Optimal low-rank approximations of posteriors for linear Gaussian inverse problems on Hilbert spaces
For linear inverse problems with Gaussian priors and observation noise, the posterior is Gaussian, with mean and covariance determined by the conditioning formula. We analyse measure approximation problems of finding the best approximation to the posterior in a family of Gaussians with approximate covariance or approximate mean, for Hilbert parameter spaces and finite-dimensional observations. We quantify the error of the approximating Gaussian either with the Kullback-Leibler divergence or the family of R\'{e}nyi divergences. Using the Feldman-Hajek theorem and recent results on reduced-rank operator approximations, we identify optimal solutions to these measure approximation problems. Our results extend those of Spantini et al. (SIAM J. Sci. Comput. 2015) to Hilbertian parameter spaces. In addition, our results show that the posterior differs from the prior only on a subspace of dimension equal to the rank of the Hessian of the negative log-likelihood, and that this subspace is a subspace of the Cameron-Martin space of the prior.
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We consider the problem of enumerating all minimal transversals (also called minimal hitting sets) of a hypergraph $\mathcal{H}$. An equivalent formulation of this problem known as the \emph{transversal hypergraph} problem (or \emph{hypergraph dualization} problem) is to decide, given two hypergraphs, whether one corresponds to the set of minimal transversals of the other. The existence of a polynomial time algorithm to solve this problem is a long standing open question. In \cite{fredman_complexity_1996}, the authors present the first sub-exponential algorithm to solve the transversal hypergraph problem which runs in quasi-polynomial time, making it unlikely that the problem is (co)NP-complete. In this paper, we show that when one of the two hypergraphs is of bounded VC-dimension, the transversal hypergraph problem can be solved in polynomial time, or equivalently that if $\mathcal{H}$ is a hypergraph of bounded VC-dimension, then there exists an incremental polynomial time algorithm to enumerate its minimal transversals. This result generalizes most of the previously known polynomial cases in the literature since they almost all consider classes of hypergraphs of bounded VC-dimension. As a consequence, the hypergraph transversal problem is solvable in polynomial time for any class of hypergraphs closed under partial subhypergraphs. We also show that the proposed algorithm runs in quasi-polynomial time in general hypergraphs and runs in polynomial time if the conformality of the hypergraph is bounded, which is one of the few known polynomial cases where the VC-dimension is unbounded.
Motivated by the need for the rigorous analysis of the numerical stability of variational least-squares kernel-based methods for solving second-order elliptic partial differential equations, we provide previously lacking stability inequalities. This fills a significant theoretical gap in the previous work [Comput. Math. Appl. 103 (2021) 1-11], which provided error estimates based on a conjecture on the stability. With the stability estimate now rigorously proven, we complete the theoretical foundations and compare the convergence behavior to the proven rates. Furthermore, we establish another stability inequality involving weighted-discrete norms, and provide a theoretical proof demonstrating that the exact quadrature weights are not necessary for the weighted least-squares kernel-based collocation method to converge. Our novel theoretical insights are validated by numerical examples, which showcase the relative efficiency and accuracy of these methods on data sets with large mesh ratios. The results confirm our theoretical predictions regarding the performance of variational least-squares kernel-based method, least-squares kernel-based collocation method, and our new weighted least-squares kernel-based collocation method. Most importantly, our results demonstrate that all methods converge at the same rate, validating the convergence theory of weighted least-squares in our proven theories.
We derive a priori and a posteriori error estimates for the discontinuous Galerkin (dG) approximation of the time-harmonic Maxwell's equations. Specifically, we consider an interior penalty dG method, and establish error estimates that are valid under minimal regularity assumptions and involving constants that do not depend on the frequency for sufficiently fine meshes. The key result of our a priori error analysis is that the dG solution is asymptotically optimal in an augmented energy norm that contains the dG stabilization. Specifically, up to a constant that tends to one as the mesh is refined, the dG solution is as accurate as the best approximation in the same norm. The main insight is that the quantities controlling the smallness of the mesh size are essentially those already appearing in the conforming setting. We also show that for fine meshes, the inf-sup stability constant is as good as the continuous one up to a factor two. Concerning the a posteriori analysis, we consider a residual-based error estimator under the assumption of piecewise constant material properties. We derive a global upper bound and local lower bounds on the error with constants that (i) only depend on the shape-regularity of the mesh if it is sufficiently refined and (ii) are independent of the stabilization bilinear form.
Detection of abrupt spatial changes in physical properties representing unique geometric features such as buried objects, cavities, and fractures is an important problem in geophysics and many engineering disciplines. In this context, simultaneous spatial field and geometry estimation methods that explicitly parameterize the background spatial field and the geometry of the embedded anomalies are of great interest. This paper introduces an advanced inversion procedure for simultaneous estimation using the domain independence property of the Karhunen-Lo\`eve (K-L) expansion. Previous methods pursuing this strategy face significant computational challenges. The associated integral eigenvalue problem (IEVP) needs to be solved repeatedly on evolving domains, and the shape derivatives in gradient-based algorithms require costly computations of the Moore-Penrose inverse. Leveraging the domain independence property of the K-L expansion, the proposed method avoids both of these bottlenecks, and the IEVP is solved only once on a fixed bounding domain. Comparative studies demonstrate that our approach yields two orders of magnitude improvement in K-L expansion gradient computation time. Inversion studies on one-dimensional and two-dimensional seepage flow problems highlight the benefits of incorporating geometry parameters along with spatial field parameters. The proposed method captures abrupt changes in hydraulic conductivity with a lower number of parameters and provides accurate estimates of boundary and spatial-field uncertainties, outperforming spatial-field-only estimation methods.
Most of the scientific literature on causal modeling considers the structural framework of Pearl and the potential-outcome framework of Rubin to be formally equivalent, and therefore interchangeably uses do-interventions and the potential-outcome subscript notation to write counterfactual outcomes. In this paper, we agnostically superimpose the two causal models to specify under which mathematical conditions structural counterfactual outcomes and potential outcomes need to, do not need to, can, or cannot be equal (almost surely or law). Our comparison reminds that a structural causal model and a Rubin causal model compatible with the same observations do not have to coincide, and highlights real-world problems where they even cannot correspond. Then, we examine common claims and practices from the causal-inference literature in the light of these results. In doing so, we aim at clarifying the relationship between the two causal frameworks, and the interpretation of their respective counterfactuals.
A new variant of the GMRES method is presented for solving linear systems with the same matrix and subsequently obtained multiple right-hand sides. The new method keeps such properties of the classical GMRES algorithm as follows. Both bases of the search space and its image are maintained orthonormal that increases the robustness of the method. Moreover there is no need to store both bases since they are effectively represented within a common basis. Along with it our method is theoretically equivalent to the GCR method extended for a case of multiple right-hand sides but is more numerically robust and requires less memory. The main result of the paper is a mechanism of adding an arbitrary direction vector to the search space that can be easily adopted for flexible GMRES or GMRES with deflated restarting.
Regularization is a critical technique for ensuring well-posedness in solving inverse problems with incomplete measurement data. Traditionally, the regularization term is designed based on prior knowledge of the unknown signal's characteristics, such as sparsity or smoothness. Inhomogeneous regularization, which incorporates a spatially varying exponent $p$ in the standard $\ell_p$-norm-based framework, has been used to recover signals with spatially varying features. This study introduces weighted inhomogeneous regularization, an extension of the standard approach incorporating a novel exponent design and spatially varying weights. The proposed exponent design mitigates misclassification when distinct characteristics are spatially close, while the weights address challenges in recovering regions with small-scale features that are inadequately captured by traditional $\ell_p$-norm regularization. Numerical experiments, including synthetic image reconstruction and the recovery of sea ice data from incomplete wave measurements, demonstrate the effectiveness of the proposed method.
We consider a fully discretized numerical scheme for parabolic stochastic partial differential equations with multiplicative noise. Our abstract framework can be applied to formulate a non-iterative domain decomposition approach. Such methods can help to parallelize the code and therefore lead to a more efficient implementation. The domain decomposition is integrated through the Douglas-Rachford splitting scheme, where one split operator acts on one part of the domain. For an efficient space discretization of the underlying equation, we chose the discontinuous Galerkin method as this suits the parallelization strategy well. For this fully discretized scheme, we provide a strong space-time convergence result. We conclude the manuscript with numerical experiments validating our theoretical findings.
The dynamics of magnetization in ferromagnetic materials are modeled by the Landau-Lifshitz equation, which presents significant challenges due to its inherent nonlinearity and non-convex constraint. These complexities necessitate efficient numerical methods for micromagnetics simulations. The Gauss-Seidel Projection Method (GSPM), first introduced in 2001, is among the most efficient techniques currently available. However, existing GSPMs are limited to first-order accuracy. This paper introduces two novel second-order accurate GSPMs based on a combination of the biharmonic equation and the second-order backward differentiation formula, achieving computational complexity comparable to that of solving the scalar biharmonic equation implicitly. The first proposed method achieves unconditional stability through Gauss-Seidel updates, while the second method exhibits conditional stability with a Courant-Friedrichs-Lewy constant of 0.25. Through consistency analysis and numerical experiments, we demonstrate the efficacy and reliability of these methods. Notably, the first method displays unconditional stability in micromagnetics simulations, even when the stray field is updated only once per time step.
Finite element discretization of Stokes problems can result in singular, inconsistent saddle point linear algebraic systems. This inconsistency can cause many iterative methods to fail to converge. In this work, we consider the lowest-order weak Galerkin finite element method to discretize Stokes flow problems and study a consistency enforcement by modifying the right-hand side of the resulting linear system. It is shown that the modification of the scheme does not affect the optimal-order convergence of the numerical solution. Moreover, inexact block diagonal and triangular Schur complement preconditioners and the minimal residual method (MINRES) and the generalized minimal residual method (GMRES) are studied for the iterative solution of the modified scheme. Bounds for the eigenvalues and the residual of MINRES/GMRES are established. Those bounds show that the convergence of MINRES and GMRES is independent of the viscosity parameter and mesh size. The convergence of the modified scheme and effectiveness of the preconditioners are verified using numerical examples in two and three dimensions.
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