亚洲男人的天堂2018av,欧美草比,久久久久久免费视频精选,国色天香在线看免费,久久久久亚洲av成人片仓井空

Deep Learning-based Reduced Order Models (DL-ROMs) provide nowadays a well-established class of accurate surrogate models for complex physical systems described by parametrized PDEs, by nonlinearly compressing the solution manifold into a handful of latent coordinates. Until now, design and application of DL-ROMs mainly focused on physically parameterized problems. Within this work, we provide a novel extension of these architectures to problems featuring geometrical variability and parametrized domains, namely, we propose Continuous Geometry-Aware DL-ROMs (CGA-DL-ROMs). In particular, the space-continuous nature of the proposed architecture matches the need to deal with multi-resolution datasets, which are quite common in the case of geometrically parametrized problems. Moreover, CGA-DL-ROMs are endowed with a strong inductive bias that makes them aware of geometrical parametrizations, thus enhancing both the compression capability and the overall performance of the architecture. Within this work, we justify our findings through a thorough theoretical analysis, and we practically validate our claims by means of a series of numerical tests encompassing physically-and-geometrically parametrized PDEs, ranging from the unsteady Navier-Stokes equations for fluid dynamics to advection-diffusion-reaction equations for mathematical biology.

We consider quantum circuit models where the gates are drawn from arbitrary gate ensembles given by probabilistic distributions over certain gate sets and circuit architectures, which we call stochastic quantum circuits. Of main interest in this work is the speed of convergence of stochastic circuits with different gate ensembles and circuit architectures to unitary t-designs. A key motivation for this theory is the varying preference for different gates and circuit architectures in different practical scenarios. In particular, it provides a versatile framework for devising efficient circuits for implementing $t$-designs and relevant applications including random circuit and scrambling experiments, as well as benchmarking the performance of gates and circuit architectures. We examine various important settings in depth. A key aspect of our study is an "ironed gadget" model, which allows us to systematically evaluate and compare the convergence efficiency of entangling gates and circuit architectures. Particularly notable results include i) gadgets of two-qubit gates with KAK coefficients $\left(\frac{\pi}{4}-\frac{1}{8}\arccos(\frac{1}{5}),\frac{\pi}{8},\frac{1}{8}\arccos(\frac{1}{5})\right)$ (which we call $\chi$ gates) directly form exact 2- and 3-designs; ii) the iSWAP gate family achieves the best efficiency for convergence to 2-designs under mild conjectures with numerical evidence, even outperforming the Haar-random gate, for generic many-body circuits; iii) iSWAP + complete graph achieve the best efficiency for convergence to 2-designs among all graph circuits. A variety of numerical results are provided to complement our analysis. We also derive robustness guarantees for our analysis against gate perturbations. Additionally, we provide cursory analysis on gates with higher locality and found that the Margolus gate outperforms various other well-known gates.

After nearly two decades of research, the question of a quantum PCP theorem for quantum Constraint Satisfaction Problems (CSPs) remains wide open. As a result, proving QMA-hardness of approximation for ground state energy estimation has remained elusive. Recently, it was shown [Bittel, Gharibian, Kliesch, CCC 2023] that a natural problem involving variational quantum circuits is QCMA-hard to approximate within ratio N^(1-eps) for any eps > 0 and N the input size. Unfortunately, this problem was not related to quantum CSPs, leaving the question of hardness of approximation for quantum CSPs open. In this work, we show that if instead of focusing on ground state energies, one considers computing properties of the ground space, QCMA-hardness of computing ground space properties can be shown. In particular, we show that it is (1) QCMA-complete within ratio N^(1-eps) to approximate the Ground State Connectivity problem (GSCON), and (2) QCMA-hard within the same ratio to estimate the amount of entanglement of a local Hamiltonian's ground state, denoted Ground State Entanglement (GSE). As a bonus, a simplification of our construction yields NP-completeness of approximation for a natural k-SAT reconfiguration problem, to be contrasted with the recent PCP-based PSPACE hardness of approximation results for a different definition of k-SAT reconfiguration [Karthik C.S. and Manurangsi, 2023, and Hirahara, Ohsaka, STOC 2024].

We provide abstract, general and highly uniform rates of asymptotic regularity for a generalized stochastic Halpern-style iteration, which incorporates a second mapping in the style of a Krasnoselskii-Mann iteration. This iteration is general in two ways: First, it incorporates stochasticity in a completely abstract way rather than fixing a sampling method; secondly, it includes as special cases stochastic versions of various schemes from the optimization literature, including Halpern's iteration as well as a Krasnoselskii-Mann iteration with Tikhonov regularization terms in the sense of Bo\c{t}, Csetnek and Meier. For these particular cases, we in particular obtain linear rates of asymptotic regularity, matching (or improving) the currently best known rates for these iterations in stochastic optimization, and quadratic rates of asymptotic regularity are obtained in the context of inner product spaces for the general iteration. We utilize these rates to give bounds on the oracle complexity of such iterations under suitable variance assumptions and batching strategies, again presented in an abstract style. Finally, we sketch how the schemes presented here can be instantiated in the context of reinforcement learning to yield novel methods for Q-learning.

A Riemannian geometric framework for Markov chain Monte Carlo (MCMC) is developed where using the Fisher-Rao metric on the manifold of probability density functions (pdfs), informed proposal densities for Metropolis-Hastings (MH) algorithms are constructed. We exploit the square-root representation of pdfs under which the Fisher-Rao metric boils down to the standard $L^2$ metric on the positive orthant of the unit hypersphere. The square-root representation allows us to easily compute the geodesic distance between densities, resulting in a straightforward implementation of the proposed geometric MCMC methodology. Unlike the random walk MH that blindly proposes a candidate state using no information about the target, the geometric MH algorithms move an uninformed base density (e.g., a random walk proposal density) towards different global/local approximations of the target density, allowing effective exploration of the distribution simultaneously at different granular levels of the state space. We compare the proposed geometric MH algorithm with other MCMC algorithms for various Markov chain orderings, namely the covariance, efficiency, Peskun, and spectral gap orderings. The superior performance of the geometric algorithms over other MH algorithms like the random walk Metropolis, independent MH, and variants of Metropolis adjusted Langevin algorithms is demonstrated in the context of various multimodal, nonlinear, and high dimensional examples. In particular, we use extensive simulation and real data applications to compare these algorithms for analyzing mixture models, logistic regression models, spatial generalized linear mixed models and ultra-high dimensional Bayesian variable selection models. A publicly available R package accompanies the article.

In this work is considered an elliptic problem, referred to as the Ventcel problem, involvinga second order term on the domain boundary (the Laplace-Beltrami operator). A variationalformulation of the Ventcel problem is studied, leading to a finite element discretization. Thefocus is on the construction of high order curved meshes for the discretization of the physicaldomain and on the definition of the lift operator, which is aimed to transform a functiondefined on the mesh domain into a function defined on the physical one. This lift is definedin a way as to satisfy adapted properties on the boundary, relatively to the trace operator.The Ventcel problem approximation is investigated both in terms of geometrical error and offinite element approximation error. Error estimates are obtained both in terms of the meshorder r $\ge$ 1 and to the finite element degree k $\ge$ 1, whereas such estimates usually have beenconsidered in the isoparametric case so far, involving a single parameter k = r. The numericalexperiments we led, both in dimension 2 and 3, allow us to validate the results obtained andproved on the a priori error estimates depending on the two parameters k and r. A numericalcomparison is made between the errors using the former lift definition and the lift defined inthis work establishing an improvement in the convergence rate of the error in the latter case.

Nowadays, we have seen that dual quaternion algorithms are used in 3D coordinate transformation problems due to their advantages. 3D coordinate transformation problem is one of the important problems in geodesy. This transformation problem is encountered in many application areas other than geodesy. Although there are many coordinate transformation methods (similarity, affine, projective, etc.), similarity transformation is used because of its simplicity. The asymmetric transformation is preferred to the symmetric coordinate transformation because of its ease of use. In terms of error theory, the symmetric transformation should be preferred. In this study, the topic of symmetric similarity 3D coordinate transformation based on the dual quaternion algorithm is discussed, and the bottlenecks encountered in solving the problem and the solution method are discussed. A new iterative algorithm based on the dual quaternion is presented. The solution is implemented in two different models: with constraint equations and without constraint equations. The advantages and disadvantages of the two models compared to each other are also evaluated. Not only the transformation parameters but also the errors of the transformation parameters are determined. The detailed derivation of the formulas for estimating the symmetric similarity of 3D transformation parameters is presented step by step. Since symmetric transformation is the general form of asymmetric transformation, we can also obtain asymmetric transformation results with a simple modification of the model we developed for symmetric transformation. The proposed algorithm is capable of performing both 2D and 3D symmetric and asymmetric similarity transformations. For the 2D transformation, it is sufficient to replace the z and Z coordinates in both systems with zero.

The paper studies asymptotic properties of estimators of multidimensional stochastic differential equations driven by Brownian motions from high-frequency discrete data. Consistency and central limit properties of a class of estimators of the diffusion parameter and an approximate maximum likelihood estimator of the drift parameter based on a discretized likelihood function have been established in a suitable scaling regime involving the time-gap between the observations and the overall time span. Our framework is more general than that typically considered in the literature and, thus, has the potential to be applicable to a wider range of stochastic models.

The gradient bounds of generalized barycentric coordinates play an essential role in the $H^1$ norm approximation error estimate of generalized barycentric interpolations. Similarly, the $H^k$ norm, $k>1$, estimate needs upper bounds of high-order derivatives, which are not available in the literature. In this paper, we derive such upper bounds for the Wachspress generalized barycentric coordinates on simple convex $d$-dimensional polytopes, $d\ge 1$. The result can be used to prove optimal convergence for Wachspress-based polytopal finite element approximation of, for example, fourth-order elliptic equations. Another contribution of this paper is to compare various shape-regularity conditions for simple convex polytopes, and to clarify their relations using knowledge from convex geometry.

Large Language Models (LLMs) demonstrate exceptional capabilities in various scenarios. However, they suffer from much redundant information and are sensitive to the position of key information (relevant to the input question) in long context scenarios, leading to inferior performance. To address these challenges, we present Perception Compressor, a training-free prompt compression method. It includes a perception retriever that leverages guiding questions and instruction to retrieve the most relevant demonstrations, a dual-slope ratio allocator to dynamically allocate compression ratios and open-book ratios, and a semi-guided iterative compression that retains key information at the token level while removing tokens that distract the LLM. We conduct extensive experiments on long context benchmarks, i.e., NaturalQuestions, LongBench, and MuSiQue. Experiment results show that Perception Compressor outperforms existing methods by a large margin, achieving state-of-the-art performance.