Modern scientific and engineering challenges, from biomedical imaging to materials design, often require solving partial differential equations (PDEs) that depend on many parameters. As the number of parameters grows, the size of the resulting linear systems after discretization increases exponentially, rendering classical numerical methods computationally infeasible. Addressing this challenge requires data-sparse representations and solvers that are both computationally efficient and supported by rigorous theoretical guarantees. Iterative solvers combined with low-rank tensor arithmetic offer a promising approach to mitigate this growth by representing the parameter-dependent linear system using low-rank tensor formats and applying truncation to maintain a data-sparse structure. However, several open challenges remain. Preconditioning in low-rank tensor formats is often limited, as existing approaches may ignore parameter dependence or require iterative solutions of auxiliary systems of comparable complexity. Similarly, computing extremal eigenvalues in low-rank tensor formats, when only operator applications with truncation are feasible, remains insufficiently explored. These challenges have hindered the extension of multigrid methods to low-rank tensor representations. In particular, such methods rely, e.g., on approximate inverses of the parameter-dependent diagonal, which are nontrivial to compute directly within low-rank formats. This thesis develops theoretically grounded and practically effective methods to solve high-dimensional, parameter-dependent elliptic PDEs, overcoming these limitations. We derive the analytical solution and numerical approximation of a model problem and prove the existence of an exact data-sparse representation using low-rank tensor formats. We present the Low-Rank Tensor Toolkit, a versatile and efficient software package that makes low-rank tensor computations accessible for a broad range of applications. All methods developed in this thesis are implemented within this toolkit. To compute extremal eigenvalues of low-rank tensor operators, we introduce a Lanczos-based approach. By linking classical rounding-error analysis with low-rank truncation, we establish principles for applying truncation without compromising the accuracy of the approximated eigenvalues, as confirmed by numerical experiments. This leads to a restarted Lanczos method for low-rank tensor operators capable of estimating extremal eigenvalues efficiently and reliably using only operator applications. We develop a novel approach to compute approximate inverses of low-rank tensor operators based on exponential sums, with rigorously controlled error bounds. For the CP format, we derive error and rank bounds for computing the inverse of a parameter-dependent diagonal operator. We further develop techniques to efficiently compute matrix exponentials within low-rank tensor formats, allowing us to generalize the approximate inverse from the diagonal to the full operator within the hierarchical Tucker format, again with guaranteed error bounds. Numerical experiments indicate that the conservative theoretical bounds can be relaxed in practice, enabling efficient computation of approximate inverses directly within low-rank tensor formats. These approximate inverses enable the construction of parameter-dependent multigrid components, such as a parameter-dependent Jacobi smoother as well as parameter-dependent operator-dependent prolongation and restriction, all computed with fully controlled approximation errors. By transferring classical matrix multigrid results to the parameter-dependent case, we further prove that the approximated smoothers satisfy the parameter-dependent smoothing property, and the approximated prolongation and restriction operators satisfy the parameter-dependent approximation property. Explicit bounds are derived to determine how accurately these approximations must be computed to guarantee the corresponding property. Based on this, a complete convergence theory for the parameter-dependent multigrid method is established, explicitly accounting for low-rank truncation and operator approximations. Numerical experiments support the theoretical findings.
Tim Andreas Werthmann (Thu,) studied this question.