In this article we apply reduced order techniques for the approximation of parametric eigenvalue problems. The effect of the choice of sampling points is investigated. Here we use the standard proper orthogonal decomposition technique to obtain the basis of the reduced space and Galerking orthogonal technique is used to get the reduced problem. We present some numerical results and observe that the use of sparse sampling is a good idea for sampling if the dimension of parameter space is high.
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We propose an efficient, accurate and robust implicit solver for the incompressible Navier-Stokes equations, based on a DG spatial discretization and on the TR-BDF2 method for time discretization. The effectiveness of the method is demonstrated in a number of classical benchmarks, which highlight its superior efficiency with respect to other widely used implicit approaches. The parallel implementation of the proposed method in the framework of the deal.II software package allows for accurate and efficient adaptive simulations in complex geometries, which makes the proposed solver attractive for large scale industrial applications.
Answer Set Programming with Quantifiers ASP(Q) extends Answer Set Programming (ASP) to allow for declarative and modular modeling of problems from the entire polynomial hierarchy. The first implementation of ASP(Q), called qasp, was based on a translation to Quantified Boolean Formulae (QBF) with the aim of exploiting the well-developed and mature QBF-solving technology. However, the implementation of the QBF encoding employed in qasp is very general and might produce formulas that are hard to evaluate for existing QBF solvers because of the large number of symbols and sub-clauses. In this paper, we present a new implementation that builds on the ideas of qasp and features both a more efficient encoding procedure and new optimized encodings of ASP(Q) programs in QBF. The new encodings produce smaller formulas (in terms of the number of quantifiers, variables, and clauses) and result in a more efficient evaluation process. An algorithm selection strategy automatically combines several QBF-solving back-ends to further increase performance. An experimental analysis, conducted on known benchmarks, shows that the new system outperforms qasp.
We consider a general difference-in-differences model in which the treatment variable of interest may be non-binary and its value may change in each time period. It is generally difficult to estimate treatment parameters defined with the potential outcome given the entire path of treatment adoption, because each treatment path may be experienced by only a small number of observations. We propose an alternative approach using the concept of effective treatment, which summarizes the treatment path into an empirically tractable low-dimensional variable, and develop doubly robust identification, estimation, and inference methods. We also provide a companion R software package.
We discuss the approximation of the eigensolutions associated with the Maxwell eigenvalues problem in the framework of least-squares finite elements. We write the Maxwell curl curl equation as a system of two first order equation and design a novel least-squares formulation whose minimum is attained at the solution of the system. The eigensolution are then approximated by considering the eigenmodes of the underlying solution operator. We study the convergence of the finite element approximation and we show several numerical tests confirming the good behavior of the method. It turns out that nodal elements can be successfully employed for the approximation of our problem also in presence of singular solutions.
We give the first numerical calculation of the spectrum of the Laplacian acting on bundle-valued forms on a Calabi-Yau three-fold. Specifically, we show how to compute the approximate eigenvalues and eigenmodes of the Dolbeault Laplacian acting on bundle-valued $(p,q)$-forms on K\"ahler manifolds. We restrict our attention to line bundles over complex projective space and Calabi-Yau hypersurfaces therein. We give three examples. For two of these, $\mathbb{P}^3$ and a Calabi-Yau one-fold (a torus), we compare our numerics with exact results available in the literature and find complete agreement. For the third example, the Fermat quintic three-fold, there are no known analytic results, so our numerical calculations are the first of their kind. The resulting spectra pass a number of non-trivial checks that arise from Serre duality and the Hodge decomposition. The outputs of our algorithm include all the ingredients one needs to compute physical Yukawa couplings in string compactifications.
Optimal control problems driven by evolutionary partial differential equations arise in many industrial applications and their numerical solution is known to be a challenging problem. One approach to obtain an optimal feedback control is via the Dynamic Programming principle. Nevertheless, despite many theoretical results, this method has been applied only to very special cases since it suffers from the curse of dimensionality. Our goalis to mitigate this crucial obstruction developing a new version of dynamic programming algorithms based on a tree structure and exploiting the compact representation of the dynamical systems based on tensors notations via a model reduction approach. Here, we want to show how this algorithm can be constructed for general nonlinear control problems and to illustrate its performances on a number of challenging numerical tests. Our numerical results indicate a large decrease in memory requirements, as well as computational time, for the proposed problems. Moreover, we prove the convergence of the algorithm and give some hints on its implementation
The present work proposes a framework for nonlinear model order reduction based on a Graph Convolutional Autoencoder (GCA-ROM). In the reduced order modeling (ROM) context, one is interested in obtaining real-time and many-query evaluations of parametric Partial Differential Equations (PDEs). Linear techniques such as Proper Orthogonal Decomposition (POD) and Greedy algorithms have been analyzed thoroughly, but they are more suitable when dealing with linear and affine models showing a fast decay of the Kolmogorov n-width. On one hand, the autoencoder architecture represents a nonlinear generalization of the POD compression procedure, allowing one to encode the main information in a latent set of variables while extracting their main features. On the other hand, Graph Neural Networks (GNNs) constitute a natural framework for studying PDE solutions defined on unstructured meshes. Here, we develop a non-intrusive and data-driven nonlinear reduction approach, exploiting GNNs to encode the reduced manifold and enable fast evaluations of parametrized PDEs. We show the capabilities of the methodology for several models: linear/nonlinear and scalar/vector problems with fast/slow decay in the physically and geometrically parametrized setting. The main properties of our approach consist of (i) high generalizability in the low-data regime even for complex regimes, (ii) physical compliance with general unstructured grids, and (iii) exploitation of pooling and un-pooling operations to learn from scattered data.
Importance sampling (IS) is a popular technique in off-policy evaluation, which re-weights the return of trajectories in the replay buffer to boost sample efficiency. However, training with IS can be unstable and previous attempts to address this issue mainly focus on analyzing the variance of IS. In this paper, we reveal that the instability is also related to a new notion of Reuse Bias of IS -- the bias in off-policy evaluation caused by the reuse of the replay buffer for evaluation and optimization. We theoretically show that the off-policy evaluation and optimization of the current policy with the data from the replay buffer result in an overestimation of the objective, which may cause an erroneous gradient update and degenerate the performance. We further provide a high-probability upper bound of the Reuse Bias, and show that controlling one term of the upper bound can control the Reuse Bias by introducing the concept of stability for off-policy algorithms. Based on these analyses, we finally present a novel Bias-Regularized Importance Sampling (BIRIS) framework along with practical algorithms, which can alleviate the negative impact of the Reuse Bias. Experimental results show that our BIRIS-based methods can significantly improve the sample efficiency on a series of continuous control tasks in MuJoCo.
We derive the exact asymptotic distribution of the maximum likelihood estimator $(\hat{\alpha}_n, \hat{\theta}_n)$ of $(\alpha, \theta)$ for the Ewens--Pitman partition in the regime of $0<\alpha<1$ and $\theta>-\alpha$: we show that $\hat{\alpha}_n$ is $n^{\alpha/2}$-consistent and converges to a variance mixture of normal distributions, i.e., $\hat{\alpha}_n$ is asymptotically mixed normal, while $\hat{\theta}_n$ is not consistent and converges to a transformation of the generalized Mittag-Leffler distribution. As an application, we derive a confidence interval of $\alpha$ and propose a hypothesis testing of sparsity for network data. In our proof, we define an empirical measure induced by the Ewens--Pitman partition and prove a suitable convergence of the measure in some test functions, aiming to derive asymptotic behavior of the log likelihood.
Standard perturbation theory of eigenvalue problems consists of obtaining approximations of eigenmodes in the neighborhood of a Hamiltonian where the corresponding eigenmode is known. Nevertheless, if the corresponding eigenmodes of several nearby Hamiltonians are known, standard perturbation theory cannot simultaneously use all this knowledge to provide a better approximation. We derive a formula enabling such an approximation result, and provide numerical examples for which this method is more competitive than standard perturbation theory.
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