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Cascades of Scales: Applications and Mathematical Methodologies
No full textThe present special collection presents reviews and genuine research articles addressing real-life applications involving many interacting scales and related methodological developments. Specifically, applications from the biomolecular, material, atmospheric, and geophysical sciences form the core of the dedicated collection, while this does not exclude selected ventures into other areas.
Of particular interest here are problems involving couplings across their scales from small to large and vice versa,1–3 and mathematical concepts and methodologies that have proven to be transferable from specific applications to a generalized framework guided by mathematical abstraction.4–9 In many cases, these abstractions result in related computational codes and software libraries for specific applications of interest to the different disciplines involved.10–13
Being, in this way, inherently interdisciplinary in nature, contributions to this volume give high priority to explaining relations between the application-oriented domain-specific languages used to frame their scientific problem in an application of mathematical physics on the one hand, and the complementary unifying abstract mathematical language that enables the generalization of methodologies to other fields on the other hand.
Thus, the mission of the special issue is two-pronged: It reports on exciting recent methodological developments for challenging problems in mathematical physics involving cascades of interacting scales, and it showcases the importance of precise language(s) and the power abstraction in fostering efficient interdisciplinary research. May it become a well-visited point of reference and point of departure for further conceptual developments. With solid foundations in concrete applications and phrased in the general language of mathematics, they may gain substantial momentum beyond their original discipline of application.
The contributions are published as regular articles of the Journal of Mathematical Physics, while they are linked at the same time in the present themed (online) collection.
Physical systems featuring multiple spatio-temporal scales have posed notoriously difficult challenges to theory and mathematical analysis in past decades, as witnessed, e.g., by a myriad of articles published in the present Journal of Mathematical Physics (JMP), by the establishment in 2003 of the Journal of Multiscale Modeling and Simulation of the Society of Industrial and Applied Mathematics (SIAM), or the references listed below. For a long time, the complexities of real-life physical problems were beyond the reach of mathematical analysis, both formal and rigorous. Thus, theoreticians developed and honed their tools and techniques on physically motivated but substantially simplified models, while researchers from the applied sciences (Physics, Chemistry, Biology, and the Geo-Sciences) pragmatically bridged gaps in theoretical knowledge by well-informed and intuitive closure schemes that would address the ubiquitous multiscale-effects. Over the past 1 1/2 decades, however, mathematical/theoretical multiscale techniques have matured to levels which increasingly allow their transfer to real-life applications, see, e.g., Refs. 4, 6, 7, 14, and 15. In addition, the wealth of observational data available today for physical processes from laboratory setups to the Earth system jointly with the exciting capabilities of recent machine learning techniques have lended tremendous thrust to data-based modeling activities.16–18 The recent rapid rise of data-based modeling in the physical and chemical sciences offers a promising path to accessing a large class of problems across scales. This path comes with the danger, however, of artificial results due to insufficient and/or corrupted data—especially in the face of the curse of dimensionality. For this reason, current research calls strongly for physical models that must be as rigorous as possible to ensure scientific consistency and avoidance of any misuse of data (see, e.g., Refs. 17 and 18). One aim of the present collection is to highlight this aspect, among the others, and build a path to a sustainable research based also on data but with the control of rigor and consistency of physico-mathematical models.
These and similar developments have motivated JMP’s special collection on “Cascades of Scales: Applications and Mathematical Methodologies” of which we provide an overview in this paper.
A very important aspect in this context, besides impressive recent methodological advances and particularly successful real-life applications, concerns the challenges of interdisciplinary cooperation. To a large extent, these challenges are due to the fact that the scientific disciplines participating in a project rely on fine-tuned domain-specific languages for efficient communication. This is compounded by often similar vocabulary, which carries different meanings in different disciplines. A prominent example is the notion of “multiscale” itself, which, depending on the scientific community considered, can be associated with “broad and continuous Fourier spectra” (e.g., in turbulence theory), with the presence of large (asymptotic) scale separations [e.g., in (stiff) chemistry or geophysical fluid dynamics], or with non-asymptotic metastabilities (e.g., in molecular dynamics). Progress in interdisciplinary projects is often hampered by subtle misunderstandings arising from such discipline-dependent “concept overloading.” The authors who have contributed to the present special collection of papers have taken particular care to use precise and unambiguous language that will hopefully help make their contributions efficiently accessible to the readers of the Journal of Mathematical Physics
Generalized Dimension Truncation Error Analysis for High-Dimensional Numerical Integration: Lognormal Setting and Beyond
No full textAbstract.
Partial differential equations (PDEs) with uncertain or random inputs have been considered in many studies of uncertainty quantification. In forward uncertainty quantification, one is interested in analyzing the stochastic response of the PDE subject to input uncertainty, which usually involves solving high-dimensional integrals of the PDE output over a sequence of stochastic variables. In practical computations, one typically needs to discretize the problem in several ways: approximating an infinite-dimensional input random field with a finite-dimensional random field, spatial discretization of the PDE using, e.g., finite elements, and approximating high-dimensional integrals using cubatures such as quasi–Monte Carlo methods. In this paper, we focus on the error resulting from dimension truncation of an input random field. We show how Taylor series can be used to derive theoretical dimension truncation rates for a wide class of problems and we provide a simple checklist of conditions that a parametric mathematical model needs to satisfy in order for our dimension truncation error bound to hold. Some of the novel features of our approach include that our results are applicable to nonaffine parametric operator equations, dimensionally truncated conforming finite element discretized solutions of parametric PDEs, and even compositions of PDE solutions with smooth nonlinear quantities of interest. As a specific application of our method, we derive an improved dimension truncation error bound for elliptic PDEs with lognormally parameterized diffusion coefficients. Numerical examples support our theoretical findings
Deterministic Fokker-Planck Transport -- With Applications to Sampling, Variational Inference, Kernel Mean Embeddings & Sequential Monte Carlo
No full textThe Fokker-Planck equation can be reformulated as a continuity equation, which naturally suggests using the associated velocity field in particle flow methods. While the resulting probability flow ODE offers appealing properties - such as defining a gradient flow of the Kullback-Leibler divergence between the current and target densities with respect to the 2-Wasserstein distance - it relies on evaluating the current probability density, which is intractable in most practical applications. By closely examining the drawbacks of approximating this density via kernel density estimation, we uncover opportunities to turn these limitations into advantages in contexts such as variational inference, kernel mean embeddings, and sequential Monte Carlo
Less interaction with forward models in Langevin dynamics: Enrichment and Homotopy Martin , Robert , David
No full textEnsemble methods have become ubiquitous for the solution of Bayesian inference problems. State-of-the-art Langevin samplers such as the Ensemble Kalman Sampler (EKS), Affine Invariant Langevin Dynamics (ALDI) or its extension using weighted covariance estimates rely on successive evaluations of the forward model or its gradient. A main drawback of these methods hence is their vast number of required forward calls as well as their possible lack of convergence in the case of more involved posterior measures such as multimodal distributions. The goal of this paper is to address these challenges to some extend. First, several possible adaptive ensemble enrichment strategies that successively enlarge the number of particles in the underlying Langevin dynamics are discusses that in turn lead to a significant reduction of the total number of forward calls. Second, analytical consistency guarantees of the ensemble enrichment method are provided for linear forward models. Third, to address more involved target distributions, the method is extended by applying adapted Langevin dynamics based on a homotopy formalism for which convergence is proved. Finally, numerical investigations of several benchmark problems illustrates the possible gain of the proposed method, comparing it to state-of-the-art Langevin samplers
Gendered Gatekeeping in the Recruitment and Support of (Prospective) PhDs and Postdocs in a Mathematical Cluster of Excellence
No full textGender disparities persist in the field of mathematics in German science and academia. From one scientific career level to the next, the proportion of female scientists decreases and women are still underrepresented in the top positions in science/academia. Gatekeeping is assumed to be one reason for the persistence of this disparity. Gatekeepers influence access to and advancement in the science system: They recruit researchers and provide support in the form of knowledge relevant for career advancement and open the way for further career steps, i.e., they hold an important decision-making position regarding the future of prospective female scientists. The study investigates if and if so, how gender disparities are reinforced in recruitment and support processes by gatekeepers in a mathematical cluster of excellence in Germany. Qualitative semi-structured interviews were conducted with 44 scientific gatekeepers in leadership positions. The results show how recruitment and support practices, perceptions, and criteria of scientific potential are interwoven with gender stereotypes, thereby creating potential barriers for female PhD students and postdocs
Efficient Mapping of Phase Diagrams with Conditional Boltzmann Generators
No full textThe accurate prediction of phase diagrams is of central importance for both the fundamental understanding of materials as well as for technological applications in material sciences. However, the computational prediction of the relative stability between phases based on their free energy is a daunting task, as traditional free energy estimators require a large amount of simulation data to obtain uncorrelated equilibrium samples over a grid of thermodynamic states. In this work, we develop deep generative machine learning models based on the Boltzmann Generator approach for entire phase diagrams, employing normalizing flows conditioned on the thermodynamic states, e.g., temperature and pressure, that they map to. By training a single normalizing flow to transform the equilibrium distribution sampled at only one reference thermodynamic state to a wide range of target temperatures and pressures, we can efficiently generate equilibrium samples across the entire phase diagram. Using a permutation-equivariant architecture allows us, thereby, to treat solid and liquid phases on the same footing. We demonstrate our approach by predicting the solid-liquid coexistence line for a Lennard-Jones system in excellent agreement with state-of-the-art free energy methods while significantly reducing the number of energy evaluations needed
Computationally feasible bounds for the free energy of nonequilibrium steady states, applied to simple models of heat conduction
No full textIn this paper, we study computationally feasible bounds for relative free energies between two many-particle systems. Specifically, we consider systems out of equilibrium that admit a nonequilibrium steady state that is reached asymptotically in the long-time limit. The bounds that we suggest are based on the well-known Bogoliubov inequality and variants of Gibbs' and Donsker–Varadhan variational principles. As a general paradigm, we consider systems of oscillators coupled to heat baths at different temperatures. For such systems, we define the free energy of the system relative to any given reference system in terms of the Kullback–Leibler divergence between steady states. By employing a two-sided Bogoliubov inequality and a mean-variance approximation of the free energy (or cumulant generating function), we can efficiently estimate the free energy cost needed in passing from the reference system to the system out of equilibrium (characterised by a temperature gradient). A specific test case to validate our bounds are harmonic oscillator chains with ends that are coupled to Langevin thermostats at different temperatures; such a system is simple enough to allow for analytic calculations and general enough to be used as a prototype to estimate, e.g. heat fluxes or interface effects in a larger class of nonequilibrium particle systems
Lattice-based kernel approximation and serendipitous weights for parametric PDEs in very high dimensions
No full textWe describe a fast method for solving elliptic partial differential equations (PDEs) with uncertain coefficients using kernel interpolation at a lattice point set. By representing the input random field of the system using the model proposed by Kaarnioja, Kuo, and Sloan (SIAM J.~Numer.~Anal.~2020), in which a countable number of independent random variables enter the random field as periodic functions, it was shown by Kaarnioja, Kazashi, Kuo, Nobile, and Sloan (Numer.~Math.~2022) that the lattice-based kernel interpolant can be constructed for the PDE solution as a function of the stochastic variables in a highly efficient manner using fast Fourier transform (FFT). In this work, we discuss the connection between our model and the popular ``affine and uniform model'' studied widely in the literature of uncertainty quantification for PDEs with uncertain coefficients. We also propose a new class of weights entering the construction of the kernel interpolant -- \emph{serendipitous weights} -- which dramatically improve the computational performance of the kernel interpolant for PDE problems with uncertain coefficients, and allow us to tackle function approximation problems up to very high dimensionalities. Numerical experiments are presented to showcase the performance of the serendipitous weights
Kinetic Trapping of Charge-Transfer Molecules at Metal Interfaces
No full textDespite the common expectation that conjugated organic molecules on metals adsorb in a flat-lying layer, several recent studies have found coverage-dependent transitions to upright-standing phases, which exhibit notably different physical properties. In this work, we argue that from an energetic perspective, thermodynamically stable upright-standing phases may be more common than hitherto thought. However, for kinetic reasons, this phase may often not be observed experimentally. Using first-principles kinetic Monte Carlo simulations, we find that the structure with lower molecular density is (almost) always formed first, reminiscent of Ostwald’s rule of stages. The phase transitions to the upright-standing phase are likely to be kinetically hindered under the conditions typically used in surface science. The simulation results are experimentally confirmed for the adsorption of tetracyanoethylene on Cu(111) using infrared and X-ray photoemission spectroscopy. Investigating both the role of the growth conditions and the energetics of the interface, we find that the time for the phase transition is determined mostly by the deposition rate and, thus, is mostly independent of the nature of the molecule
Decoil: Reconstructing Extrachromosomal DNA Structural Heterogeneity from Long-Read Sequencing Data
No full textCircular extrachromosomal DNA (ecDNA) is a form of oncogene amplification found across cancer types and associated with poor outcome in patients. EcDNA can be structurally complex and contain rearranged DNA sequences derived from multiple chromosome locations. As the structure of ecDNA can impact oncogene regulation and may indicate mechanisms of its formation, disentangling it at high resolution from sequencing data is essential. Even though methods have been developed to identify and reconstruct ecDNA in cancer genome sequencing, it remains challenging to resolve complex ecDNA structures, in particular amplicons with shared genomic footprints. We here introduce Decoil, a computational method which combines a breakpoint-graph approach with LASSO regression to reconstruct complex ecDNA and deconvolve co-occurring ecDNA elements with overlapping genomic footprints from long-read nanopore sequencing. Decoil outperforms de-novo assembly and alignment-based methods in simulated long-read sequencing data for both simple and complex ecDNAs. Applying Decoil on whole genome sequencing data uncovered different ecDNA topologies and explored ecDNA structure heterogeneity in neuroblastoma tumors and cell lines, indicating that this method may improve ecDNA structural analyzes in cancer