Kristin Kirchner

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Kirchner

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Kristin

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0000-0002-3609-9431

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Publications 1 - 10 of 10

Multilevel approximation of Gaussian random fields: Covariance compression, estimation, and spatial prediction
Harbrecht, Helmut; Herrmann, Lukas; Kirchner, Kristin; et al. (2024)
Advances in Computational Mathematics
The distribution of centered Gaussian random fields (GRFs) indexed by compacta such as smooth, bounded Euclidean domains or smooth, compact and orientable manifolds is determined by their covariance operators. We consider centered GRFs given as variational solutions to coloring operator equations driven by spatial white noise, with an elliptic self-adjoint pseudodifferential coloring operator from the Hörmander class. This includes the Matérn class of GRFs as a special case. Using biorthogonal multiresolution analyses on the manifold, we prove that the precision and covariance operators, respectively, may be identified with bi-infinite matrices and finite sections may be diagonally preconditioned rendering the condition number independent of the dimension p of this section. We prove that a tapering strategy by thresholding applied on finite sections of the bi-infinite precision and covariance matrices results in optimally numerically sparse approximations. That is, asymptotically only linearly many nonzero matrix entries are sufficient to approximate the original section of the bi-infinite covariance or precision matrix using this tapering strategy to arbitrary precision. The locations of these nonzero matrix entries can be determined a priori. The tapered covariance or precision matrices may also be optimally diagonally preconditioned. Analysis of the relative size of the entries of the tapered covariance matrices motivates novel, multilevel Monte Carlo (MLMC) oracles for covariance estimation, in sample complexity that scales log-linearly with respect to the number p of parameters. In addition, we propose and analyze novel compressive algorithms for simulating and kriging of GRFs. The complexity (work and memory vs. accuracy) of these three algorithms scales near-optimally in terms of the number of parameters p of the sample-wise approximation of the GRF in Sobolev scales.
Multilevel approximation of Gaussian random fields: fast simulation
Herrmann, Lukas; Kirchner, Kristin; Schwab, Christoph (2019)
SAM Research Report
Monte Carlo convergence rates for kth moments in Banach spaces
Kirchner, Kristin; Schwab, Christoph (2022)
SAM Research Report
The Rational SPDE Approach for Gaussian Random Fields With General Smoothness
Bolin, David; Kirchner, Kristin (2020)
Journal of Computational and Graphical Statistics
Multilevel approximation of Gaussian random fields: Covariance compression, estimation and spatial prediction
Harbrecht, Helmut; Herrmann, Lukas; Kirchner, Kristin; et al. (2021)
SAM Research Report
Centered Gaussian random fields (GRFs) indexed by compacta such as smooth, bounded domains in Euclidean space or smooth, compact and orientable manifolds are determined by their covariance operators. We consider centered GRFs given sample-wise as variational solutions to coloring operator equations driven by spatial white noise, with pseudodifferential coloring operator being elliptic, self-adjoint and positive from the Hörmander class. This includes the Mat\'ern class of GRFs as a special case. Using microlocal tools and biorthogonal multiresolution analyses on the manifold, we prove that the precision and covariance operators, respectively, may be identified with bi-infinite matrices and finite sections may be diagonally preconditioned rendering the condition number independent of the dimension of this section. We prove that a tapering strategy by thresholding as e.g. in [Bickel, P.J. and Levina, E. Covariance regularization by thresholding, Ann. Statist., 36 (2008), 2577-2604] applied on finite sections of the bi-infinite precision and covariance matrices results in optimally numerically sparse approximations. Numerical sparsity signifies that only asymptotically linearly many nonzero matrix entries are sufficient to approximate the original section of the bi-infinite covariance or precision matrix using this tapering strategy to arbitrary precision. This tapering strategy is non-adaptive and the locations of these nonzero matrix entries are known a priori. The tapered covariance or precision matrices may also be optimally diagonal preconditioned. Analysis of the relative size of the entries of the tapered covariance matrices motivates novel, multilevel Monte Carlo (MLMC) oracles for covariance estimation, in sample complexity that scales log-linearly with respect to the number of parameters. This extends [Bickel, P.J. and Levina, E. Regularized Estimation of Large Covariance Matrices, Ann. Stat., 36 (2008), pp. 199-227] to estimation of (finite sections of) pseudodifferential covariances for GRFs by this fast MLMC method. Assuming at hand sections of the bi-infinite covariance matrix in wavelet coordinates, we propose and analyze a novel compressive algorithm for simulating and kriging of GRFs. The complexity (work and memory vs. accuracy) of these three algorithms scales near-optimally in terms of the number of parameters of the sample-wise approximation of the GRF in Sobolev scales.
Numerical methods for the deterministic second moment equation of parabolic stochastic PDEs
Kirchner, Kristin (2020)
Mathematics of Computation
Numerical methods for stochastic partial differential equations typically estimate moments of the solution from sampled paths. Instead, we shall directly target the deterministic equations satisfied by the mean and the spatio-temporal covariance structure of the solution process. In the first part, we focus on stochastic ordinary differential equations. For the canonical examples with additive noise (Ornstein-Uhlenbeck process) or multiplicative noise (geometric Brownian motion) we derive these deterministic equations in variational form and discuss their well-posedness in detail. Notably, the second moment equation in the multiplicative case is naturally posed on projective-injective tensor product spaces as trial-test spaces. We then propose numerical approximations based on Petrov-Galerkin discretizations with tensor product piecewise polynomials and analyze their stability and convergence in the natural tensor norms. In the second part, we proceed with parabolic stochastic partial differential equations with affine multiplicative noise. We prove well-posedness of the deterministic variational problem for the second moment, improving an earlier result. We then propose conforming space-time Petrov-Galerkin discretizations, which we show to be stable and quasi-optimal. In both parts, the outcomes are validated by numerical examples. © 2020 American Mathematical Society.
Monte Carlo convergence rates for kth moments in Banach spaces
Kirchner, Kristin; Schwab, Christoph (2024)
Journal of Functional Analysis
We formulate standard and multilevel Monte Carlo methods for the kth moment Mεk[ξ] of a Banach space valued random variable ξ:Ω→E, interpreted as an element of the k-fold injective tensor product space ⊗εkE. For the standard Monte Carlo estimator of Mεk[ξ], we prove the k-independent convergence rate 1 - 1/p in the Lq(Ω;⊗εkE)-norm, provided that (i) ξ∈Lkq(Ω;E) and (ii) q∈[p,∞), where p∈[1,2] is the Rademacher type of E. By using the fact that Rademacher averages are dominated by Gaussian sums combined with a version of Slepian's inequality for Gaussian processes due to Fernique, we moreover derive corresponding results for multilevel Monte Carlo methods, including a rigorous error estimate in the Lq(Ω;⊗εkE)-norm and the optimization of the computational cost for a given accuracy. Whenever the type of the Banach space E is p=2, our findings coincide with known results for Hilbert space valued random variables. We illustrate the abstract results by three model problems: second-order elliptic PDEs with random forcing or random coefficient, and stochastic evolution equations. In these cases, the solution processes naturally take values in non-Hilbertian Banach spaces. Further applications, where physical modeling constraints impose a setting in Banach spaces of type p<2, are indicated.
Regularity and convergence analysis in Sobolev and Hölder spaces for generalized Whittle-Matérn fields
Cox, Sonja Gisela; Kirchner, Kristin (2019)
SAM Research Report
Multilevel approximation of Gaussian random fields: Fast simulation
Herrmann, Lukas; Kirchner, Kristin; Schwab, Christoph (2020)
Mathematical Models and Methods in Applied Sciences
Regularity and convergence analysis in Sobolev and Hölder spaces for generalized Whittle–Matérn fields
Cox, Sonja G.; Kirchner, Kristin (2020)
Numerische Mathematik
We analyze several types of Galerkin approximations of a Gaussian random field Z: D× Ω→ R indexed by a Euclidean domain D⊂ Rd whose covariance structure is determined by a negative fractional power L-2β of a second-order elliptic differential operator L: = - ∇ · (A∇) + κ2. Under minimal assumptions on the domain D, the coefficients A: D→ Rd×d, κ: D→ R, and the fractional exponent β> 0 , we prove convergence in Lq(Ω; Hσ(D)) and in Lq(Ω; Cδ(D¯)) at (essentially) optimal rates for (1) spectral Galerkin methods and (2) finite element approximations. Specifically, our analysis is solely based on H1+α(D) -regularity of the differential operator L, where 0 < α≤ 1. For this setting, we furthermore provide rigorous estimates for the error in the covariance function of these approximations in L∞(D× D) and in the mixed Sobolev space Hσ,σ(D× D) , showing convergence which is more than twice as fast compared to the corresponding Lq(Ω; Hσ(D)) -rate. We perform several numerical experiments which validate our theoretical results for (a) the original Whittle–Matérn class, where A≡IdRd and κ≡ const. , and (b) an example of anisotropic, non-stationary Gaussian random fields in d= 2 dimensions, where A: D→ R2 × 2 and κ: D→ R are spatially varying.

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Kristin Kirchner

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