@article{KellerMuench2019,
  author    = {Keller, Matthias and M{\"u}nch, Florentin},
  title     = {A new discrete Hopf-Rinow theorem},
  series = {Discrete Mathematics},
  volume    = {342},
  journal   = {Discrete Mathematics},
  number    = {9},
  publisher = {Elsevier},
  address   = {Amsterdam},
  issn      = {0012-365X},
  doi       = {10.1016/j.disc.2019.03.014},
  pages     = {2751 -- 2757},
  year      = {2019},
  abstract  = {We prove a version of the Hopf-Rinow theorem with respect to path metrics on discrete spaces. The novel aspect is that we do not a priori assume local finiteness but isolate a local finiteness type condition, called essentially locally finite, that is indeed necessary. As a side product we identify the maximal weight, called the geodesic weight, generating the path metric in the situation when the space is complete with respect to any of the equivalent notions of completeness proven in the Hopf-Rinow theorem. As an application we characterize the graphs for which the resistance metric is a path metric induced by the graph structure.},
  language  = {en}
}
@article{RaschSchindler2005,
  author    = {Rasch, T. and Schindler, R.},
  title     = {A new condensation principle},
  issn      = {1432-0665},
  year      = {2005},
  abstract  = {We generalize del(A), which was introduced in [Schinfinity], to larger cardinals. For a regular cardinal kappa>N-0 we denote by del(kappa)(A) the statement that Asubset of or equal tokappa and for all regular theta>kappa(o), {X is an element of[L-theta[A]](<) : X \&AND; \&ISIN; \&AND; otp (X \&AND; Ord) \&ISIN; Card (L[A\&AND;X\&AND;])} is stationary in [L-[A]](<). It was shown in [Sch\&INFIN;] that \&DEL;(N1) (A) can hold in a set-generic extension of L. We here prove that \&DEL;(N2) (A) can hold in a set-generic extension of L as well. In both cases we in fact get equiconsistency theorems. This strengthens results of [Ra00] and [Ran01]. \&DEL;(N3) () is equivalent with the existence of 0\#},
  language  = {en}
}
@book{KamaniMansouri1995,
  author    = {Kamani, D. and Mansouri, R.},
  title     = {A new class of inhomogeneous cosmological solutions},
  series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik},
  volume    = {1995, 07},
  journal   = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik},
  publisher = {Univ.},
  address   = {Potsdam},
  pages     = {12 S.},
  year      = {1995},
  language  = {en}
}
@article{HedayatMahmoudiSchulze2018,
  author    = {Hedayat Mahmoudi, Mahdi and Schulze, Bert-Wolfgang},
  title     = {A new approach to the second order edge calculus},
  series = {Journal of pseudo-differential operators and applications},
  volume    = {9},
  journal   = {Journal of pseudo-differential operators and applications},
  number    = {2},
  publisher = {Springer},
  address   = {Basel},
  issn      = {1662-9981},
  doi       = {10.1007/s11868-017-0191-2},
  pages     = {265 -- 300},
  year      = {2018},
  abstract  = {We establish essential steps of an iterative approach to operator algebras, ellipticity and Fredholm property on stratified spaces with singularities of second order. We cover, in particular, corner-degenerate differential operators. Our constructions are focused on the case where no additional conditions of trace and potential type are posed, but this case works well and will be considered in a forthcoming paper as a conclusion of the present calculus.},
  language  = {en}
}
@article{ShinZoellerHolschneideretal.2011,
  author    = {Shin, Seoleun and Z{\"o}ller, Gert and Holschneider, Matthias and Reich, Sebastian},
  title     = {A multigrid solver for modeling complex interseismic stress fields},
  series = {Computers \& geosciences : an international journal devoted to the publication of papers on all aspects of geocomputation and to the distribution of computer programs and test data sets ; an official journal of the International Association for Mathematical Geology},
  volume    = {37},
  journal   = {Computers \& geosciences : an international journal devoted to the publication of papers on all aspects of geocomputation and to the distribution of computer programs and test data sets ; an official journal of the International Association for Mathematical Geology},
  number    = {8},
  publisher = {Elsevier},
  address   = {Oxford},
  issn      = {0098-3004},
  doi       = {10.1016/j.cageo.2010.11.011},
  pages     = {1075 -- 1082},
  year      = {2011},
  abstract  = {We develop a multigrid, multiple time stepping scheme to reduce computational efforts for calculating complex stress interactions in a strike-slip 2D planar fault for the simulation of seismicity. The key elements of the multilevel solver are separation of length scale, grid-coarsening, and hierarchy. In this study the complex stress interactions are split into two parts: the first with a small contribution is computed on a coarse level, and the rest for strong interactions is on a fine level. This partition leads to a significant reduction of the number of computations. The reduction of complexity is even enhanced by combining the multigrid with multiple time stepping. Computational efficiency is enhanced by a factor of 10 while retaining a reasonable accuracy, compared to the original full matrix-vortex multiplication. The accuracy of solution and computational efficiency depend on a given cut-off radius that splits multiplications into the two parts. The multigrid scheme is constructed in such a way that it conserves stress in the entire half-space.},
  language  = {en}
}
@phdthesis{Zass2021,
  author    = {Zass, Alexander},
  title     = {A multifaceted study of marked Gibbs point processes},
  doi       = {10.25932/publishup-51277},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-512775},
  school      = {Universit{\"a}t Potsdam},
  pages     = {vii, 104},
  year      = {2021},
  abstract  = {This thesis focuses on the study of marked Gibbs point processes, in particular presenting some results on their existence and uniqueness, with ideas and techniques drawn from different areas of statistical mechanics: the entropy method from large deviations theory, cluster expansion and the Kirkwood--Salsburg equations, the Dobrushin contraction principle and disagreement percolation. We first present an existence result for infinite-volume marked Gibbs point processes. More precisely, we use the so-called entropy method (and large-deviation tools) to construct marked Gibbs point processes in R^d under quite general assumptions. In particular, the random marks belong to a general normed space S and are not bounded. Moreover, we allow for interaction functionals that may be unbounded and whose range is finite but random. The entropy method relies on showing that a family of finite-volume Gibbs point processes belongs to sequentially compact entropy level sets, and is therefore tight. We then present infinite-dimensional Langevin diffusions, that we put in interaction via a Gibbsian description. In this setting, we are able to adapt the general result above to show the existence of the associated infinite-volume measure. We also study its correlation functions via cluster expansion techniques, and obtain the uniqueness of the Gibbs process for all inverse temperatures β and activities z below a certain threshold. This method relies in first showing that the correlation functions of the process satisfy a so-called Ruelle bound, and then using it to solve a fixed point problem in an appropriate Banach space. The uniqueness domain we obtain consists then of the model parameters z and β for which such a problem has exactly one solution. Finally, we explore further the question of uniqueness of infinite-volume Gibbs point processes on R^d, in the unmarked setting. We present, in the context of repulsive interactions with a hard-core component, a novel approach to uniqueness by applying the discrete Dobrushin criterion to the continuum framework. We first fix a discretisation parameter a>0 and then study the behaviour of the uniqueness domain as a goes to 0. With this technique we are able to obtain explicit thresholds for the parameters z and β, which we then compare to existing results coming from the different methods of cluster expansion and disagreement percolation. Throughout this thesis, we illustrate our theoretical results with various examples both from classical statistical mechanics and stochastic geometry.},
  language  = {en}
}
@book{BoeckmannNiebsch1996,
  author    = {B{\"o}ckmann, Christine and Niebsch, Jenny},
  title     = {A mollifier method for aerosol size},
  series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik},
  volume    = {1996, 07},
  journal   = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik},
  publisher = {Univ.},
  address   = {Potsdam},
  pages     = {[5] Bl.},
  year      = {1996},
  language  = {en}
}
@article{BergemannReich2010,
  author    = {Bergemann, Kay and Reich, Sebastian},
  title     = {A mollified ensemble Kalman filter},
  issn      = {0035-9009},
  doi       = {10.1002/Qj.672},
  year      = {2010},
  abstract  = {It is well recognized that discontinuous analysis increments of sequential data assimilation systems, such as ensemble Kalman filters, might lead to spurious high-frequency adjustment processes in the model dynamics. Various methods have been devised to spread out the analysis increments continuously over a fixed time interval centred about the analysis time. Among these techniques are nudging and incremental analysis updates (IAU). Here we propose another alternative, which may be viewed as a hybrid of nudging and IAU and which arises naturally from a recently proposed continuous formulation of the ensemble Kalman analysis step. A new slow-fast extension of the popular Lorenz-96 model is introduced to demonstrate the properties of the proposed mollified ensemble Kalman filter.},
  language  = {en}
}
@book{Ramlau1997,
  author    = {Ramlau, Ronny},
  title     = {A modified Landweber-method for inverse problems},
  series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik},
  volume    = {1997, 03},
  journal   = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik},
  publisher = {Univ.},
  address   = {Potsdam},
  pages     = {24 Bl.},
  year      = {1997},
  language  = {en}
}
@article{PornsawadSapsakulBoeckmann2019,
  author    = {Pornsawad, Pornsarp and Sapsakul, Nantawan and B{\"o}ckmann, Christine},
  title     = {A modified asymptotical regularization of nonlinear ill-posed problems},
  series = {Mathematics},
  volume    = {7},
  journal   = {Mathematics},
  edition   = {5},
  publisher = {MDPI},
  address   = {Basel, Schweiz},
  issn      = {2227-7390},
  doi       = {10.3390/math7050419},
  pages     = {19},
  year      = {2019},
  abstract  = {In this paper, we investigate the continuous version of modified iterative Runge-Kutta-type methods for nonlinear inverse ill-posed problems proposed in a previous work. The convergence analysis is proved under the tangential cone condition, a modified discrepancy principle, i.e., the stopping time T is a solution of ∥𝐹(𝑥𝛿(𝑇))-𝑦𝛿∥=𝜏𝛿+ for some 𝛿+>𝛿, and an appropriate source condition. We yield the optimal rate of convergence.},
  language  = {en}
}