necessary information from subdomains solved at the current iteration to their next neighboring subdomains. Although, subproblems in these subdomains can be solved concurrently at each step, the sequential nature of the process of the sweeping approaches still exists, which limits their potential for parallelization. We propose a block Jacobi sweeping preconditioner, which is an improved variant of sweeping-type preconditioners. The new feature of these improved variants can be interpreted as several partial sweeps, which can be thought of as sweeps that operate on a subset of the subdomains in parallel. We present several two-dimensional finite element results to study and compare the sweeping preconditioner and the block Jacobi sweeping preconditioner.
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Neural ordinary differential equations (ODEs) are an emerging class of deep learning models for dynamical systems. They are particularly useful for learning an ODE vector field from observed trajectories (i.e., inverse problems). We here consider aspects of these models relevant for their application in science and engineering. Scientific predictions generally require structured uncertainty estimates. As a first contribution, we show that basic and lightweight Bayesian deep learning techniques like the Laplace approximation can be applied to neural ODEs to yield structured and meaningful uncertainty quantification. But, in the scientific domain, available information often goes beyond raw trajectories, and also includes mechanistic knowledge, e.g., in the form of conservation laws. We explore how mechanistic knowledge and uncertainty quantification interact on two recently proposed neural ODE frameworks - symplectic neural ODEs and physical models augmented with neural ODEs. In particular, uncertainty reflects the effect of mechanistic information more directly than the predictive power of the trained model could. And vice versa, structure can improve the extrapolation abilities of neural ODEs, a fact that can be best assessed in practice through uncertainty estimates. Our experimental analysis demonstrates the effectiveness of the Laplace approach on both low dimensional ODE problems and a high dimensional partial differential equation.
The geometric optimization of crystal structures is a procedure widely used in Chemistry that changes the geometrical placement of the particles inside a structure. It is called structural relaxation and constitutes a local minimization problem with a non-convex objective function whose domain complexity increases according to the number of particles involved. In this work we study the performance of the two most popular first order optimization methods in structural relaxation. Although frequently employed, there is a lack of their study in this context from an algorithmic point of view. We run each algorithm in combination with a constant step size, which provides a benchmark for the methods' analysis and direct comparison. We also design dynamic step size rules and study how these improve the two algorithms' performance. Our results show that there is a trade-off between convergence rate and the possibility of an experiment to succeed, hence we construct a function to assign utility to each method based on our respective preference. The function is built according to a recently introduced model of preference indication concerning algorithms with deadline and their run time. Finally, building on all our insights from the experimental results, we provide algorithmic recipes that best correspond to each of the presented preferences and select one recipe as the optimal for equally weighted preferences. Alongside our results we present our open source Python software veltiCRYS, which was used to perform the geometric optimization experiments. Our implementation, can be easily edited to accommodate other energy functions and is especially targeted for testing different methods in structural relaxation.
The numerical solution of the generalized eigenvalue problem for a singular matrix pencil is challenging due to the discontinuity of its eigenvalues. Classically, such problems are addressed by first extracting the regular part through the staircase form and then applying a standard solver, such as the QZ algorithm. Recently, several novel approaches have been proposed to transform the singular pencil into a regular pencil by relatively simple randomized modifications. In this work, we analyze three such methods by Hochstenbach, Mehl, and Plestenjak that modify, project, or augment the pencil using random matrices. All three methods rely on the normal rank and do not alter the finite eigenvalues of the original pencil. In this work we analyze these methods and show that the eigenvalue condition numbers of the transformed pencils are unlikely to be much larger than the $\delta$-weak eigenvalue condition numbers, introduced by Lotz and Noferini, of the original pencil. This not only indicates favorable numerical stability but also shows that these condition numbers are a reliable criterion for detecting finite eigenvalues. We also provide evidence that, from a numerical stability perspective, the use of complex instead of real random matrices is preferable even for real singular matrix pencils and real eigenvalues.
Despite recent progress in text-to-SQL parsing, current semantic parsers are still not accurate enough for practical use. In this paper, we investigate how to build automatic text-to-SQL error correction models. Noticing that token-level edits are out of context and sometimes ambiguous, we propose building clause-level edit models instead. Besides, while most language models of code are not specifically pre-trained for SQL, they know common data structures and their operations in programming languages such as Python. Thus, we propose a novel representation for SQL queries and their edits that adheres more closely to the pre-training corpora of language models of code. Our error correction model improves the exact set match accuracy of different parsers by 2.4-6.5 and obtains up to 4.3 point absolute improvement over two strong baselines. Our code and data are available at //github.com/OSU-NLP-Group/Auto-SQL-Correction.
Physics-informed neural networks (PINNs) [4, 10] are an approach for solving boundary value problems based on differential equations (PDEs). The key idea of PINNs is to use a neural network to approximate the solution to the PDE and to incorporate the residual of the PDE as well as boundary conditions into its loss function when training it. This provides a simple and mesh-free approach for solving problems relating to PDEs. However, a key limitation of PINNs is their lack of accuracy and efficiency when solving problems with larger domains and more complex, multi-scale solutions. In a more recent approach, finite basis physics-informed neural networks (FBPINNs) [8] use ideas from domain decomposition to accelerate the learning process of PINNs and improve their accuracy. In this work, we show how Schwarz-like additive, multiplicative, and hybrid iteration methods for training FBPINNs can be developed. We present numerical experiments on the influence of these different training strategies on convergence and accuracy. Furthermore, we propose and evaluate a preliminary implementation of coarse space correction for FBPINNs.
Deep Neural Networks are increasingly adopted in critical tasks that require a high level of safety, e.g., autonomous driving. While state-of-the-art verifiers can be employed to check whether a DNN is unsafe w.r.t. some given property (i.e., whether there is at least one unsafe input configuration), their yes/no output is not informative enough for other purposes, such as shielding, model selection, or training improvements. In this paper, we introduce the #DNN-Verification problem, which involves counting the number of input configurations of a DNN that result in a violation of a particular safety property. We analyze the complexity of this problem and propose a novel approach that returns the exact count of violations. Due to the #P-completeness of the problem, we also propose a randomized, approximate method that provides a provable probabilistic bound of the correct count while significantly reducing computational requirements. We present experimental results on a set of safety-critical benchmarks that demonstrate the effectiveness of our approximate method and evaluate the tightness of the bound.
Nowadays many real-world datasets can be considered as functional, in the sense that the processes which generate them are continuous. A fundamental property of this type of data is that in theory they belong to an infinite-dimensional space. Although in practice we usually receive finite observations, they are still high-dimensional and hence dimensionality reduction methods are crucial. In this vein, the main state-of-the-art method for functional data analysis is Functional PCA. Nevertheless, this classic technique assumes that the data lie in a linear manifold, and hence it could have problems when this hypothesis is not fulfilled. In this research, attention has been placed on a non-linear manifold learning method: Diffusion Maps. The article explains how to extend this multivariate method to functional data and compares its behavior against Functional PCA over different simulated and real examples.
The innovations in reactive synthesis from {\em Linear Temporal Logics over finite traces} (LTLf) will be amplified by the ability to verify the correctness of the strategies generated by LTLf synthesis tools. This motivates our work on {\em LTLf model checking}. LTLf model checking, however, is not straightforward. The strategies generated by LTLf synthesis may be represented using {\em terminating} transducers or {\em non-terminating} transducers where executions are of finite-but-unbounded length or infinite length, respectively. For synthesis, there is no evidence that one type of transducer is better than the other since they both demonstrate the same complexity and similar algorithms. In this work, we show that for model checking, the two types of transducers are fundamentally different. Our central result is that LTLf model checking of non-terminating transducers is \emph{exponentially harder} than that of terminating transducers. We show that the problems are EXPSPACE-complete and PSPACE-complete, respectively. Hence, considering the feasibility of verification, LTLf synthesis tools should synthesize terminating transducers. This is, to the best of our knowledge, the \emph{first} evidence to use one transducer over the other in LTLf synthesis.
The Kaczmarz method is a popular iterative method for solving consistent, overdetermined linear system such as medical imaging in computerized tomography. The Kaczmarz's iteration repeatedly scans all equations in order, which leads to lower computational efficiency especially in solving a large scale problem. The standard form of Kaczmarz-Tanabe's iteration proposed recently effectively overcomes the computational redundancy problem of the Kaczmarz method. In this paper, we introduce relaxation parameters ${\bf u}=(\mu_1,\ldots,\mu_m)$ into the Kaczmarz-Tanabe method based on the relaxation Kaczmarz method, and consider the standard form and convergence of this combination. Moreover, we analyze and prove the sufficient conditions for convergence of the relaxation Kaczmarz-Tanabe method, i.e., $\mu_i\in (0,2)$. Numerical experiments show the convergence characteristics of the relaxation Kaczmarz-Tanabe method corresponding to these parameters.
In recent years, researchers have proposed a number of automated tools to identify and improve floating-point rounding error in mathematical expressions. However, users struggle to effectively apply these tools. In this paper, we describe an iterative design process, working with novices, experts, and tool developers, to investigate user needs during the expression rewriting process. We find that users want to compare expressions on multiple input ranges, integrate and guide various rewriting tools, and understand where errors come from. We organize this investigation's results into a three-stage workflow and implement that workflow in a new, extensible workbench dubbed Odyssey. Odyssey enables users to: (1) diagnose problems in an expression, (2) generate solutions automatically or by hand, and (3) tune their results. Odyssey tracks a working set of expressions and turns a state-of-the-art automated tool ``inside out,'' giving the user access to internal heuristics, algorithms, and functionality. In a user study, Odyssey enabled five expert numerical analysts to solve challenging rewriting problems where state-of-the-art automated tools fail. In particular, the experts unanimously praised Odyssey's novel support for interactive range modification and local error visualization.
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