Stability of explicit one-step methods for P1-finite element approximation of linear diffusion equations on anisotropic meshes

We study the stability of explicit one-step integration schemes for the linear finite element approximation of linear parabolic equations. The derived bound on the largest permissible time step is tight for any mesh and any diffusion matrix within a factor of $2(d+1)$, where $d$ is the spatial dimension. Both full mass matrix and mass lumping are considered. The bound reveals that the stability condition is affected by two factors. The first one depends on the number of mesh elements and corresponds to the classic bound for the Laplace operator on a uniform mesh. The other factor reflects the effects of the interplay of the mesh geometry and the diffusion matrix. It is shown that it is not the mesh geometry itself but the mesh geometry in relation to the diffusion matrix that is crucial to the stability of explicit methods. When the mesh is uniform in the metric specified by the inverse of the diffusion matrix, the stability condition is comparable to the situation with the Laplace operator on a uniform mesh. Numerical results are presented to verify the theoretical findings.Comment: Revised WIAS Preprin

Fractional Calculus and the Future of Science

Newton foresaw the limitations of geometry’s description of planetary behavior and developed fluxions (differentials) as the new language for celestial mechanics and as the way to implement his laws of mechanics. Two hundred years later Mandelbrot introduced the notion of fractals into the scientific lexicon of geometry, dynamics, and statistics and in so doing suggested ways to see beyond the limitations of Newton’s laws. Mandelbrot’s mathematical essays suggest how fractals may lead to the understanding of turbulence, viscoelasticity, and ultimately to end of dominance of the Newton’s macroscopic world view.Fractional Calculus and the Future of Science examines the nexus of these two game-changing contributions to our scientific understanding of the world. It addresses how non-integer differential equations replace Newton’s laws to describe the many guises of complexity, most of which lay beyond Newton’s experience, and many had even eluded Mandelbrot’s powerful intuition. The book’s authors look behind the mathematics and examine what must be true about a phenomenon’s behavior to justify the replacement of an integer-order with a noninteger-order (fractional) derivative. This window into the future of specific science disciplines using the fractional calculus lens suggests how what is seen entails a difference in scientific thinking and understanding

Analysis of contrast-enhanced medical images.

Early detection of human organ diseases is of great importance for the accurate diagnosis and institution of appropriate therapies. This can potentially prevent progression to end-stage disease by detecting precursors that evaluate organ functionality. In addition, it also assists the clinicians for therapy evaluation, tracking diseases progression, and surgery operations. Advances in functional and contrast-enhanced (CE) medical images enabled accurate noninvasive evaluation of organ functionality due to their ability to provide superior anatomical and functional information about the tissue-of-interest. The main objective of this dissertation is to develop a computer-aided diagnostic (CAD) system for analyzing complex data from CE magnetic resonance imaging (MRI). The developed CAD system has been tested in three case studies: (i) early detection of acute renal transplant rejection, (ii) evaluation of myocardial perfusion in patients with ischemic heart disease after heart attack; and (iii), early detection of prostate cancer. However, developing a noninvasive CAD system for the analysis of CE medical images is subject to multiple challenges, including, but are not limited to, image noise and inhomogeneity, nonlinear signal intensity changes of the images over the time course of data acquisition, appearances and shape changes (deformations) of the organ-of-interest during data acquisition, determination of the best features (indexes) that describe the perfusion of a contrast agent (CA) into the tissue. To address these challenges, this dissertation focuses on building new mathematical models and learning techniques that facilitate accurate analysis of CAs perfusion in living organs and include: (i) accurate mathematical models for the segmentation of the object-of-interest, which integrate object shape and appearance features in terms of pixel/voxel-wise image intensities and their spatial interactions; (ii) motion correction techniques that combine both global and local models, which exploit geometric features, rather than image intensities to avoid problems associated with nonlinear intensity variations of the CE images; (iii) fusion of multiple features using the genetic algorithm. The proposed techniques have been integrated into CAD systems that have been tested in, but not limited to, three clinical studies. First, a noninvasive CAD system is proposed for the early and accurate diagnosis of acute renal transplant rejection using dynamic contrast-enhanced MRI (DCE-MRI). Acute rejection–the immunological response of the human immune system to a foreign kidney–is the most sever cause of renal dysfunction among other diagnostic possibilities, including acute tubular necrosis and immune drug toxicity. In the U.S., approximately 17,736 renal transplants are performed annually, and given the limited number of donors, transplanted kidney salvage is an important medical concern. Thus far, biopsy remains the gold standard for the assessment of renal transplant dysfunction, but only as the last resort because of its invasive nature, high cost, and potential morbidity rates. The diagnostic results of the proposed CAD system, based on the analysis of 50 independent in-vivo cases were 96% with a 95% confidence interval. These results clearly demonstrate the promise of the proposed image-based diagnostic CAD system as a supplement to the current technologies, such as nuclear imaging and ultrasonography, to determine the type of kidney dysfunction. Second, a comprehensive CAD system is developed for the characterization of myocardial perfusion and clinical status in heart failure and novel myoregeneration therapy using cardiac first-pass MRI (FP-MRI). Heart failure is considered the most important cause of morbidity and mortality in cardiovascular disease, which affects approximately 6 million U.S. patients annually. Ischemic heart disease is considered the most common underlying cause of heart failure. Therefore, the detection of the heart failure in its earliest forms is essential to prevent its relentless progression to premature death. While current medical studies focus on detecting pathological tissue and assessing contractile function of the diseased heart, this dissertation address the key issue of the effects of the myoregeneration therapy on the associated blood nutrient supply. Quantitative and qualitative assessment in a cohort of 24 perfusion data sets demonstrated the ability of the proposed framework to reveal regional perfusion improvements with therapy, and transmural perfusion differences across the myocardial wall; thus, it can aid in follow-up on treatment for patients undergoing the myoregeneration therapy. Finally, an image-based CAD system for early detection of prostate cancer using DCE-MRI is introduced. Prostate cancer is the most frequently diagnosed malignancy among men and remains the second leading cause of cancer-related death in the USA with more than 238,000 new cases and a mortality rate of about 30,000 in 2013. Therefore, early diagnosis of prostate cancer can improve the effectiveness of treatment and increase the patient’s chance of survival. Currently, needle biopsy is the gold standard for the diagnosis of prostate cancer. However, it is an invasive procedure with high costs and potential morbidity rates. Additionally, it has a higher possibility of producing false positive diagnosis due to relatively small needle biopsy samples. Application of the proposed CAD yield promising results in a cohort of 30 patients that would, in the near future, represent a supplement of the current technologies to determine prostate cancer type. The developed techniques have been compared to the state-of-the-art methods and demonstrated higher accuracy as shown in this dissertation. The proposed models (higher-order spatial interaction models, shape models, motion correction models, and perfusion analysis models) can be used in many of today’s CAD applications for early detection of a variety of diseases and medical conditions, and are expected to notably amplify the accuracy of CAD decisions based on the automated analysis of CE images

Left-invariant Stochastic Evolution Equations on SE(2) and its Applications to Contour Enhancement and Contour Completion via Invertible Orientation Scores

We provide the explicit solutions of linear, left-invariant, (convection)-diffusion equations and the corresponding resolvent equations on the 2D-Euclidean motion group SE(2). These diffusion equations are forward Kolmogorov equations for stochastic processes for contour enhancement and completion. The solutions are group-convolutions with the corresponding Green's function, which we derive in explicit form. We mainly focus on the Kolmogorov equations for contour enhancement processes which, in contrast to the Kolmogorov equations for contour completion, do not include convection. The Green's functions of these left-invariant partial differential equations coincide with the heat-kernels on SE(2), which we explicitly derive. Then we compute completion distributions on SE(2) which are the product of a forward and a backward resolvent evolved from resp. source and sink distribution on SE(2). On the one hand, the modes of Mumford's direction process for contour completion coincide with elastica curves minimizing $\int \kappa^{2} + \epsilon ds$, related to zero-crossings of 2 left-invariant derivatives of the completion distribution. On the other hand, the completion measure for the contour enhancement concentrates on geodesics minimizing $\int \sqrt{\kappa^{2} + \epsilon} ds$. This motivates a comparison between geodesics and elastica, which are quite similar. However, we derive more practical analytic solutions for the geodesics. The theory is motivated by medical image analysis applications where enhancement of elongated structures in noisy images is required. We use left-invariant (non)-linear evolution processes for automated contour enhancement on invertible orientation scores, obtained from an image by means of a special type of unitary wavelet transform

Recent Advances in Single-Particle Tracking: Experiment and Analysis

This Special Issue of Entropy, titled “Recent Advances in Single-Particle Tracking: Experiment and Analysis”, contains a collection of 13 papers concerning different aspects of single-particle tracking, a popular experimental technique that has deeply penetrated molecular biology and statistical and chemical physics. Presenting original research, yet written in an accessible style, this collection will be useful for both newcomers to the field and more experienced researchers looking for some reference. Several papers are written by authorities in the field, and the topics cover aspects of experimental setups, analytical methods of tracking data analysis, a machine learning approach to data and, finally, some more general issues related to diffusion

Dynamics and distribution of immunoglobolin E receptors : a dialog between experiment and theory

This dissertation explores the dynamics and distribution of immunoglobulin E receptors (FceRI) on mast cells by drawing on the techniques of experimental and theoretical physics. The motivation for these investigations is provided by a considerable interest in the transmembrane signaling mechanisms of immunoreceptors, especially when triggered with membrane-bound ligands. Experimental investigations quantify the spatiotemporal dynamics of the redistribution of FceRI due to membrane-bound monovalent ligands, using total internal reflection fluorescence microscopy and single-particle tracking. When mast cells contact such substrates, receptor clusters form at cell-substrate contact points. The initial rate of accumulation of receptors into these contact points or cell protrusions is consistent with diffusion-limited trapping. Over longer timescales (\u3e10 s), individual clusters move with both diffusive and directed motion components and eventually coalesce to form a large central receptor patch surrounded by a receptor cluster depletion zone. Detailed analysis of single-particle trajectories show that receptors maintain their diffusivity when confined within receptor clusters, and increase their diffusivity (above that of monomeric unliganded FceRI) in central patches. To study the kinetics of central patch formation, a new coalescence theory described by a melding process, which is not instantaneous, was developed. In these theoretical investigations, the difficult problem of moving boundaries is encountered. To handle the complexity, which stems from boundary growth due to particle melding, the study is divided into three parts. The first is about stationary trapping problems investigated by the standard defect technique, and the second is about a validity study of an adiabatic approximation for moving boundaries. In the last part of this dissertation, a new coalescence theory is developed, which is based on a completely self-consistent approach. Here, the time dependence of the moving boundary is not prescribed but obtained through feedback. Comparison of experiment and theory shows that observed biological cluster coalescence is delayed at early times and occurs at a faster rate at later times than predicted by a simple theory. The incompatibility at early times is addressed by a generalization of the theory to incorporate a time-dependent melding process by a memory concept, which quantitatively explains the observed delay

Laboratory experiments to evaluate the joint effect between heterogeneity and head fluctuation on mixing, effective porosity and tailing

Tracer visualization in the laboratory at high spatial and temporal resolution can help advance the study of mixing processes. However, grain borders, fluctuations in lighting and non-uniform brightness contribute to produce noisy images of concentrations that cannot be used to directly estimate mixing at the local scale. We present a new methodology to visualize local values of mixing from noisy images of optical tracers based on a nonparametric regression algorithm. The methodology is used to provide a full visualization of the mixing dynamics that occur in a tracer experiment performed in a reconstructed heterogeneous aquifer consisting of two materials with contrasting hydraulic properties. Results show that the method is capable of providing optimal images of mixing of high quality even at complex material edges between bodies of different sand. In transport problems, dispersivity control the mixing. Understand the behaviour of this parameter allow characterizing the mixing. Characterize this parameter is complex due to the heterogeneity and the spatial and temporal fluctuations, in that many assumptions (about the definition of dispersion, how to represent it locally, and how to handle medium heterogeneities or velocity fluctuations) are required to make it tractable. Surprisingly the results go in opposite directions. All agree that fluctuations of velocity transverse to the mean flow direction enhance transverse dispersion; some authors conclude that the enhancement is very large; others conclude that it is not so important. Researchers generally agree the effects of velocity fluctuations on longitudinal dispersion is much smaller than on transverse dispersion. However, they have found it to decrease. Others increase, others remain unchanged ironically, and despite of all the above research, little experimental work has been performed. For this reason, we presents an experiment specifically aimed at evaluating the effect of temporal fluctuations of velocity on transport through a heterogeneous medium and, specifically, on dispersion, mobile porosity and tailing. The results show that Flow fluctuations can enhance transverse mixing and mobile porosity. These phenomena transfer more mass to the plume front, which is now more mobile, and to the low permeability inclusions. As a result, the apparent longitudinal macrodispersion coefficient is substantially increased due to flow fluctuations. Experimental results determined that the effect of flow fluctuations is maximum (enhancement of transverse mixing and macrodispersion) when the kubo number is close to one. Coastal aquifers are complex zones due to the combined influences of heterogeneity (scale, shape or structure of the aquifer), inland groundwater forces and oceanic oscillations. These produce complex hydrodynamic that effect the location, shape and extent of the dispersion zone. The dynamics (fluctuations, width, and location) of the mixing zone between freshwater and seawater in coastal aquifers is essential for detailed understanding of seawater intrusion and bio-geo-chemical processes. Yet, results to date are contradictory. We perform seawater intrusion experiments heterogeneous sand box subject to groundwater flow oscillations. We find that the coupling of heterogeneity and fluctuations leads to complex flow patterns. Freshwater may flow beneath saltwater isolated in low permeability zones when the seawater wedge recedes, but itself become surrounded by seawater when the wedge advances landwards. Fluctuations also disrupt the traditional seawater convection cell, with most of the seawater flow being horizontal, and mixed water displaying some vertical component back to the sea when the seawater wedge recedes seawards. As a result, the mixing zone becomes broad when seawater advances, but narrows down when it recedes. Surprisingly, despite all this complexity the average center of mass of the wedge is very similar to that of the steady-state wedge.El transporte en un medio heterogéneo sometido a actuaciones temporales del campo de velocidades está siendo objeto de varias investigaciones, ya que los flujos de agua subterránea varían naturalmente en el tiempo (ciclos de evapotranspiración, variaciones estacionales en la lluvia, secuencias de años secos y húmedos o fluctuaciones locales por los bombeos, etc.) y por qué es intuitivo que las fluctuaciones temporales de la velocidad deberían mejorar la mezcla y dispersión del soluto. La mezcla se reconoce cada vez más como el proceso crítico para comprender y modelar el transporte reactivo. Sin embargo, como la mezcla está altamente influenciada por la variabilidad espacial de la velocidad y depende no linealmente de las concentraciones, su caracterización por ahora está lejos de ser la óptima. La visualización de un trazador en el laboratorio a alta resolución espacial y temporal puede ayudar a avanzar en el estudio de los procesos de mezcla. Sin embargo, los bordes irregulares de los granos de arena, las fluctuaciones en la iluminación y el brillo no uniforme contribuyen a producir imágenes con ruido en las concentraciones que finalmente no se pueden ser usar para estimar directamente la mezcla a una escala local. Presentamos una nueva metodología para visualizar los valores locales de la mezcla a partir de imágenes con el ruido proveniente de trazadores ópticos basados en un algoritmo de regresión no paramétrico. La metodología se utiliza para proporcionar una visualización de la dinámica de la mezcla que se produce en un experimento con un trazador óptico realizado en un acuífero heterogéneo construido en laboratorio que consta de dos materiales con contraste en sus propiedades hidráulicas. Los resultados muestran que el método es capaz de proporcionar imágenes óptimas de la mezcla de una alta calidad incluso en las zonas donde se localizan los bordes de materiales complejos como entre cuerpos de arena diferente. En los problemas de transporte, la dispersividad controla la mezcla. Por lo tanto, comprender el comportamiento de este parámetro permitiría caracterizar la mezcla. Caracterizar este parámetro es complejo debido a la heterogeneidad y las fluctuaciones espaciales y temporales, de hecho, una cantidad de suposiciones (sobre la definición de la dispersión, cómo representarlo localmente y cómo manejar el medio o las fluctuaciones de la velocidad) son requeridas para hacer manejable el problema. Sorprendentemente, los resultados van en direcciones opuestas. Todos coinciden en que las fluctuaciones de la velocidad transversal a la dirección media del flujo mejoran la dispersión transversal, algunos autores concluyen que la mejora es muy grande, otros concluyen que no es tan importante. Los investigadores generalmente están de acuerdo en que los efectos de las fluctuaciones de la velocidad sobre la dispersión longitudinal son mucho más pequeños que en la dispersión transversal. Pero ellos han encontrado que disminuye. Otros que aumenta, otros que permanece sin cambios. Irónicamente, a pesar de todas las investigaciones anteriores, se ha realizado poco trabajo experimental. Por esta razón, presentamos un experimento específicamente dirigido a evaluar el efecto de las fluctuaciones temporales de la velocidad en el transporte a través de un medio heterogéneo y, específicamente, en la dispersión, la porosidad móvil y las colas. Los resultados muestran que las fluctuaciones en los flujos pueden mejorar la mezcla transversal y la porosidad móvil. Estos fenómenos transfieren más masa al frente de la pluma, que ahora, es más móvil en las inclusiones de baja permeabilidad. Como resultado, el coeficiente de macrodispersión longitudinal aparente aumenta sustancialmente debido a las fluctuaciones del flujo. Los resultados experimentales determinaron que el efecto de las fluctuaciones de flujo es máximo (mejora de la mezcla transversal y macrodispersión) cuando el número de kubo está cerca de uno. Los acuíferos costeros son zonas complejas debido a la influencia que se tiene entre de la heterogeneidad (escala, forma o estructura del acuífero), las fuerzas de las aguas subterráneas continentales y las oscilaciones oceánicas. Esta interacción produce una hidrodinámica compleja que afecta la ubicación, la forma y la extensión de la zona de dispersión. La dinámica (fluctuaciones, ancho y ubicación) de la zona de mezcla entre el agua dulce y el agua de mar en los acuíferos costeros es esencial para una comprensión detallada de la intrusión marina y los procesos bio-geoquímicos. Sin embargo, los resultados hasta la fecha son algo contradictorios. Por este motivo, realizamos experimentos de intrusión marina en una caja de arena heterogénea sujeta a oscilaciones en el flujo de agua subterránea. Encontramos que el acoplamiento entre la heterogeneidad y las fluctuaciones conduce a patrones de flujo complejos. El agua dulce puede fluir debajo del agua salada aislada en zonas de baja permeabilidad cuando la cuña de agua de mar retrocede, pero en sí misma queda rodeada de agua de mar cuando la cuña avanza hacia el interior. Las fluctuaciones también interrumpen la celda tradicional de convección de agua de mar, ya que la mayor parte del flujo de agua de mar es horizontal, y el agua mezclada muestra algún componente vertical de regreso al mar cuando la cuña de agua de mar retrocede hacia el mar. Como resultado, la zona de mezcla se ensancha cuando avanza el agua de mar. pero se estrecha cuando retrocede. Sorprendentemente, a pesar de toda esta complejidad, el centro de masa promedio de la cuña es muy similar al de la cuña en estado estacionarioEl transport en mitjans heterogenis sota camps de velocitat temporalment fluctuant ha estat objecte de molta investigació tant perquè els fluxos d'aigües subterrànies varien de forma natural en el temps (cicles d'evapotranspiració, variacions estacionals de precipitacions, seqüències d'anys secs i humits o fluctuacions de bombeig) i perquè és intuïtiu les fluctuacions temporals de la velocitat haurien de millorar la barreja i la dispersió del solut. La mesura es reconeix cada cop més com el procés crític per comprendre i modelar el transport reactiu. Tanmateix, atès que la barreja està molt influenciada per la variabilitat espacial de la velocitat i depèn no linealment de les concentracions, la seva caracterització adequada queda lluny de ser òptima. La visualització del traçador al laboratori amb una alta resolució espacial i temporal pot ajudar a avançar en l'estudi dels processos de mescla. Tanmateix, els límits del gra, les fluctuacions de la il·luminació i la brillantor no uniforme contribueixen a produir imatges sorolloses de concentracions que no es poden utilitzar per estimar directament la barreja a escala local. Presentem una nova metodologia per visualitzar valors locals de barreja a partir d'imatges sorolloses de traçadors òptics basats en un algorisme de regressió no paramètrica. La metodologia s'utilitza per proporcionar una visualització completa de la dinàmica de mescla que es produeix en un experiment de traçador realitzat en un aqüífer heterogeni reconstruït que consta de dos materials amb propietats hidràuliques contrastants. Els resultats mostren que el mètode és capaç de proporcionar imatges òptimes de mescla d'alta qualitat fins i tot en vores de material complex entre cossos de sorra diferent. En els problemes de transport, la dispersió controla la barreja. Comprendre el comportament d'aquest paràmetre permet caracteritzar la barreja. Caracteritzar aquest paràmetre és complex a causa de l'heterogeneïtat i les fluctuacions espacials i temporals. En aquest cas, molts supòsits (sobre la definició de dispersió, com representar-lo localment i com manejar el mitjà heterogeneïtats o fluctuacions de la velocitat) per fer que sigui tractable. Sorprenentment, els resultats van en direccions oposades. Tots coincideixen que les fluctuacions de la velocitat transversals a la direcció mitjana del flux augmenten la dispersió transversal, alguns autors conclouen que la millora és molt gran, altres conclouen que no és tan important. Els investigadors generalment estan d'acord que els efectes de les fluctuacions de la velocitat en la dispersió longitudinal són molt més petits que en la dispersió transversal. Però el van trobar que es va reduir. Els altres augmenten, altres romanen sense canvis. Irònicament, malgrat tota la investigació anterior, s'ha realitzat poc treball experimental. Per aquest motiu, es presenta un experiment específicament destinat a avaluar l'efecte de les fluctuacions temporals de la velocitat en el transport a través d'un mitjà heterogeni i, específicament, en la dispersió, la porositat mòbil i el tailing. Els resultats mostren que el mètode és capaç de proporcionar imatges òptimes de mescla d'alta qualitat fins i tot en vores de material complex entre cossos de sorra diferent. Els aqüífers costaners són zones complexes a causa de les influències combinades d'heterogeneïtat (escala, forma o estructura de l'aqüífer), forces d'aigua subterrànies i oscil·lacions oceàniques. Aquests productes produeixen hidrodinàmica complexa que condiciona la ubicació, la forma i l'extensió de la zona de dispersió. La dinàmica (fluctuacions, amplada i ubicació) de la zona de barreja entre aigua dolça i aigua de mar en aqüífers costaners és essencial per a una comprensió detallada de la intrusió de l'aigua marina i els processos bioquimicoquímics. Tot i així, els resultats fins a la data són una cosa contradictoris. Realitzem experiments d'intrusió d'aigua marina, heterogènia caixa de sorra subjecte a oscil·lacions de flux d'aigües subterrànies. Trobem que l'acoblament de l'heterogeneïtat i les fluctuacions condueix a patrons de flux complex. L'aigua dolça pot fluir sota aigua salada aïllada en zones de baixa permeabilitat quan la falca de l'aigua marina es retira, però es troba envoltada d'aigua de mar quan la falca avança cap a terra. Les fluctuacions també interrompen la cèl·lula tradicional de convecció de l'aigua de mar, ja que la major part del flux d'aigua de mar és horitzontal i l'aigua mixta que mostra un component vertical al mar quan la falca de mar se separa. Com a resultat, la zona de barreja es fa àmplia quan avança l'aigua del mar, però es redueix quan es retira. Sorprenentment, malgrat tota aquesta complexitat, el centre mitjà de massa de la falca és molt similar al de la falca d'estat estable.Postprint (published version