We study Galerkin model reduction for unconstrained linear-quadratic optimal control problems and show that state-space reduction alone already induces a reduced control structure via the optimality conditions. As a result, the solely state-reduced and the combined control- and state-reduced problems are equivalent, allowing fast optimization over a reduced control space without introducing additional approximation error. We derive lower and upper a posteriori error bounds for the optimal control and use them within an online-adaptive algorithm that constructs sufficiently accurate reduced spaces while solving the control problem. Convergence of the algorithm is proven, and numerical results demonstrate that combined control and state-space reduction yields significant speed-ups without loss of accuracy compared to state-space reduction alone.
Quantized tensor trains (QTTs) are a low-rank and multiscale framework that allows for efficient approximation and manipulation of multi-dimensional, high resolution data. One area of active research is their use in the numerical simulation of hyperbolic systems such as the Navier–Stokes equations and the Vlasov equations. One popular time integration scheme is the dynamical low-rank approximation (DLRA), in which the time integration is constrained to a low-rank manifold. However, until recently, DLRA typically uses orthogonal projectors to project the original dynamical system into a reduced space, which is best-suited for linear partial differential equations (PDEs). For nonlinear PDEs requiring element-wise evaluations, one can instead consider interpolative DLRA, in which the low-rank manifold is constructed from aptly chosen interpolation points and interpolating polynomials. Thus far, interpolative DLRA has only been investigated in the context of non-quantized tensor trains. This work extends the method to QTTs, and combines the interpolative representation with various basis expansion schemes to achieve higher order time integration. Through various examples, its performance is compared to its orthogonal counterpart, demonstrating that interpolative DLRA is indeed suitable for nonlinear systems and time integrators requiring nonlinear element-wise operations, such as upwind time integration schemes.
Modeling multiphysics processes in porous media requires preconditioned iterative linear solvers to enable efficient simulations at industry-relevant scales. These solvers are typically composed of sub-algorithms that target individual physical processes. Various options are available for each algorithm, with the corresponding ranges of numerical parameters. The choices of sub-algorithms and their parameters significantly affect simulation performance and robustness. Optimizing these choices for each simulation is challenging due to the vast number of possible combinations. Moreover, optimization relies on performance data from past simulations, which becomes less representative as the simulation setup changes. This paper addresses the problem of automated selection and tuning of preconditioned linear solvers for multiphysics simulations, targeting a scenario where numerous linear systems of similar structure must be solved successively, generated by a single discretized set of equations. The proposed solver selection algorithm collects performance data during the run of the target simulation and continuously updates a machine learning model responsible for solver selection, resulting in an adaptively refined selection policy. The algorithm is evaluated on two time-dependent nonlinear model problems: (i) coupled fluid flow and heat transfer in porous media and (ii) thermo-poromechanics in porous media with fractures, governed by frictional contact mechanics. These experiments demonstrate that the algorithm selects efficient and robust solvers with negligible overhead and performs comparably to a reference selection policy that has full access to the performance data of prior simulations. Our results indicate that the proposed approach effectively addresses the challenge of solver selection and tuning, providing particular value to simulation engineers and researchers, especially when expert knowledge on linear solver tuning is not readily available.
Kernel methods are versatile tools for function approximation and surrogate modeling. In particular, greedy techniques offer computational efficiency and reliability through inherent sparsity and provable convergence. Inspired by the success of deep neural networks and structured deep kernel networks, we consider deep, multilayer kernels for greedy approximation. This multilayer structure, consisting of linear kernel layers and optimizable kernel activation function layers in an alternating fashion, increases the expressiveness of the kernels and thus of the resulting approximants. Compared to standard kernels, deep kernels are able to adapt kernel intrinsic shape parameters automatically, incorporate transformations of the input space and induce a data-dependent reproducing kernel Hilbert space. For this, deep kernels need to be pretrained using a specifically tailored optimization objective. In this work, we not only introduce deep kernel greedy models, but also present numerical investigations and comparisons with neural networks, which clearly show the advantages in terms of approximation accuracies. As applications we consider the approximation of model problems, the prediction of breakthrough curves for reactive flow through porous media and the approximation of solutions for parameterized ordinary differential equation systems.
Bayesian inverse problems use observed data to update a prior probability distribution for an unknown state or parameter of a scientific system to a posterior distribution conditioned on the data. In many applications, the unknown parameter is high-dimensional, making computation of the posterior expensive due to the need to sample in a high-dimensional space and the need to evaluate an expensive high-dimensional forward model relating the unknown parameter to the data. However, inverse problems often exhibit low-dimensional structure due to the fact that the available data are only informative in a low-dimensional subspace of the parameter space. Dimension reduction approaches exploit this structure by restricting inference to the low-dimensional subspace informed by the data, which can be sampled more efficiently. Further computational cost reductions can be achieved by replacing expensive high-dimensional forward models with cheaper lower-dimensional reduced models. In this work, we propose new dimension and model reduction approaches for linear Bayesian inverse problems with rank-deficient prior covariances, which arise in many practical inference settings. The dimension reduction approach is applicable to general linear Bayesian inverse problems whereas the model reduction approaches are specific to the problem of inferring the initial condition of a linear dynamical system. We provide theoretical approximation guarantees as well as numerical experiments demonstrating the accuracy and efficiency of the proposed approaches.
