H.N. Kekkonen
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1
We study a full waveform inversion (FWI) method in the frequency domain penalized by Besov space norms. In particular, we combine frequency marching with regularization in the Besov space, motivated by its edge-preserving and multiscale properties. First, we establish the well-posedness of the associated PDE-constrained optimization problem under Besov regularization. Second, we propose a combined approach based on frequency marching and Besov penalization, aimed at reconstructing fine-scale features by utilizing higher frequency data. Finally, we present numerical experiments to demonstrate the performance of our proposed method.
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We study a full waveform inversion (FWI) method in the frequency domain penalized by Besov space norms. In particular, we combine frequency marching with regularization in the Besov space, motivated by its edge-preserving and multiscale properties. First, we establish the well-posedness of the associated PDE-constrained optimization problem under Besov regularization. Second, we propose a combined approach based on frequency marching and Besov penalization, aimed at reconstructing fine-scale features by utilizing higher frequency data. Finally, we present numerical experiments to demonstrate the performance of our proposed method.
Journal article
(2026)
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Leon Riccius, Iuri B.C.M. Rocha, Joris Bierkens, Hanne Kekkonen, Frans P. van der Meer
Recent advancements in Markov chain Monte Carlo (MCMC) sampling and surrogate modelling have significantly enhanced the feasibility of Bayesian analysis across engineering fields. However, the selection and integration of surrogate models and cutting-edge MCMC algorithms, often depend on ad-hoc decisions. A systematic assessment of their combined influence on accuracy and efficiency is notably lacking. The present work offers a comprehensive comparative study, employing a scalable case study in computational mechanics focused on the inference of spatially varying material parameters, that sheds light on the impact of methodological choices for surrogate modelling and sampling. We show that a priori training of the surrogate model introduces large errors in the posterior estimation even in low to moderate dimensions. We introduce a simple active learning strategy based on the path of the MCMC algorithm that is superior to all a priori trained models, and determine its training data requirements. We demonstrate that the choice of the MCMC algorithm has only a small influence on the amount of training data but no significant influence on the accuracy of the resulting surrogate model. Further, we show that the accuracy of the posterior estimation largely depends on the surrogate model, but not even a tailored surrogate guarantees convergence of the MCMC. Finally, we identify the forward model as the bottleneck in the inference process, not the MCMC algorithm. While related works focus on employing advanced MCMC algorithms, we demonstrate that the training data requirements render the surrogate modelling approach infeasible before the benefits of these gradient-based MCMC algorithms on cheap models can be reaped.
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Recent advancements in Markov chain Monte Carlo (MCMC) sampling and surrogate modelling have significantly enhanced the feasibility of Bayesian analysis across engineering fields. However, the selection and integration of surrogate models and cutting-edge MCMC algorithms, often depend on ad-hoc decisions. A systematic assessment of their combined influence on accuracy and efficiency is notably lacking. The present work offers a comprehensive comparative study, employing a scalable case study in computational mechanics focused on the inference of spatially varying material parameters, that sheds light on the impact of methodological choices for surrogate modelling and sampling. We show that a priori training of the surrogate model introduces large errors in the posterior estimation even in low to moderate dimensions. We introduce a simple active learning strategy based on the path of the MCMC algorithm that is superior to all a priori trained models, and determine its training data requirements. We demonstrate that the choice of the MCMC algorithm has only a small influence on the amount of training data but no significant influence on the accuracy of the resulting surrogate model. Further, we show that the accuracy of the posterior estimation largely depends on the surrogate model, but not even a tailored surrogate guarantees convergence of the MCMC. Finally, we identify the forward model as the bottleneck in the inference process, not the MCMC algorithm. While related works focus on employing advanced MCMC algorithms, we demonstrate that the training data requirements render the surrogate modelling approach infeasible before the benefits of these gradient-based MCMC algorithms on cheap models can be reaped.
Mathematics and crochet might not appear the most likely pairing for most people. However, crocheting is an inherently mathematical process. You can create various shapes using stitches with different heights and increasing or decreasing the number of stitches in certain places. Crochet also makes it possible to create many shapes that are very difficult to make with any other technique. One can crochet shapes such as the Klein bottle and Seifert surfaces of knots in freestyle, without following exact instructions, since these surfaces do not have a strict shape. A Klein bottle can be short and wide or tall and narrow. But there are many surfaces, for example spheres and disks, that have a specific shape. To crochet such models, you need crochet instructions, and to create such instructions, you need a good understanding of the underlying mathematical model.
The idea of knitting or crocheting mathematical or scientific models is not new, though it has not been used very widely. The Scottish chemist Alexander Crum Brown knitted several interlinked surfaces to visualize the ideas presented in the late nineteenth century in his paper “On a Case of Interlacing Surfaces” [2]. Miles Reid wrote a paper on knitting mathematical surfaces in the 1970s [13] that inspired several new patterns, including a Möbius scarf and a Klein bottle. The crocheted hyperbolic surfaces were introduced by Daina Taimina in 1997 [5], and her idea led to a bloom of so-called hyperbolic crochet. A few years after the paper on hyperbolic crochet appeared, Hinke Osinga and Bernd Krauskopf described how to crochet an approximation of the Lorenz manifold [10]. See also [14] for further examples of mathematical crochet.
Both the hyperbolic plane and the Lorenz manifold require precise crochet instructions. The hyperbolic plane has constant negative Gaussian curvature, and so it looks the same at every point. This allows for a rather simple pattern that can be worked in rounds in which after a few setup rounds, every nth stitch is doubled. The Lorenz manifold is a less-regular surface, and it requires a much more complex pattern of stitches. The model is also worked in rounds, but unlike the hyperbolic surface, it requires detailed instructions on when to add or remove stitches. It takes full advantage of the versatility of crocheting, requiring three different types of stitches, which allows different parts of a round to have different heights. In this paper we consider Bour’s minimal surfaces
, which are “crochet symmetric,” allowing for simple crochet instructions (excluding possible intersections) and requiring only one type of stitch, with the added or removed stitches evenly spaced across a round. ...
The idea of knitting or crocheting mathematical or scientific models is not new, though it has not been used very widely. The Scottish chemist Alexander Crum Brown knitted several interlinked surfaces to visualize the ideas presented in the late nineteenth century in his paper “On a Case of Interlacing Surfaces” [2]. Miles Reid wrote a paper on knitting mathematical surfaces in the 1970s [13] that inspired several new patterns, including a Möbius scarf and a Klein bottle. The crocheted hyperbolic surfaces were introduced by Daina Taimina in 1997 [5], and her idea led to a bloom of so-called hyperbolic crochet. A few years after the paper on hyperbolic crochet appeared, Hinke Osinga and Bernd Krauskopf described how to crochet an approximation of the Lorenz manifold [10]. See also [14] for further examples of mathematical crochet.
Both the hyperbolic plane and the Lorenz manifold require precise crochet instructions. The hyperbolic plane has constant negative Gaussian curvature, and so it looks the same at every point. This allows for a rather simple pattern that can be worked in rounds in which after a few setup rounds, every nth stitch is doubled. The Lorenz manifold is a less-regular surface, and it requires a much more complex pattern of stitches. The model is also worked in rounds, but unlike the hyperbolic surface, it requires detailed instructions on when to add or remove stitches. It takes full advantage of the versatility of crocheting, requiring three different types of stitches, which allows different parts of a round to have different heights. In this paper we consider Bour’s minimal surfaces
, which are “crochet symmetric,” allowing for simple crochet instructions (excluding possible intersections) and requiring only one type of stitch, with the added or removed stitches evenly spaced across a round. ...
Mathematics and crochet might not appear the most likely pairing for most people. However, crocheting is an inherently mathematical process. You can create various shapes using stitches with different heights and increasing or decreasing the number of stitches in certain places. Crochet also makes it possible to create many shapes that are very difficult to make with any other technique. One can crochet shapes such as the Klein bottle and Seifert surfaces of knots in freestyle, without following exact instructions, since these surfaces do not have a strict shape. A Klein bottle can be short and wide or tall and narrow. But there are many surfaces, for example spheres and disks, that have a specific shape. To crochet such models, you need crochet instructions, and to create such instructions, you need a good understanding of the underlying mathematical model.
The idea of knitting or crocheting mathematical or scientific models is not new, though it has not been used very widely. The Scottish chemist Alexander Crum Brown knitted several interlinked surfaces to visualize the ideas presented in the late nineteenth century in his paper “On a Case of Interlacing Surfaces” [2]. Miles Reid wrote a paper on knitting mathematical surfaces in the 1970s [13] that inspired several new patterns, including a Möbius scarf and a Klein bottle. The crocheted hyperbolic surfaces were introduced by Daina Taimina in 1997 [5], and her idea led to a bloom of so-called hyperbolic crochet. A few years after the paper on hyperbolic crochet appeared, Hinke Osinga and Bernd Krauskopf described how to crochet an approximation of the Lorenz manifold [10]. See also [14] for further examples of mathematical crochet.
Both the hyperbolic plane and the Lorenz manifold require precise crochet instructions. The hyperbolic plane has constant negative Gaussian curvature, and so it looks the same at every point. This allows for a rather simple pattern that can be worked in rounds in which after a few setup rounds, every nth stitch is doubled. The Lorenz manifold is a less-regular surface, and it requires a much more complex pattern of stitches. The model is also worked in rounds, but unlike the hyperbolic surface, it requires detailed instructions on when to add or remove stitches. It takes full advantage of the versatility of crocheting, requiring three different types of stitches, which allows different parts of a round to have different heights. In this paper we consider Bour’s minimal surfaces
, which are “crochet symmetric,” allowing for simple crochet instructions (excluding possible intersections) and requiring only one type of stitch, with the added or removed stitches evenly spaced across a round.
The idea of knitting or crocheting mathematical or scientific models is not new, though it has not been used very widely. The Scottish chemist Alexander Crum Brown knitted several interlinked surfaces to visualize the ideas presented in the late nineteenth century in his paper “On a Case of Interlacing Surfaces” [2]. Miles Reid wrote a paper on knitting mathematical surfaces in the 1970s [13] that inspired several new patterns, including a Möbius scarf and a Klein bottle. The crocheted hyperbolic surfaces were introduced by Daina Taimina in 1997 [5], and her idea led to a bloom of so-called hyperbolic crochet. A few years after the paper on hyperbolic crochet appeared, Hinke Osinga and Bernd Krauskopf described how to crochet an approximation of the Lorenz manifold [10]. See also [14] for further examples of mathematical crochet.
Both the hyperbolic plane and the Lorenz manifold require precise crochet instructions. The hyperbolic plane has constant negative Gaussian curvature, and so it looks the same at every point. This allows for a rather simple pattern that can be worked in rounds in which after a few setup rounds, every nth stitch is doubled. The Lorenz manifold is a less-regular surface, and it requires a much more complex pattern of stitches. The model is also worked in rounds, but unlike the hyperbolic surface, it requires detailed instructions on when to add or remove stitches. It takes full advantage of the versatility of crocheting, requiring three different types of stitches, which allows different parts of a round to have different heights. In this paper we consider Bour’s minimal surfaces
, which are “crochet symmetric,” allowing for simple crochet instructions (excluding possible intersections) and requiring only one type of stitch, with the added or removed stitches evenly spaced across a round.
We propose alternatives to Bayesian prior distributions that are frequently used in the study of inverse problems. Our aim is to construct priors that have similar good edge-preserving properties as total variation or Mumford-Shah priors but correspond to well-defined infinite-dimensional random variables, and can be approximated by finite-dimensional random vari-ables. We introduce a new wavelet-based model, where the non-zero coefficients are chosen in a systematic way so that prior draws have certain fractal behaviour. We show that realisations of this new prior take values in Besov spaces and have singularities only on a small set τ with a certain Hausdorff dimension. We also introduce an efficient algorithm for calculating the MAP estimator, arising from the the new prior, in the denoising problem.
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We propose alternatives to Bayesian prior distributions that are frequently used in the study of inverse problems. Our aim is to construct priors that have similar good edge-preserving properties as total variation or Mumford-Shah priors but correspond to well-defined infinite-dimensional random variables, and can be approximated by finite-dimensional random vari-ables. We introduce a new wavelet-based model, where the non-zero coefficients are chosen in a systematic way so that prior draws have certain fractal behaviour. We show that realisations of this new prior take values in Besov spaces and have singularities only on a small set τ with a certain Hausdorff dimension. We also introduce an efficient algorithm for calculating the MAP estimator, arising from the the new prior, in the denoising problem.
Crochet provides a superior method for the production of two-dimensional surfaces from one-dimensional material. Compared to any of the other known processes to generate constant flat, spherical or hyperbolic shapes, it is the most flexible and precise way to build a dynamical system with very simple local rules and with very high precision of the intrinsic curvature.
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Crochet provides a superior method for the production of two-dimensional surfaces from one-dimensional material. Compared to any of the other known processes to generate constant flat, spherical or hyperbolic shapes, it is the most flexible and precise way to build a dynamical system with very simple local rules and with very high precision of the intrinsic curvature.
Building blocks and tiles are an excellent way of learning about geometry and mathematics in general. There are several versions of tiles that are either snapped together or connected with magnets that can be used to introduce topics like volume, tessellations, and Platonic solids. However, since these tiles are made of hard plastic, they are not very suitable for creating hyperbolic surfaces or shapes where the tiles need to bend. Curvagons are flexible regular polygon building blocks that allow you to quickly build anything from hyperbolic surfaces and tori to dinosaurs and shoes. They can be used to introduce mathematical concepts from Archimedean solids to Gauss-Bonnet theorem. You can also let your imagination run free and build whatever comes to mind.
...
Building blocks and tiles are an excellent way of learning about geometry and mathematics in general. There are several versions of tiles that are either snapped together or connected with magnets that can be used to introduce topics like volume, tessellations, and Platonic solids. However, since these tiles are made of hard plastic, they are not very suitable for creating hyperbolic surfaces or shapes where the tiles need to bend. Curvagons are flexible regular polygon building blocks that allow you to quickly build anything from hyperbolic surfaces and tori to dinosaurs and shoes. They can be used to introduce mathematical concepts from Archimedean solids to Gauss-Bonnet theorem. You can also let your imagination run free and build whatever comes to mind.
We consider the statistical non-linear inverse problem of recovering the absorption term f > 0 in the heat equation {∂tu-12Δu+fu=0onO×(0,T)u=gon∂ O×(0,T)u(·,0)=u0onO, where O ϵ ℝd is a bounded domain, T < ∞ is a fixed time, and g, u 0 are given sufficiently smooth functions describing boundary and initial values respectively. The data consists of N discrete noisy point evaluations of the solution u f on O×(0,T) . We study the statistical performance of Bayesian nonparametric procedures based on a large class of Gaussian process priors. We show that, as the number of measurements increases, the resulting posterior distributions concentrate around the true parameter generating the data, and derive a convergence rate for the reconstruction error of the associated posterior means. We also consider the optimality of the contraction rates and prove a lower bound for the minimax convergence rate for inferring f from the data, and show that optimal rates can be achieved with truncated Gaussian priors.
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We consider the statistical non-linear inverse problem of recovering the absorption term f > 0 in the heat equation {∂tu-12Δu+fu=0onO×(0,T)u=gon∂ O×(0,T)u(·,0)=u0onO, where O ϵ ℝd is a bounded domain, T < ∞ is a fixed time, and g, u 0 are given sufficiently smooth functions describing boundary and initial values respectively. The data consists of N discrete noisy point evaluations of the solution u f on O×(0,T) . We study the statistical performance of Bayesian nonparametric procedures based on a large class of Gaussian process priors. We show that, as the number of measurements increases, the resulting posterior distributions concentrate around the true parameter generating the data, and derive a convergence rate for the reconstruction error of the associated posterior means. We also consider the optimality of the contraction rates and prove a lower bound for the minimax convergence rate for inferring f from the data, and show that optimal rates can be achieved with truncated Gaussian priors.
How can we convince students, who have mainly learned to follow given mathematical rules, that mathematics can also be fascinating, creative, and beautiful? In this paper I discuss different ways of introducing non-Euclidean geometry to students and the general public using different physical models, including chalksphere, crocheted hyperbolic surfaces, curved folding, and polygon tilings. Spherical geometry offers a simple yet surprising introduction to the topic, whereas hyperbolic geometry is an entirely new and exciting concept to most. Non-Euclidean geometry demonstrates how crafts and art can be used to make complex mathematical concepts more accessible, and how mathematics itself can be beautiful, not just useful.
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How can we convince students, who have mainly learned to follow given mathematical rules, that mathematics can also be fascinating, creative, and beautiful? In this paper I discuss different ways of introducing non-Euclidean geometry to students and the general public using different physical models, including chalksphere, crocheted hyperbolic surfaces, curved folding, and polygon tilings. Spherical geometry offers a simple yet surprising introduction to the topic, whereas hyperbolic geometry is an entirely new and exciting concept to most. Non-Euclidean geometry demonstrates how crafts and art can be used to make complex mathematical concepts more accessible, and how mathematics itself can be beautiful, not just useful.
Journal article
(2020)
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Matteo Giordano, Hanne Kekkonen
We consider the statistical inverse problem of recovering an unknown function f from a linear measurement corrupted by additive Gaussian white noise. We employ a nonparametric Bayesian approach with standard Gaussian priors, for which the posterior-based reconstruction of f corresponds to a Tikhonov regulariser f¯ with a reproducing kernel Hilbert space norm penalty. We prove a semiparametric Bernstein-von Mises theorem for a large collection of linear functionals of f, implying that semiparametric posterior estimation and uncertainty quantification are valid and optimal from a frequentist point of view. The result is applied to study three concrete examples that cover both the mildly and severely ill-posed cases: specifically, an elliptic inverse problem, an elliptic boundary value problem and the heat equation. For the elliptic boundary value problem, we also obtain a nonparametric version of the theorem that entails the convergence of the posterior distribution to a prior-independent infinite-dimensional Gaussian probability measure with minimal covariance. As a consequence, it follows that the Tikhonov regulariser f¯ is an efficient estimator of f, and we derive frequentist guarantees for certain credible balls centred at f¯.
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We consider the statistical inverse problem of recovering an unknown function f from a linear measurement corrupted by additive Gaussian white noise. We employ a nonparametric Bayesian approach with standard Gaussian priors, for which the posterior-based reconstruction of f corresponds to a Tikhonov regulariser f¯ with a reproducing kernel Hilbert space norm penalty. We prove a semiparametric Bernstein-von Mises theorem for a large collection of linear functionals of f, implying that semiparametric posterior estimation and uncertainty quantification are valid and optimal from a frequentist point of view. The result is applied to study three concrete examples that cover both the mildly and severely ill-posed cases: specifically, an elliptic inverse problem, an elliptic boundary value problem and the heat equation. For the elliptic boundary value problem, we also obtain a nonparametric version of the theorem that entails the convergence of the posterior distribution to a prior-independent infinite-dimensional Gaussian probability measure with minimal covariance. As a consequence, it follows that the Tikhonov regulariser f¯ is an efficient estimator of f, and we derive frequentist guarantees for certain credible balls centred at f¯.
Journal article
(2018)
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Martin Burger, Tapio Helin, Hanne Kekkonen
In this paper we consider variational regularization methods for inverse problems with large noise that is in general unbounded in the image space of the forward operator. We introduce a Banach space setting that allows to define a reasonable notion of solutions for more general noise in a larger space provided one has sufficient mapping properties of the forward operators.
A key observation, which guides us through the subsequent analysis, is that such a general noise model can be understood with the same setting as approximate source conditions (while a standard model of bounded noise is related directly to classical source conditions). Based on this insight we obtain a quite general existence result for regularized variational problems and derive error estimates in terms of Bregman distances. The latter are specialized for the particularly important cases of one- and p-homogeneous regularization functionals.
As a natural further step we study stochastic noise models and in particular white noise, for which we derive error estimates in terms of the expectation of the Bregman distance. The finiteness of certain expectations leads to a novel class of abstract smoothness conditions on the forward operator, which can be easily interpreted in the Hilbert space case. We finally exemplify the approach and in particular the conditions for popular examples of regularization functionals given by squared norm, Besov norm and total variation, respectively. ...
A key observation, which guides us through the subsequent analysis, is that such a general noise model can be understood with the same setting as approximate source conditions (while a standard model of bounded noise is related directly to classical source conditions). Based on this insight we obtain a quite general existence result for regularized variational problems and derive error estimates in terms of Bregman distances. The latter are specialized for the particularly important cases of one- and p-homogeneous regularization functionals.
As a natural further step we study stochastic noise models and in particular white noise, for which we derive error estimates in terms of the expectation of the Bregman distance. The finiteness of certain expectations leads to a novel class of abstract smoothness conditions on the forward operator, which can be easily interpreted in the Hilbert space case. We finally exemplify the approach and in particular the conditions for popular examples of regularization functionals given by squared norm, Besov norm and total variation, respectively. ...
In this paper we consider variational regularization methods for inverse problems with large noise that is in general unbounded in the image space of the forward operator. We introduce a Banach space setting that allows to define a reasonable notion of solutions for more general noise in a larger space provided one has sufficient mapping properties of the forward operators.
A key observation, which guides us through the subsequent analysis, is that such a general noise model can be understood with the same setting as approximate source conditions (while a standard model of bounded noise is related directly to classical source conditions). Based on this insight we obtain a quite general existence result for regularized variational problems and derive error estimates in terms of Bregman distances. The latter are specialized for the particularly important cases of one- and p-homogeneous regularization functionals.
As a natural further step we study stochastic noise models and in particular white noise, for which we derive error estimates in terms of the expectation of the Bregman distance. The finiteness of certain expectations leads to a novel class of abstract smoothness conditions on the forward operator, which can be easily interpreted in the Hilbert space case. We finally exemplify the approach and in particular the conditions for popular examples of regularization functionals given by squared norm, Besov norm and total variation, respectively.
A key observation, which guides us through the subsequent analysis, is that such a general noise model can be understood with the same setting as approximate source conditions (while a standard model of bounded noise is related directly to classical source conditions). Based on this insight we obtain a quite general existence result for regularized variational problems and derive error estimates in terms of Bregman distances. The latter are specialized for the particularly important cases of one- and p-homogeneous regularization functionals.
As a natural further step we study stochastic noise models and in particular white noise, for which we derive error estimates in terms of the expectation of the Bregman distance. The finiteness of certain expectations leads to a novel class of abstract smoothness conditions on the forward operator, which can be easily interpreted in the Hilbert space case. We finally exemplify the approach and in particular the conditions for popular examples of regularization functionals given by squared norm, Besov norm and total variation, respectively.
Journal article
(2016)
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Hanne Kekkonen, Matti Lassas, Samuli Siltanen
The Bayesian approach to inverse problems is studied in the case where the forward map is a linear hypoelliptic pseudodifferential operator and measurement error is additive white Gaussian noise. The measurement model for an unknown Gaussian random variable U (x, w) is Mδ (y, w) = A (U (x, w)) + δ>(y, w), where A is a finitely many orders smoothing linear hypoelliptic operator and δ> 0 is the noise magnitude. The covariance operator CU of U is smoothing of order 2r, self-adjoint, injective and elliptic pseudodifferential operator. If ϵ was taking values in L2 then in Gaussian case solving the conditional mean (and maximum a posteriori) estimate is linked to solving the minimisation problem Tδ (mδ) = arg min μϵHr{ Au-mδ;2L 2+δ2CU -1/2u2L 2} However, Gaussian white noise does not take values in L2but in H-s where s>0 is big enough. A modification of the above approach to solve the inverse problem is presented, covering the case of white Gaussian measurement noise. Furthermore, the convergence of the conditional mean estimate to the correct solution as δ → 0 is proven in appropriate function spaces using microlocal analysis. Also the frequentist posterior contractions rates are studied.
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The Bayesian approach to inverse problems is studied in the case where the forward map is a linear hypoelliptic pseudodifferential operator and measurement error is additive white Gaussian noise. The measurement model for an unknown Gaussian random variable U (x, w) is Mδ (y, w) = A (U (x, w)) + δ>(y, w), where A is a finitely many orders smoothing linear hypoelliptic operator and δ> 0 is the noise magnitude. The covariance operator CU of U is smoothing of order 2r, self-adjoint, injective and elliptic pseudodifferential operator. If ϵ was taking values in L2 then in Gaussian case solving the conditional mean (and maximum a posteriori) estimate is linked to solving the minimisation problem Tδ (mδ) = arg min μϵHr{ Au-mδ;2L 2+δ2CU -1/2u2L 2} However, Gaussian white noise does not take values in L2but in H-s where s>0 is big enough. A modification of the above approach to solve the inverse problem is presented, covering the case of white Gaussian measurement noise. Furthermore, the convergence of the conditional mean estimate to the correct solution as δ → 0 is proven in appropriate function spaces using microlocal analysis. Also the frequentist posterior contractions rates are studied.
Journal article
(2014)
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Hanne Kekkonen, Matti Lassas, Samuli Siltanen
Tikhonov regularization is studied in the case of linear pseudodifferential operator as the forward map and additive white Gaussian noise as the measurement error. The measurement model for an unknown function u(x) is m(x) = Au(x) + δε(x) where δ > 0 is the noise magnitude. If ε was an L 2 -function, Tikhonov regularization gives an estimate T α (m) = arg min u∈Hr {∥Au - m∥ 2 L2 + α∥u∥ 2 Hr } for u where α = α(δ) is the regularization parameter. Here penalization of the Sobolev norm ∥u∥ Hr covers the cases of standard Tikhonov regularization (r = 0) and first derivative penalty (r = 1). Realizations of white Gaussian noise are almost never in L 2 , but do belong to H s with probability one if s < 0 is small enough. A modification of Tikhonov regularization theory is presented, covering the case of white Gaussian measurement noise. Furthermore, the convergence of regularized reconstructions to the correct solution as δ → 0 is proven in appropriate function spaces using microlocal analysis. The convergence of the related finite-dimensional problems to the infinite-dimensional problem is also analysed.
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Tikhonov regularization is studied in the case of linear pseudodifferential operator as the forward map and additive white Gaussian noise as the measurement error. The measurement model for an unknown function u(x) is m(x) = Au(x) + δε(x) where δ > 0 is the noise magnitude. If ε was an L 2 -function, Tikhonov regularization gives an estimate T α (m) = arg min u∈Hr {∥Au - m∥ 2 L2 + α∥u∥ 2 Hr } for u where α = α(δ) is the regularization parameter. Here penalization of the Sobolev norm ∥u∥ Hr covers the cases of standard Tikhonov regularization (r = 0) and first derivative penalty (r = 1). Realizations of white Gaussian noise are almost never in L 2 , but do belong to H s with probability one if s < 0 is small enough. A modification of Tikhonov regularization theory is presented, covering the case of white Gaussian measurement noise. Furthermore, the convergence of regularized reconstructions to the correct solution as δ → 0 is proven in appropriate function spaces using microlocal analysis. The convergence of the related finite-dimensional problems to the infinite-dimensional problem is also analysed.