Search

Motivation: We consider continuous-time Markov chains that describe the stochastic evolution of a dynamical system by a transition-rate matrix $Q$ which depends on a parameter $\theta$. Computing the probability distribution over states at time $t$ requires the matrix exponential $\exp(tQ)$, and inferring $\theta$ from data requires its derivative $\partial\exp\!(tQ)/\partial\theta$. Both are challenging to compute when the state space and hence the size of $Q$ is huge. This can happen when the state space consists of all combinations of the values of several interacting discrete variables. Often it is even impossible to store $Q$. However, when $Q$ can be written as a sum of tensor products, computing $\exp(tQ)$ becomes feasible by the uniformization method, which does not require explicit storage of $Q$. Results: Here we provide an analogous algorithm for computing $\partial\exp\!(tQ)/\partial\theta$, the differentiated uniformization method. We demonstrate our algorithm for the stochastic SIR model of epidemic spread, for which we show that $Q$ can be written as a sum of tensor products. We estimate monthly infection and recovery rates during the first wave of the COVID-19 pandemic in Austria and quantify their uncertainty in a full Bayesian analysis. Availability: Implementation and data are available at https://github.com/spang-lab/TenSIR.

The solution of parameter-dependent linear systems, by classical methods, leads to an arithmetic effort that grows exponentially in the number of parameters. This renders the multigrid method, which has a well understood convergence theory, infeasible. A parameter-dependent representation, e.g., a low-rank tensor format, can avoid this exponential dependence, but in these it is unknown how to calculate the inverse directly within the representation. The combination of these representations with the multigrid method requires a parameter-dependent version of the classical multigrid theory and a parameter-dependent representation of the linear system, the smoother, the prolongation and the restriction. A derived parameter-dependent version of the smoothing property, fulfilled by parameter-dependent versions of the Richardson and Jacobi methods, together with the approximation property prove the convergence of the multigrid method for arbitrary parameter-dependent representations. For a model problem low-rank tensor formats represent the parameter-dependent linear system, prolongation and restriction. The smoother, a damped Jacobi method, is directly approximated in the low-rank tensor format by using exponential sums. Proving the smoothing property for this approximation guarantees the convergence of the parameter-dependent method. Numerical experiments for the parameter-dependent model problem, with bounded parameter value range, indicate a grid size independent convergence rate.

Continuous-time Markov chains describing interacting processes exhibit a state space that grows exponentially in the number of processes. This state-space explosion renders the computation or storage of the time-marginal distribution, which is defined as the solution of a certain linear system, infeasible using classical methods. We consider Markov chains whose transition rates are separable functions, which allows for an efficient low-rank tensor representation of the operator of this linear system. Typically, the right-hand side also has low-rank structure, and thus we can reduce the cost for computation and storage from exponential to linear. Previously known iterative methods also allow for low-rank approximations of the solution but are unable to guarantee that its entries sum up to one as required for a probability distribution. We derive a convergent iterative method using low-rank formats satisfying this condition. We also perform numerical experiments illustrating that the marginal distribution is well approximated with low rank., Comment: 20 pages, 10 figures

Low-rank tensor approximations have shown great potential for uncertainty quantification in high dimensions, for example, to build surrogate models that can be used to speed up large-scale inference problems (Eigel et al., Inverse Problems 34, 2018; Dolgov et al., Statistics & Computing 30, 2020). The feasibility and efficiency of such approaches depends critically on the rank that is necessary to represent or approximate the underlying distribution. In this paper, a-priori rank bounds for approximations in the functional tensor-train representation for the case of Gaussian models are developed. It is shown that under suitable conditions on the precision matrix, the Gaussian density can be approximated to high accuracy without suffering from an exponential growth of complexity as the dimension increases. These results provide a rigorous justification of the suitability and the limitations of low-rank tensor methods in a simple but important model case. Numerical experiments confirm that the rank bounds capture the qualitative behavior of the rank structure when varying the parameters of the precision matrix and the accuracy of the approximation. Finally, the practical relevance of the theoretical results is demonstrated in the context of a Bayesian filtering problem., Comment: 25 pages, 6 figures

The reconstruction of the structure of biological tissue using electromyographic data is a non-invasive imaging method with diverse medical applications. Mathematically, this process is an inverse problem. Furthermore, electromyographic data are highly sensitive to changes in the electrical conductivity that describes the structure of the tissue. Modeling the inevitable measurement error as a stochastic quantity leads to a Bayesian approach. Solving the discretized Bayes-inverse problem means drawing samples from the posterior distribution of parameters, e.g., the conductivity, given measurement data. Using, e.g., a Metropolis-Hastings algorithm for this purpose involves solving the forward problem for different parameter combinations which requires a high computational effort. Low-rank tensor formats can reduce this effort by providing a data-sparse representation of all occurring linear systems of equations simultaneously and allow for their efficient solution. The application of Bayes' theorem proves the well-posedness of the Bayes-inverse problem. The derivation and proof of a low-rank representation of the forward problem allow for the precomputation of all solutions of this problem under certain assumptions, resulting in an efficient and theory-based sampling algorithm. Numerical experiments support the theoretical results, but also indicate that a high number of samples is needed to obtain reliable estimates for the parameters. The Metropolis-Hastings sampling algorithm, using the precomputed forward solution in a tensor format, draws this high number of samples and therefore enables solving problems which are infeasible using classical methods.

We present an iterative method for the search of extreme entries in low-rank tensors which is based on a power iteration combined with a binary search. In this work we use the HT-format for low-rank tensors but other low-rank formats can be used verbatim. We present two different approaches to accelerate the basic power iteration: an orthogonal projection of Rayleigh-Ritz type, as well as an acceleration of the power iteration itself which can be achieved due to the diagonal structure of the underlying eigenvalue problem. Finally the maximizing index is determined by a binary search based on the proposed iterative method for the approximation of the maximum norm. The iterative method for the maximum norm estimation inherits the linear complexity of the involved tensor arithmetic in the HT-format w.r.t. the tensor order, which is also verified by numerical tests.