arXiv subjects

Jonas Sauer

Publications and source records attributed to Jonas Sauer.

At least 19 recordsLinked to original sources

arXiv2026

Pathwise Global-in-Time Existence for the generalised KPZ Equation in the Full Subcritical Regime

We provide a pathwise proof of global-in-time well-posedness for the generalised KPZ equation in the full subcritical regime by an adaptation of the strategy recently applied to the generalised Parabolic Anderson Model in [ES26]. Since this strategy relies crucially on the assumption that control of the supremum norm is sufficient to continue the solution, the main additional ingredient required is a treatment of the initial layer for gKPZ with merely $L^\infty$ initial data, rather than the more usual setting of $C^\theta$ initial data with $\theta > 0$. Our approach is based on an expansion around a deterministic profile followed by the introduction of an integrating factor in order to remove the terms which have critical scaling at time $0$. For pedagogical purposes, we first demonstrate the proof techniques in the case of the standard $(1+1)$-dimensional KPZ equation before turning to the more computationally involved case of generalised KPZ.

math.AP↗

arXiv2026

An $L^p$-Theory for Time-Periodic Mixed-Order Partial Differential Equations under General Boundary Conditions

We develop an $L^p$-theory for time-periodic boundary value problems associated with partial differential equations and systems of mixed order. Our approach is based on anisotropic function spaces described by order functions and their associated Newton polygons. We develop a trace theory on the half-space, including a characterization of the trace spaces by real interpolation. For general boundary conditions, we introduce an abstract framework in which well-posed\-ness is characterized by a complementing condition formulated in terms of the traces of solutions. In particular, the admissible data space, including the compatibility conditions induced by the boundary operators, emerges naturally from the abstract framework. For mixed-order differential operators, the complementing condition is reduced to the invertibility of a complemented boundary matrix, yielding an explicit criterion for well-posedness in $L^p$-based spaces. The resulting theory applies to general Newton polygon structures and allows for boundary operators involving time derivatives. As applications, we establish time-periodic $L^p$-well-posedness for the Cahn--Hilliard--Gurtin system and for parabolic problems with dynamic boundary conditions.

math.AP↗

arXiv2026

Homogenization of regularized Oldroyd-type fluids

We study homogenization of a regularized viscoelastic Oldroyd-type model in a periodically perforated bounded domain. The system describes an incompressible non-Newtonian fluid coupled to an elastic extra stress tensor and includes both nonlinear viscosity and nonlinear stress diffusion effects. The governing model, introduced by Kreml, Pokorn\'y, and \v{S}alom (2015), covers Oldroyd-A- and Oldroyd-B-type constitutive laws. We establish qualitative and quantitative homogenization results in suitable scaling regimes and show convergence toward an effective Darcy law on the macroscopic domain. In particular, we prove that, under appropriate assumptions on the scaling parameters, the polymeric stress does not contribute to the effective limit equation. The analysis combines uniform estimates, oscillating test-function techniques, and a relative energy method, and additionally yields a weak-strong uniqueness principle for the viscoelastic system.

math.AP↗

arXiv2026

T-REX: Fast and Dynamic Journey Planning for Continental-Scale Public Transit Networks

We present T-REX (Transfer-Ranked EXploration), a new algorithm for journey planning in public transit networks on the country and continental scale. Our algorithm applies the principles of multi-level overlays to Trip-Based Public Transit Routing (TB). Using a multi-level partition of the network, T-REX identifies transfers between trips that are relevant for long-distance travel in a short precomputation phase. This information is then used to prune irrelevant local transfers during a query. Like other state-of-the-art algorithms, T-REX Pareto-optimizes arrival time and the number of used trips. T-REX dramatically outperforms previous overlay-based algorithms for three key reasons: (1) a better partition, (2) reducing the search space by focusing on transfers rather than trips, and (3) a redesigned query algorithm with improved memory efficiency and throughput. As a result, T-REX answers queries in less than 10ms on a network of Europe, including local and long-distance transit. This constitutes a speedup of 20 compared to TB and 80 compared to algorithms without preprocessing. The memory footprint is moderate and the precomputation takes only two minutes, while real-time schedule updates can be incorporated in a few seconds. These properties make T-REX the first public transit journey planning algorithm that fulfills the requirements of interactive real-time applications on the continental scale.

cs.SI↗

arXiv2026

Graph-Based Nearest-Neighbor Search without the Spread

$\renewcommand{\Re}{\mathbb{R}}$Recent work showed how to construct nearest-neighbor graphs of linear size, on a given set $P$ of $n$ points in $\Re^d$, such that one can answer approximate nearest-neighbor queries in logarithmic time in the spread. Unfortunately, the spread might be unbounded in $n$, and an interesting theoretical question is how to remove the dependency on the spread. Here, we show how to construct an external linear-size data structure that, combined with the linear-size graph, allows us to answer ANN queries in logarithmic time in $n$.

cs.CG↗

arXiv2025

The Time-Periodic Cahn-Hilliard-Gurtin System on the Half Space as a Mixed-Order System with General Boundary Conditions

A well-posedness and maximal regularity result for the time-periodic Cahn-Hilliard-Gurtin system in the half space is proved. For this purpose, we introduce a novel class of complementing boundary conditions, extending the classical Lopatinski\u{\i}-Shapiro conditions from elliptic and parabolic theory to time-periodic mixed-order systems with general boundary conditions. Moreover, we show that the classical Lopatinski\u{\i}-Shapiro conditions are in general insufficient for well-posedness of mixed-order systems.

math.AP↗

arXiv2025

Limited-Range Multilinear Off-Diagonal Extrapolation and Weighted Transference Principle

Multilinear $L^p$ extrapolation results are established in a limited-range, multilinear, and off-diagonal setting for mixed-norm Lebesgue spaces over $\sigma$-finite measure spaces. Integrability exponents are allowed in the full range $(0,\infty]$. We detach the exponents for the weight classes completely from the exponents for the initial and target spaces for the extrapolation except for the basic consistency condition. This enables to cover the full range $(0,\infty]$ for all integrability exponents and provides new insights into the dependency of the extrapolated bounds on the weight characteristic. Certain endpoint results are new even for $\mathbb{R}^d$. Additionally, in the setting of compact abelian groups, a weighted transference principle is established.

math.AP↗

arXiv2025

The Expiration Streaming Model: Diameter, $k$-Center, Counting, Sampling, and Friends

An important thread in the study of data-stream algorithms focuses on settings where stream items are active only for a limited time. We introduce a new expiration model, where each item arrives with its own arbitrary expiration time. The special case where items expire in the order that they arrive, which we call consistent expirations, contains the classical sliding-window model of Datar, Gionis, Indyk, and Motwani [SICOMP 2002] and its timestamp-based variant of Braverman and Ostrovsky [FOCS 2007]. Our first set of results explores the expiration streaming model and presents algorithms for several fundamental problems, including approximate counting, uniform sampling, and weighted sampling by efficiently tracking active items without explicitly storing them all. Naturally, these algorithms have many immediate applications, e.g., to range counting. Our second and main set of results for the expiration model designs algorithms for the diameter and k-center problems, where items are points in a metric space. Our results significantly extend those known for the special case of sliding-window streams by Cohen-Addad, Schwiegelshohn, and Sohler [ICALP 2016], and obtain a strictly better approximation factor for the diameter in the important special case of high-dimensional Euclidean metrics. We develop new decomposition and coordination techniques along with a geometric dominance framework to filter out redundant points based on both temporal and spatial proximity.

cs.DS↗

arXiv2025

Bicriteria Polygon Aggregation with Arbitrary Shapes

We study the problem of aggregating a set of polygons by covering them with disjoint representative regions, thereby inducing a clustering of the polygons. Equivalently, this can be seen as a fence enclosure problem, where the goal is to surround the polygons with a set of closed curves. Our objective is to minimize a weighted sum of the total area and the total perimeter of the regions, which naturally extends other fencing problems and has applications in geographical information systems. Previously, this objective was only studied in a restricted variant, in which the boundary curves of the regions must be selected from a fixed subdivision of the plane. It is natural to ask whether the problem is still tractable if this restriction is removed, allowing output regions to be bounded by arbitrary curves. We provide a positive answer in the form of an algorithm with runtime $\mathcal{\tilde{O}}(n^4)$, where $n$ is the number of input vertices. To achieve this, we fully characterize the optimal solutions by showing that their boundaries are composed of input edges and circular arcs of constant radius. Additionally, we consider the parametric problem, where for every weighting factor we seek a solution that is optimal for that trade-off of area and perimeter. We show that $\mathcal{O}(n^2)$ combinatorial solutions suffice to describe all optimal solutions across all weighting factors, and provide both an exact algorithm and an approximation scheme. To make the algorithms scalable in practice, we develop engineering techniques that exploit structural properties of the solutions. Our experimental evaluation on real-world data shows linear runtime in practice, even for the parametric variant.

cs.CG↗

arXiv2025

Weisfeiler and Leman Follow the Arrow of Time: Expressive Power of Message Passing in Temporal Event Graphs

An important characteristic of temporal graphs is how the directed arrow of time influences their causal topology, i.e., which nodes can possibly influence each other causally via time-respecting paths. The resulting patterns are often neglected by temporal graph neural networks (TGNNs). To formally analyze the expressive power of TGNNs, we lack a generalization of graph isomorphism to temporal graphs that fully captures their causal topology. Addressing this gap, we introduce the notion of consistent event graph isomorphism, which utilizes a time-unfolded representation of time-respecting paths in temporal graphs. We compare this definition with existing notions of temporal graph isomorphisms. We illustrate and highlight the advantages of our approach and develop a temporal generalization of the Weisfeiler-Leman algorithm to heuristically distinguish non-isomorphic temporal graphs. Building on this theoretical foundation, we derive a novel message passing scheme for temporal graph neural networks that operates on the event graph representation of temporal graphs. An experimental evaluation shows that our approach performs well in a temporal graph classification experiment.

cs.LG↗

arXiv2024

Schauder Estimates for Germs by Scaling

In this expository note, we show that the blow-up arguments of L. Simon adapt well to the corresponding Schauder theory of germs used in the study of singular SPDEs. We illustrate this through some representative examples. As in the classical PDE framework, the argument relies only on the scaling properties of the germ semi-norms and the Liouville principle for the operator.

math.AP↗

arXiv2024

Well-Posedness and Regularity of the Heat Equation with Robin Boundary Conditions in the Two-Dimensional Wedge

Well-posedness and higher regularity of the heat equation with Robin boundary conditions in an unbounded two-dimensional wedge is established in an $L^{2}$-setting of monomially weighted spaces. A mathematical framework is developed which allows to obtain arbitrarily high regularity without a smallness assumption on the opening angle of the wedge. The challenging aspect is that the resolvent problem exhibits two breakings of the scaling invariance, one in the equation and one in the boundary condition.

math.AP↗

arXiv2024

A Simpler Approach for Monotone Parametric Minimum Cut: Finding the Breakpoints in Order

We present parametric breadth-first search (PBFS), a new algorithm for solving the parametric minimum cut problem in a network with source-sink-monotone capacities. The objective is to find the set of breakpoints, i.e., the points at which the minimum cut changes. It is well known that this problem can be solved in the same asymptotic runtime as the static minimum cut problem. However, existing algorithms that achieve this runtime bound involve fairly complicated steps that are inefficient in practice. PBFS uses a simpler approach that discovers the breakpoints in ascending order, which allows it to achieve the desired runtime bound while still performing well in practice. We evaluate our algorithm on benchmark instances from polygon aggregation and computer vision. Polygon aggregation was recently proposed as an application for parametric minimum cut, but the monotonicity property has not been exploited fully. PBFS outperforms the state of the art on most benchmark instances, usually by a factor of 2-3. It is particularly strong on instances with many breakpoints, which is the case for polygon aggregation. Compared to the existing min-cut-based approach for polygon aggregation, PBFS scales much better with the instance size. On large instances with millions of vertices, it is able to compute all breakpoints in a matter of seconds.

cs.DS↗

arXiv2024

Well-posedness of the Stokes equations on a wedge with Navier-slip boundary conditions

We consider the incompressible and stationary Stokes equations on an infinite two-dimensional wedge with non-scaling invariant Navier-slip boundary conditions. We prove well-posedness and higher regularity of the Stokes problem in a certain class of weighted Sobolev spaces. The novelty of this work is the occurrence of two different scalings in the boundary condition, which is not treated so far for the Stokes system in unbounded wedge-type domains. These difficulties are overcome by first constructing a variational solution in a second order weighted Sobolev space and subsequently proving higher regularity up to the tip of the wedge by employing an iterative scheme. We believe that this method can be used for other problems with variational structure and multiple scales.

math.AP↗

arXiv2023

FLASH-TB: Integrating Arc-Flags and Trip-Based Public Transit Routing

We present FLASH-TB, a journey planning algorithm for public transit networks that combines Trip-Based Public Transit Routing (TB) with the Arc-Flags speedup technique. The basic idea is simple: The network is partitioned into a configurable number of cells. For each cell and each possible transfer between two vehicles, the algorithm precomputes a flag that indicates whether the transfer is required to reach the cell. During a query, only flagged transfers are explored. Our algorithm improves upon previous attempts to apply Arc-Flags to public transit networks, which saw limited success due to conflicting rules for pruning the search space. We show that these rules can be reconciled while still producing correct results. Because the number of cells is configurable, FLASH-TB offers a tradeoff between query time and memory consumption. It is significantly more space-efficient than existing techniques with a comparable preprocessing time, which store generalized shortest-path trees: to match their query performance, it requires up to two orders of magnitude less memory. The fastest configuration of FLASH-TB achieves a speedup of more than two orders of magnitude over TB, offering sub-millisecond query times even on large countrywide networks.

cs.DS↗

arXiv2023

Optimal Regularity in Time and Space for Nonlocal Porous Medium Type Equations

A broad class of possibly non-unique generalized kinetic solutions to hyperbolic-parabolic PDEs is introduced. Optimal regularity estimates in time and space for such solutions to nonlocal, and spatially inhomogeneous variants of the porous medium equation are shown in the scale of Sobolev spaces. The optimality of these results is shown by comparison to the nonlocal Barenblatt solution. The regularity results are used in order to obtain existence of generalized kinetic solutions.

math.AP↗

arXiv2023

Fast and Delay-Robust Multimodal Journey Planning

We study journey planning in multimodal networks consisting of public transit plus an unrestricted transfer mode (e.g., walking or cycling). In order to provide good results in practice, algorithms must account for vehicle delays. Delay-responsive algorithms receive a continuous stream of delay updates and must return optimal journeys in the currently known delay scenario. Updates are incorporated in an update phase, which must be fast (e.g., a few seconds). The fastest known approach for multimodal journey planning is ULTRA, which precomputes shortcuts representing transfers between vehicles. This allows query algorithms to find Pareto-optimal journeys regarding arrival time and the number of public transit trips without any performance loss compared to pure public transit networks. However, the precomputation phase does not account for delays and is too slow to rerun during the update phase. We present Delay-ULTRA, a delay-responsive variant of ULTRA. Since accounting for all theoretically possible delays would yield an impractically large set of shortcuts, our approach precomputes shortcuts that are provably sufficient as long as delays do not exceed a configurable limit (e.g., 5 minutes). To handle delays above the limit (which are less frequent in practice), we propose a heuristic search for missing shortcuts that is fast enough to be run during the update phase. Our experimental evaluation on real-world data shows that Delay-ULTRA fails to find less than 0.02% of optimal journeys on metropolitan and mid-sized country networks, and 0.16% on the much larger Germany network. Considering that the available delay information in realistic applications is never perfectly accurate, these error rates are negligible. Query speed is at most twice as slow as ULTRA without delay information, and up to 8 times faster than the fastest exact algorithm, which does not require a preprocessing phase.

cs.DS↗

arXiv2023

Arc-Flags Meet Trip-Based Public Transit Routing

We present Arc-Flag TB, a journey planning algorithm for public transit networks which combines Trip-Based Public Transit Routing (TB) with the Arc-Flags speedup technique. Compared to previous attempts to apply Arc-Flags to public transit networks, which saw limited success, our approach uses stronger pruning rules to reduce the search space. Our experiments show that Arc-Flag TB achieves a speedup of up to two orders of magnitude over TB, offering query times of less than a millisecond even on large countrywide networks. Compared to the state-of-the-art speedup technique Trip-Based Public Transit Routing Using Condensed Search Trees (TB-CST), our algorithm achieves similar query times but requires significantly less additional memory. Other state-of-the-art algorithms which achieve even faster query times, e.g., Public Transit Labeling, require enormous memory usage. In contrast, Arc-Flag TB offers a tradeoff between query performance and memory usage due to the fact that the number of regions in the network partition required by our algorithm is a configurable parameter. We also identify an issue in the transfer precomputation of TB that affects both TB-CST and Arc-Flag TB, leading to incorrect answers for some queries. This has not been previously recognized by the author of TB-CST. We provide discussion on how to resolve this issue in the future. Currently, Arc-Flag TB answers 1-6% of queries incorrectly, compared to over 20% for TB-CST on some networks.

cs.DS↗