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Abstract: Abstract We consider the Navier–Stokes–Fourier system governing the motion of a general compressible, heat conducting, Newtonian fluid driven by random initial/boundary data. Convergence of the stochastic collocation and Monte Carlo numerical methods is shown under the hypothesis that approximate solutions are bounded in probability. Abstract results are illustrated by numerical experiments for the Rayleigh–Bénard convection problem. PubDate: 2024-08-06 DOI: 10.1007/s10208-024-09666-7
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Abstract: Abstract We describe fast algorithms for approximating the connection coefficients between a family of orthogonal polynomials and another family with a polynomially or rationally modified measure. The connection coefficients are computed via infinite-dimensional banded matrix factorizations and may be used to compute the modified Jacobi matrices all in linear complexity with respect to the truncation degree. A family of orthogonal polynomials with modified classical weights is constructed that support banded differentiation matrices, enabling sparse spectral methods with modified classical orthogonal polynomials. We present several applications and numerical experiments using an open source implementation which make direct use of these results. PubDate: 2024-08-05 DOI: 10.1007/s10208-024-09671-w
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Abstract: Abstract We study the computational complexity of (deterministic or randomized) algorithms based on point samples for approximating or integrating functions that can be well approximated by neural networks. Such algorithms (most prominently stochastic gradient descent and its variants) are used extensively in the field of deep learning. One of the most important problems in this field concerns the question of whether it is possible to realize theoretically provable neural network approximation rates by such algorithms. We answer this question in the negative by proving hardness results for the problems of approximation and integration on a novel class of neural network approximation spaces. In particular, our results confirm a conjectured and empirically observed theory-to-practice gap in deep learning. We complement our hardness results by showing that error bounds of a comparable order of convergence are (at least theoretically) achievable. PubDate: 2024-08-01 DOI: 10.1007/s10208-023-09607-w
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Abstract: Abstract We present a Lohner-type algorithm for rigorous integration of systems of delay differential equations (DDEs) with multiple delays, and its application in computation of Poincaré maps, to study the dynamics of some bounded, eternal solutions. The algorithm is based on a piecewise Taylor representation of the solutions in the phase space, and it exploits the smoothing of solutions occurring in DDEs to produce enclosures of solutions of a high order. We apply the topological techniques to prove various kinds of dynamical behaviour, for example, existence of (apparently) unstable periodic orbits in Mackey–Glass equation (in the regime of parameters where chaos is numerically observed) and persistence of symbolic dynamics in a delay-perturbed chaotic ODE (the Rössler system). PubDate: 2024-08-01 DOI: 10.1007/s10208-023-09614-x
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Abstract: Abstract We consider the problem of estimating expectations with respect to a target distribution with an unknown normalising constant, and where even the un-normalised target needs to be approximated at finite resolution. This setting is ubiquitous across science and engineering applications, for example in the context of Bayesian inference where a physics-based model governed by an intractable partial differential equation (PDE) appears in the likelihood. A multi-index sequential Monte Carlo (MISMC) method is used to construct ratio estimators which provably enjoy the complexity improvements of multi-index Monte Carlo (MIMC) as well as the efficiency of sequential Monte Carlo (SMC) for inference. In particular, the proposed method provably achieves the canonical complexity of \(\hbox {MSE}^{-1}\) , while single-level methods require \(\hbox {MSE}^{-\xi }\) for \(\xi >1\) . This is illustrated on examples of Bayesian inverse problems with an elliptic PDE forward model in 1 and 2 spatial dimensions, where \(\xi =5/4\) and \(\xi =3/2\) , respectively. It is also illustrated on more challenging log-Gaussian process models, where single-level complexity is approximately \(\xi =9/4\) and multilevel Monte Carlo (or MIMC with an inappropriate index set) gives \(\xi = 5/4 + \omega \) , for any \(\omega > 0\) , whereas our method is again canonical. We also provide novel theoretical verification of the product-form convergence results which MIMC requires for Gaussian processes built in spaces of mixed regularity defined in the spectral domain, which facilitates acceleration with fast Fourier transform methods via a cumulant embedding strategy, and may be of independent interest in the context of spatial statistics and machine learning. PubDate: 2024-08-01 DOI: 10.1007/s10208-023-09612-z
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Abstract: Abstract Let A be an \(m\times n\) matrix with nonnegative real entries. The psd rank of A is the smallest k for which there exist two families \((P_1,\ldots ,P_m)\) and \((Q_1,\ldots ,Q_n)\) of positive semidefinite Hermitian \(k\times k\) matrices such that \(A(i j)={\text {tr}}(P_i Q_j)\) for all i, j. Several questions on the algorithmic complexity of related matrix invariants were posed in recent literature: (i) by Stark (for the psd rank as defined above), (ii) by Goucha, Gouveia (for phaseless rank, which appears if the matrices \(P_i\) and \(Q_j\) are required to be of rank one in the above definition), (iii) by Gribling, de Laat, Laurent (for cpsd rank, which corresponds to the situation when A is symmetric and \(P_i=Q_i\) for all i). We solve these questions by proving that the decision versions of the corresponding invariants are \(\exists {\mathbb {R}}\) -complete. In addition, we give a polynomial time recognition algorithm for matrices of bounded cpsd rank. PubDate: 2024-08-01 DOI: 10.1007/s10208-023-09610-1
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Abstract: Abstract We investigate some Bernstein–Gelfand–Gelfand complexes consisting of Sobolev spaces on bounded Lipschitz domains in \({\mathbb {R}}^{n}\) . In particular, we compute the cohomology of the conformal deformation complex and the conformal Hessian complex in the Sobolev setting. The machinery does not require algebraic injectivity/surjectivity conditions between the input spaces, and allows multiple input complexes. As applications, we establish a conformal Korn inequality in two space dimensions with the Cauchy–Riemann operator and an additional third-order operator with a background in Möbius geometry. We show that the linear Cosserat elasticity model is a Hodge–Laplacian problem of a twisted de Rham complex. From this cohomological perspective, we propose potential generalizations of continuum models with microstructures. PubDate: 2024-08-01 DOI: 10.1007/s10208-023-09608-9
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Abstract: Abstract We study the representative volume element (RVE) method, which is a method to approximately infer the effective behavior \(a_{\textrm{hom}}\) of a stationary random medium. The latter is described by a coefficient field a(x) generated from a given ensemble \(\langle \cdot \rangle \) and the corresponding linear elliptic operator \(-\nabla \cdot a\nabla \) . In line with the theory of homogenization, the method proceeds by computing \(d=3\) correctors (d denoting the space dimension). To be numerically tractable, this computation has to be done on a finite domain: the so-called representative volume element, i.e., a large box with, say, periodic boundary conditions. The main message of this article is: Periodize the ensemble instead of its realizations. By this, we mean that it is better to sample from a suitably periodized ensemble than to periodically extend the restriction of a realization a(x) from the whole-space ensemble \(\langle \cdot \rangle \) . We make this point by investigating the bias (or systematic error), i.e., the difference between \(a_{\textrm{hom}}\) and the expected value of the RVE method, in terms of its scaling w.r.t. the lateral size L of the box. In case of periodizing a(x), we heuristically argue that this error is generically \(O(L^{-1})\) . In case of a suitable periodization of \(\langle \cdot \rangle \) , we rigorously show that it is \(O(L^{-d})\) . In fact, we give a characterization of the leading-order error term for both strategies and argue that even in the isotropic case it is generically non-degenerate. We carry out the rigorous analysis in the convenient setting of ensembles \(\langle \cdot \rangle \) of Gaussian type, which allow for a straightforward periodization, passing via the (integrable) covariance function. This setting has also the advantage of making the Price theorem and the Malliavin calculus available for optimal stochastic estimates of correctors. We actually need control of second-order correctors to capture the leading-order error term. This is due to inversion symmetry when applying the two-scale expansion to the Green function. As a bonus, we present a stream-lined strategy to estimate the error in a higher-order two-scale expansion of the Green function. PubDate: 2024-08-01 DOI: 10.1007/s10208-023-09613-y
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Abstract: Abstract We study symmetries of bases and spanning sets in finite element exterior calculus, using representation theory. We want to know which vector-valued finite element spaces have bases invariant under permutation of vertex indices. The permutations of vertex indices correspond to the symmetry group of the simplex. That symmetry group is represented on simplicial finite element spaces by the pullback action. We determine a natural notion of invariance and sufficient conditions on the dimension and polynomial degree for the existence of invariant bases. We conjecture that these conditions are necessary too. We utilize Djoković and Malzan’s classification of monomial irreducible representations of the symmetric group and show new symmetries of the geometric decomposition and canonical isomorphisms of the finite element spaces. Explicit invariant bases with complex coefficients are constructed in dimensions two and three for different spaces of finite element differential forms. PubDate: 2024-08-01 DOI: 10.1007/s10208-023-09609-8
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Abstract: Abstract On the reference tetrahedron \(K\) , we construct, for each \(k \in {\mathbb {N}}_0\) , a right inverse for the trace operator \(u \mapsto (u, \partial _{\textbf{n}} u, \ldots , \partial _{\textbf{n}}^k u) _{\partial K}\) . The operator is stable as a mapping from the trace space of \(W^{s, p}(K)\) to \(W^{s, p}(K)\) for all \(p \in (1, \infty )\) and \(s \in (k+1/p, \infty )\) . Moreover, if the data is the trace of a polynomial of degree \(N \in {\mathbb {N}}_0\) , then the resulting lifting is a polynomial of degree N. One consequence of the analysis is a novel characterization for the range of the trace operator. PubDate: 2024-07-29 DOI: 10.1007/s10208-024-09670-x
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Abstract: Abstract Classically, the continuous-time Langevin diffusion converges exponentially fast to its stationary distribution \(\pi \) under the sole assumption that \(\pi \) satisfies a Poincaré inequality. Using this fact to provide guarantees for the discrete-time Langevin Monte Carlo (LMC) algorithm, however, is considerably more challenging due to the need for working with chi-squared or Rényi divergences, and prior works have largely focused on strongly log-concave targets. In this work, we provide the first convergence guarantees for LMC assuming that \(\pi \) satisfies either a Latała–Oleszkiewicz or modified log-Sobolev inequality, which interpolates between the Poincaré and log-Sobolev settings. Unlike prior works, our results allow for weak smoothness and do not require convexity or dissipativity conditions. PubDate: 2024-07-26 DOI: 10.1007/s10208-024-09667-6
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Abstract: Abstract A large toolbox of numerical schemes for dispersive equations has been established, based on different discretization techniques such as discretizing the variation-of-constants formula (e.g., exponential integrators) or splitting the full equation into a series of simpler subproblems (e.g., splitting methods). In many situations these classical schemes allow a precise and efficient approximation. This, however, drastically changes whenever non-smooth phenomena enter the scene such as for problems at low regularity and high oscillations. Classical schemes fail to capture the oscillatory nature of the solution, and this may lead to severe instabilities and loss of convergence. In this article we review a new class of resonance-based schemes. The key idea in the construction of the new schemes is to tackle and deeply embed the underlying nonlinear structure of resonances into the numerical discretization. As in the continuous case, these terms are central to structure preservation and offer the new schemes strong properties at low regularity. PubDate: 2024-07-26 DOI: 10.1007/s10208-024-09665-8
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Abstract: Abstract Motivated by the problem of matching vertices in two correlated Erdős-Rényi graphs, we study the problem of matching two correlated Gaussian Wigner matrices. We propose an iterative matching algorithm, which succeeds in polynomial time as long as the correlation between the two Gaussian matrices does not vanish. Our result is the first polynomial time algorithm that solves a graph matching type of problem when the correlation is an arbitrarily small constant. PubDate: 2024-07-22 DOI: 10.1007/s10208-024-09662-x
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Abstract: Abstract We study the quantitative stability of the mapping that to a measure associates its pushforward measure by a fixed (non-smooth) optimal transport map. We exhibit a tight Hölder-behavior for this operation under minimal assumptions. Our proof essentially relies on a new bound that quantifies the size of the singular sets of a convex and Lipschitz continuous function on a bounded domain. PubDate: 2024-07-19 DOI: 10.1007/s10208-024-09669-4
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Abstract: Abstract We develop algebraic geometry for coupled cluster (CC) theory of quantum many-body systems. The high-dimensional eigenvalue problems that encode the electronic Schrödinger equation are approximated by a hierarchy of polynomial systems at various levels of truncation. The exponential parametrization of the eigenstates gives rise to truncation varieties. These generalize Grassmannians in their Plücker embedding. We explain how to derive Hamiltonians, we offer a detailed study of truncation varieties and their CC degrees, and we present the state of the art in solving the CC equations. PubDate: 2024-07-17 DOI: 10.1007/s10208-024-09657-8
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Abstract: Abstract The stein variational gradient descent (SVGD) algorithm is a deterministic particle method for sampling. However, a mean-field analysis reveals that the gradient flow corresponding to the SVGD algorithm (i.e., the Stein Variational Gradient Flow) only provides a constant-order approximation to the Wasserstein gradient flow corresponding to the KL-divergence minimization. In this work, we propose the Regularized Stein Variational Gradient Flow, which interpolates between the Stein Variational Gradient Flow and the Wasserstein gradient flow. We establish various theoretical properties of the Regularized Stein Variational Gradient Flow (and its time-discretization) including convergence to equilibrium, existence and uniqueness of weak solutions, and stability of the solutions. We provide preliminary numerical evidence of the improved performance offered by the regularization. PubDate: 2024-07-17 DOI: 10.1007/s10208-024-09663-w
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Abstract: Abstract Under certain conditions, Koszul complexes can be used to calculate relative Betti diagrams of vector space-valued functors indexed by a poset, without the explicit computation of global minimal relative resolutions. In relative homological algebra of such functors, free functors are replaced by an arbitrary family of functors. Relative Betti diagrams encode the multiplicities of these functors in minimal relative resolutions. In this article we provide conditions under which grading the chosen family of functors leads to explicit Koszul complexes whose homology dimensions are the relative Betti diagrams, thus giving a scheme for the computation of these numerical descriptors. PubDate: 2024-06-18 DOI: 10.1007/s10208-024-09660-z
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Abstract: Abstract Classical results show that gradient descent converges linearly to minimizers of smooth strongly convex functions. A natural question is whether there exists a locally nearly linearly convergent method for nonsmooth functions with quadratic growth. This work designs such a method for a wide class of nonsmooth and nonconvex locally Lipschitz functions, including max-of-smooth, Shapiro’s decomposable class, and generic semialgebraic functions. The algorithm is parameter-free and derives from Goldstein’s conceptual subgradient method. PubDate: 2024-06-14 DOI: 10.1007/s10208-024-09653-y
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Abstract: Abstract Regularization is necessary when solving inverse problems to ensure the well-posedness of the solution map. Additionally, it is desired that the chosen regularization strategy is convergent in the sense that the solution map converges to a solution of the noise-free operator equation. This provides an important guarantee that stable solutions can be computed for all noise levels and that solutions satisfy the operator equation in the limit of vanishing noise. In recent years, reconstructions in inverse problems are increasingly approached from a data-driven perspective. Despite empirical success, the majority of data-driven approaches do not provide a convergent regularization strategy. One such popular example is given by iterative plug-and-play (PnP) denoising using off-the-shelf image denoisers. These usually provide only convergence of the PnP iterates to a fixed point, under suitable regularity assumptions on the denoiser, rather than convergence of the method as a regularization technique, thatis under vanishing noise and regularization strength. This paper serves two purposes: first, we provide an overview of the classical regularization theory in inverse problems and survey a few notable recent data-driven methods that are provably convergent regularization schemes. We then continue to discuss PnP algorithms and their established convergence guarantees. Subsequently, we consider PnP algorithms with learned linear denoisers and propose a novel spectral filtering technique of the denoiser to control the strength of regularization. Further, by relating the implicit regularization of the denoiser to an explicit regularization functional, we are the first to rigorously show that PnP with a learned linear denoiser leads to a convergent regularization scheme. The theoretical analysis is corroborated by numerical experiments for the classical inverse problem of tomographic image reconstruction. PubDate: 2024-06-03 DOI: 10.1007/s10208-024-09654-x
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