Department of Mathematics
https://hdl.handle.net/1956/1065
Thu, 13 Jun 2024 08:19:39 GMT2024-06-13T08:19:39ZWho benefits from postponement in multi-period supply channel optimization?
https://hdl.handle.net/11250/3130733
Who benefits from postponement in multi-period supply channel optimization?
Azad Gholami, Reza; Sandal, Leif Kristoffer; Ubøe, Jan
Duopolistic price-setting supply channels competing in a bilevel framework have been extensively studied in single-period (static) settings. However, such supply channels typically face uncertain and time-varying demand; and thus require a dynamic analysis. Dynamic channel optimization while addressing uncertain demand has received limited attention due to the highly nested structure of the ensuing equilibrium problems. The level of complexity rises when demand is dependent on current and previous prices. We consider a decentralized (non-cooperative) supply channel whose members, a manufacturer and a retailer, competing in a Stackelberg framework, must address the demand for a perishable commodity within a multi-period discrete-time setting. In the first part of the paper, we propose a constructive theorem providing an explicit solution algorithm to obtain equilibrium states at each period. Next, we prove that the resulting equilibria are subgame perfect. In the second part, we allow the retailer (follower) to postpone the supply and pricing decisions until demand uncertainty is resolved in each period. Using subgame perfection of the equilibria, we propose solution algorithms that use the delayed information obtained by the postponement. Our comparative theorems and simulated scenarios indicate that postponement strategies are always beneficial for the follower, and, for a centralized (cooperative) channel. Whereas in a decentralized channel, due to vertical competition, there may be scenarios wherein postponement strategies, i.e., access to extra information, turn out to be detrimental to the manufacturer (leader).
Mon, 01 Jan 2024 00:00:00 GMThttps://hdl.handle.net/11250/31307332024-01-01T00:00:00ZStatistical Embedding: Beyond Principal Components
https://hdl.handle.net/11250/3130397
Statistical Embedding: Beyond Principal Components
Tjøstheim, Dag Bjarne; Jullum, Martin; Løland, Anders
There has been an intense recent activity in embedding of very high-dimensional and nonlinear data structures, much of it in the data science and machine learning literature. We survey this activity in four parts. In the first part, we cover nonlinear methods such as principal curves, multidimensional scaling, local linear methods, ISOMAP, graph-based methods and diffusion mapping, kernel based methods and random projections. The second part is concerned with topological embedding methods, in particular mapping topological properties into persistence diagrams and the Mapper algorithm. Another type of data sets with a tremendous growth is very high-dimensional network data. The task considered in part three is how to embed such data in a vector space of moderate dimension to make the data amenable to traditional techniques such as cluster and classification techniques. Arguably, this is the part where the contrast between algorithmic machine learning methods and statistical modeling, represented by the so-called stochastic block model, is at its greatest. In the paper, we discuss the pros and cons for the two approaches. The final part of the survey deals with embedding in R2, that is, visualization. Three methods are presented: t-SNE, UMAP and LargeVis based on methods in parts one, two and three, respectively. The methods are illustrated and compared on two simulated data sets; one consisting of a triplet of noisy Ranunculoid curves, and one consisting of networks of increasing complexity generated with stochastic block models and with two types of nodes.
Sun, 01 Jan 2023 00:00:00 GMThttps://hdl.handle.net/11250/31303972023-01-01T00:00:00ZSome recent trends in embeddings of time series and dynamic networks
https://hdl.handle.net/11250/3130068
Some recent trends in embeddings of time series and dynamic networks
Tjøstheim, Dag Bjarne; Jullum, Martin; Løland, Anders
We give a review of some recent developments in embeddings of time series and dynamic networks. We start out with traditional principal components and then look at extensions to dynamic factor models for time series. Unlike principal components for time series, the literature on time-varying nonlinear embedding is rather sparse. The most promising approaches in the literature is neural network based, and has recently performed well in forecasting competitions. We also touch on different forms of dynamics in topological data analysis (TDA). The last part of the article deals with embedding of dynamic networks, where we believe there is a gap between available theory and the behavior of most real world networks. We illustrate our review with two simulated examples. Throughout the review, we highlight differences between the static and dynamic case, and point to several open problems in the dynamic case.
Sun, 01 Jan 2023 00:00:00 GMThttps://hdl.handle.net/11250/31300682023-01-01T00:00:00ZThe highly nonlinear shallow water equation: local well-posedness, wave breaking data and non-existence of sech2 solutions
https://hdl.handle.net/11250/3129112
The highly nonlinear shallow water equation: local well-posedness, wave breaking data and non-existence of sech2 solutions
Hussien Elkhorbatly, Bashar
In the context of the initial data and an amplitude parameter ε, we establish a local existence result for a highly nonlinear shallow water equation on the real line. This result holds in the space Hk as long as k > 5/2. Additionally, we illustrate that the threshold time for the occurrence of wave breaking in the surging type is on the order of ε−1, while plunging breakers do not manifest. Lastly, in accordance with ODE theory, it is demonstrated that there are no exact solitary wave solutions in the form of sech and sech2.
Mon, 01 Jan 2024 00:00:00 GMThttps://hdl.handle.net/11250/31291122024-01-01T00:00:00ZLimited Link of Common Blood Parameters with Tinnitus
https://hdl.handle.net/11250/3128843
Limited Link of Common Blood Parameters with Tinnitus
Bulla, Jan; Brueggemann, Petra; Wrzosek, Małgorzata; Klasing, Sven; Boecking, Benjamin; Basso, Laura; Nyamaa, Amarjargal; Psatha, Stamatina; Rose, Matthias; Mazurek, Birgit
Background: Tinnitus severity is generally assessed by psychometric and audiological instruments. However, no objective measure exists to evaluate the subjective discomfort and suffering caused by this hearing phenomenon. The objective of this work was to determine the possible blood parameters for diagnostics and therapy. Methods: We measured tinnitus distress by using the Tinnitus Questionnaire (TQ) and collected tinnitus-related audiological measures, namely the hearing threshold (HT), tinnitus loudness (TL), and sensation level (SL, i.e., the tinnitus loudness/hearing threshold at a tinnitus frequency). Blood samples were taken from 200 outpatients of the Tinnitus Centre of the Charité, and 46 routine blood count parameters were examined. The possible interactions were determined by (robust) linear models. Results: Tinnitus distress and audiological measurements were largely uncorrelated but could partly be predicted by selected blood parameters. First, the erythrocyte counts predicted tinnitus distress to a small extent. Second, the levels of vitamin D3 explained about 6% of tinnitus loudness and, age-dependently, the hearing threshold variability. Last, the levels of uric acid explained about 5% of the sensation level variability. Conclusions: Tinnitus is a multidimensional phenomenon. The marginal influences of blood markers suggest the possible roles of inflammation and oxidative stress produced by psychological or somatic burdens. Clinically, a vitamin D substitution (in older patients) might have a hearing-protective effect.
Sun, 01 Jan 2023 00:00:00 GMThttps://hdl.handle.net/11250/31288432023-01-01T00:00:00ZQuantifying prior model complexity for subsurface reservoir models
https://hdl.handle.net/11250/3128785
Quantifying prior model complexity for subsurface reservoir models
Mioratina, Nomenjanahary Tanteliniaina; Oliver, Dean
In Bayesian approaches to history matching for subsurface inference, the prior model specifies the uncertain model parameters and the joint probability of those parameters before incorporating production-related data. A good prior model is generally complex enough to capture the future reservoir behavior in the long term, realistic enough to be plausible, consistent with geologic knowledge, and simple enough to allow calibration for data matching. Model complexity is often associated with the number of model parameters, thus the focus on finding the sufficient number of parameters needed for history matching and quantifying future uncertainty. This work explores model choice based on concepts of complexity and informativeness of models for subsurface reservoir models. It focuses on the effect of the misspecification of prior models for assimilating flow data and their predictive accuracy. The concept of the effective number of parameters is used to investigate the suitability of various types of prior models with different levels of complexity, ranging from a highly simplified polynomial trend model to a more realistic multipoint statistical model(MPS) and a family of isotropic Gaussian models and explore the effect of level of model complexity on the robustness of forecasting. The numerical experiments were performed with different combinations of data type, prior informativeness, forecast type, and model type to compare the effect of different prior models on the robustness of the results. The effective number of parameters was computed for each prior model and their accuracy for predicting future reservoir behavior was analyzed. The results suggest that effective model dimension is a useful measure of model complexity for history matching problems, although it is not independent of the data used for model calibration and the number of effective model parameters is generally much smaller than the number of model parameters. In a data-rich problem, realism of a model is much less important than the complexity of a model, while for a problem with few data, realism was beneficial for reliable forecasts.
Sun, 01 Jan 2023 00:00:00 GMThttps://hdl.handle.net/11250/31287852023-01-01T00:00:00ZRigorous derivation of some asymptotic models for free water surfaces and interfaces
https://hdl.handle.net/11250/3127351
Rigorous derivation of some asymptotic models for free water surfaces and interfaces
Paulsen, Martin Oen
Opprinnelsen til moderne matematisk hydrodynamikk kan spores tilbake til 1700-tallet. I 1757 publiserte Euler artikkelen der han introduserte ligninger, i dag kjent som Euler likningene, som kan beskrive væskedynamikk [31]. Hvordan beskrive bølger i væsker ble også studert av Lagrange, Laplace, Poisson, Cauchy og det er fortsatt et aktivt forskningsfelt i dag (for en historisk gjennomgang se [21]). Men fra et praktisk og analytisk perspektiv, er Euler ligningene veldig kompliserte. Det er derfor vanlig å introdusere forenklede modeller, som er karakterisert av dimensjonsløse parametre og som gir en god beskrivelse av den opprinnelige modellen.
I dette arbeidet, utleder vi asymptotiske modeller fra Euler likningene med en fri overflate og for irroterende flyt. Mer spesifikt, så utleder vi flere modeller, hvor vi kvantifiserer feilen med den opprinnelige modellen. Altså, vi beviser at løsningene fra modellen konvergerer til løsningen av referansemodellen med hensyn på noen dimensjonsløse parametre. Vi sier at en modell er fullstendig rettferdiggjort dersom vi kan svare på følgende punkter:
1. Løsningen av referanse modellen eksisterer på den relevante tidsskalaen.
2. Løsningen av den asymptotiske modellen eksisterer (minst) på den samme tidsskalaen.
3. Sist må vi vise at modellene er konsistente. Altså at løsningen av referansemodellen er også en løsning av den asymptotiske modellen opp til gitt toleranse. Deretter må vi vise at differansen mellom disse løsningene er "liten."
Denne oppgaven består hovedsakelig i to deler. Den første delen består av tre artikler som angår utledningen av asymptotiske modeller for å beskrive en enkel væske i grunt vann. I artikkel 1 og 2, studerer vi såkalte Whitham-type modeller som ble tidligere utledet av Emerald [27] hvor han viste at ligningene var konsistente. I dette tilfellet er referansemodellen kjent som vannbølge ligningene, hvor punkt en er bevisst i den viktige artikkelen av Alvarez-Samaniego og Lannes [7]. Dermed, for å fullstendig rettferdiggjøre Whitham-modellene, gjenstår det bare å vise at de eksisterer på den relevante tidsskalaen. I artikkel 3, utleder vi nye modeller, der vi viser at de er konsistente med vannbølge likningene, og hvor presisjonen gir en bedre beskrivelse av endringen av bunnen. Modellene har også en forbedret beskrivelse av dispersjonsforholdet til referansemodellen, og målet er å beskrive bølger som beveger seg over en bunn med bråe endringer. Dette er motivert av de eksperimentelle resultatene i den klassiske artikkelen til Dingemans [23].
I den andre delen av avhandlingen gir vi et bevis for den fullstendige rettferdiggjørelsen av Benjamin-Ono likningen. Dette er en asymptotisk modell som beskriver lange bølger som beveger seg i en retning mellom to fluider. Her er dybden til den ene væsken mye dypere enn den andre. I dette tilfellet er punkt nummer to velkjent, mens det første og tredje punktet er bevist i artikkel 4 av denne avhandlingen. Beviset for at modellene er konsistente er basert på artikkelen til Bona, Lannes, og Saut [14]. Mens eksistens resultatet for det generelle systemet for to væsker på den relevante tidsskalaen er en ikke triviell forlengelse av resultatet til Lannes [40] hvor begge fluidene har en endelig dybde.; The origins of contemporary mathematical hydrodynamics can be traced back to the 18th century. In 1757, Euler published a paper where he introduced equations that could describe the motion of fluids [31], known today as the Euler equations. Other notable figures such as Lagrange, Laplace, Poisson, and Cauchy around this time were also drawn to the study of waves in fluids, which continues to be an active field of research to this day (see [21] for a historical review). However, the Euler equations contain several difficulties related to the complexity of the system, both analytically and in applications. To overcome some of these issues, one typically considers simplified models characterized by dimensionless parameters that describe the main mechanisms involved.
In this work, we rigorously derive asymptotic models from the irrotational Euler equations with a free surface. Specifically, we derive several new models and prove that their solutions converge to the solution of its reference model with respect to the scaling parameters. We say that an asymptotic model is a fully justified if we can answer the following points in the affirmative:
1. The solutions of the reference model exist on the relevant time scale.
2. The solutions of the asymptotic model exist (at least) on the same time scale.
3. We must establish the consistency between the asymptotic model and the reference model. This means the solutions of the reference model solve the asymptotic model up to a certain precision. Then show that the error is "small" when comparing the two solutions.
This thesis consists of two main parts. The first part consists of three papers and concerns the study of asymptotic models in the case of a single fluid in shallow water. In papers 1 and 2, we study Whitham-type systems that were previously derived in the sense of consistency by Emerald [27]. The reference model, in this case, is the \textit{water waves equations}, where the first point is proved in the seminal paper by Alvarez-Samaniego and Lannes [7]. The remaining point in their full justification is to prove the well-posedness of these systems on the relevant time scale. In paper 3, we derive new models, in the sense of consistency, with an improved description of the variation of the bottom toporgaphy. Here, we derive models with improved frequency dispersion, where the goal is to describe waves passing over an obstacle that is studied experimentally in the classical paper by Dingemans [23].
For the second part of the thesis, we prove the full justification of the Benjamin-Ono equation as an asymptotic model for the unidirectional propagation of long internal water waves in a two-layer fluid, where one layer is of great depth. In this case, the second point is well-known, while the first and third point is proved in paper 4. The proof of the consistency is based on the paper by Bona, Lannes, and Saut [14]. While the existence result for the general two-layer fluid model on the relevant time scale is a nontrivial extension of the work of Lannes [40], where both fluids are of finite depth.
Fri, 26 Apr 2024 00:00:00 GMThttps://hdl.handle.net/11250/31273512024-04-26T00:00:00ZThe weight spectrum of two families of Reed-Muller codes
https://hdl.handle.net/11250/3127008
The weight spectrum of two families of Reed-Muller codes
Carlet, Claude Michael; Solé, Patrick
We determine the weight spectra of the Reed-Muller codes RM (m− 3, m) for m ≥ 6 and RM (m − 4, m) for m ≥ 8. The technique used is induction on m, using that the sum of two weights in RM (r −1, m−1) is a weight in RM (r, m), and using the characterization by Kasami and Tokura of the weights in RM (r, m) that lie between its minimum distance 2m−r and the double of this minimum distance. We also de- rive the weights of RM (3, 8), RM (4, 9), by the same technique. We conclude with a conjecture on the weights of RM (m − c, m), where c is fixed and m is large enough.
Sun, 01 Jan 2023 00:00:00 GMThttps://hdl.handle.net/11250/31270082023-01-01T00:00:00ZThe dissolution of a miscible drop rising or falling in another liquid at low Reynolds number
https://hdl.handle.net/11250/3126934
The dissolution of a miscible drop rising or falling in another liquid at low Reynolds number
Nordbotten, Jan Martin; Mossige, Joachim
“A basic and basically unsolved problem in fluid dynamics is to determine the evolution of rising bubbles and falling drops of one miscible liquid in another” [D. D. Joseph and Y. Y. Renardy, Fundamentals of Two-Fluid Dynamics: Part II: Lubricated Transport, Drops and Miscible Liquids (Springer Science & Business Media, 2013), Vol. 4.]. Here, we address this important literature gap and present the first theory predicting the velocity, volume, and composition of such drops at low Reynolds numbers. For the case where the diffusion out of the drop is negligible, we obtain a universal scaling law. For the more general case where diffusion occurs into and out of the drop, the full dynamics is governed by a parameter-free first-order ordinary differential equation, whose closed form solution exists and only depends on the initial condition. Our analysis depends primarily on “drop-scale” effective parameters for the diffusivity through the interfacial boundary layer. We validate our results against experimental data for water drops suspended in syrup, corresponding to certain regimes of the mass exchange ratio between water and syrup, and by this explicitly identify the drop-scale parameters of the theory.
Sun, 01 Jan 2023 00:00:00 GMThttps://hdl.handle.net/11250/31269342023-01-01T00:00:00ZPoroTwin: A Digital Twin for a FluidFlower Rig
https://hdl.handle.net/11250/3126622
PoroTwin: A Digital Twin for a FluidFlower Rig
Keilegavlen, Eirik; Fonn, Eivind; Johannessen, Kjetil Andre; Eikehaug, Kristoffer; Both, Jakub Wiktor; Fernø, Martin; Kvamsdal, Trond; Rasheed, Adil; Nordbotten, Jan Martin
We present a framework for integrated experiments and simulations of tracer transport in heterogeneous porous media using digital twin technology. The physical asset in our setup is a meter-scale FluidFlower rig. The digital twin consists of a traditional physics-based forward simulation tool and a correction technique which compensates for mismatches between simulation results and observations. The latter augments the range of the physics-based simulation and allows us to bridge the gap between simulation and experiments in a quantitative sense. We describe the setup of the physical and digital twin, including data transfer protocols using cloud technology. The accuracy of the digital twin is demonstrated on a case with artificially high diffusion that must be compensated by the correction approach, as well as by simulations in geologically complex media. The digital twin is then applied to control tracer transport by manipulating fluid injection and production in the experimental rig, thereby enabling two-way coupling between the physical and digital twins.
Sun, 01 Jan 2023 00:00:00 GMThttps://hdl.handle.net/11250/31266222023-01-01T00:00:00Z