Bayesian approaches to statistical inverse problems provide a powerful framework for inferring unknown parameters from noisy observations, with rigorous uncertainty quantification. The efficient sampling from the posterior distribution remains a key computational challenge. The preconditioned Metropolis adjusted Langevin algorithm (MALA) is a widely used sampler in statistical applications, however selecting the preconditioning matrix is crucial. Recently, Titsias [44] demonstrated that the inverse Fisher information matrix is the optimal preconditioner by minimizing the expected squared jump distance and proposed an adaptive scheme to estimate the Fisher matrix using the sampling history. In this paper, we establish the ergodicity and asymptotic convergence of Fisher adaptive MALA and analyze the convergence rate of the adaptive Fisher information matrix estimator. We then apply the resulting algorithm to Bayesian inverse problems. To evaluate the performance of this algorithm, we apply it to several Bayesian inverse problems and compare it with standard adaptive MALA (which uses the empirical covariance matrix of the posterior distribution as the preconditioner), the preconditioned Crank-Nicolson (pCN) algorithm and Randomize-Then-Optimize (RTO) method. Numerical experiments in moderate- and moderately high-dimensional settings demonstrate that FisherMala achieves improved sampling efficiency.
The Method of Reflections was historically introduced to obtain approximate solutions as series expansions for the motion of particles in suspension. It can however equally well be used for solving multiple scattering problems numerically. We show for Helmholtz multiple scattering problems that the Method of Reflections, whether applied in its alternating or parallel version, suffers from convergence problems when scatterers are close. We use boundary integral equations to formulate the methods, and then identify their discretization as algebraic Schwarz methods, thereby interpreting them as boundary domain decomposition techniques. This connection allows us to introduce remedies such as overlap (which can be partial, covering only the illuminating region of the obstacles) and coarse spaces from domain decomposition into the Method of Reflections. This leads to substantially accelerated variants, and also naturally makes them suitable preconditioners for GMRES. These new approaches are particularly efficient for closeby obstacles. Moreover, numerical experiments show that the number of iterations increases only very slowly with respect to the wavenumber.
This work studies the invariant measure of numerical approximation to a class of superlinear stochastic differential equations (SDEs) with periodic coefficients by the truncated Euler–Maruyama (EM) method. The existing studies on numerical invariant measures mostly focus on autonomous SDEs. To the best of our knowledge, this paper is the first devoted to the case of non-autonomous SDEs which allow superlinear growth. Technical challenges including superlinearity, periodicity and time-inhomogeneity of the system make this a non-trivial work. Not only the existence and uniqueness of a numerical invariant measure but also the convergence of the numerical invariant measure to the underlying one are deduced in the Wasserstein metric. Consequently, a case study is carried out to demonstrate the main results.
In this paper, we consider the low rank matrix feasibility problem arising in quantum marginal computing. Based on the alternating projection method and Newton’s method, we propose the Alternating-Newton algorithm to solve this problem. The convergence analysis of the Alternating-Newton algorithm is given and its quadratic convergence rate is also derived. Some numerical experiments demonstrate the feasibility and efficiency of the proposed algorithm.
In this paper, we propose three unisolvent weighted quadratic enrichments to extend classical linear histopolation on 3D tetrahedral meshes. The first combines face and interior weighted moments, the second relies exclusively on volumetric quadratic moments, and the third employs edge supported probabilistic moments. In all three cases, the degrees of freedom are defined by weighted integral functionals associated with suitable probability densities and quadratic trial spaces. We establish the unisolvence of the enriched schemes and derive explicit conditions under which this property holds. Representative density families, including two-parameter symmetric Dirichlet laws and volumetric families defined by convex combinations of densities, are examined in detail, and a general procedure for constructing the associated quadratic basis functions is presented. For the considered parametric density families, the corresponding parameters are selected via a grid search procedure. Numerical experiments show improved accuracy compared to the classical linear histopolation scheme.
