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ResearcherID
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- N-3741-2015 (1)
Analysis of discretization schemes for Fokker-Planck equations and related optimality systems
(2015)
The Fokker-Planck (FP) equation is a fundamental model in thermodynamic kinetic theories and
statistical mechanics.
In general, the FP equation appears in a number of different fields in natural sciences, for instance in solid-state physics, quantum optics, chemical physics, theoretical biology, and circuit theory. These equations also provide a powerful mean to define
robust control strategies for random models. The FP equations are partial differential equations (PDE) describing the time evolution of the probability density function (PDF) of stochastic processes.
These equations are of different types depending on the underlying stochastic process.
In particular, they are parabolic PDEs for the PDF of Ito processes, and hyperbolic PDEs for piecewise deterministic processes (PDP).
A fundamental axiom of probability calculus requires that the integral of the PDF over all the allowable state space must be equal to one, for all time. Therefore, for the purpose of accurate numerical simulation, a discretized FP equation must guarantee conservativeness of the total probability. Furthermore, since the
solution of the FP equation represents a probability density, any numerical scheme that approximates the FP equation is required to guarantee the positivity of the solution. In addition, an approximation scheme must be accurate and stable.
For these purposes, for parabolic FP equations on bounded domains, we investigate the Chang-Cooper (CC) scheme for space discretization and first- and
second-order backward time differencing. We prove that the resulting
space-time discretization schemes are accurate, conditionally stable, conservative, and preserve positivity.
Further, we discuss a finite difference discretization for the FP system corresponding to a PDP process in a bounded domain.
Next, we discuss FP equations in unbounded domains.
In this case, finite-difference or finite-element methods cannot be applied. By employing a suitable set of basis functions, spectral methods allow to treat unbounded domains. Since FP solutions decay exponentially at infinity, we consider Hermite functions as basis functions, which are Hermite polynomials multiplied by a Gaussian.
To this end, the Hermite spectral discretization is applied
to two different FP equations; the parabolic PDE corresponding to Ito processes, and the system of hyperbolic PDEs corresponding to a PDP process. The resulting discretized schemes are analyzed. Stability and spectral accuracy of the Hermite spectral discretization of the FP problems is proved. Furthermore, we investigate the conservativity of the solutions of FP equations discretized with the Hermite spectral scheme.
In the last part of this thesis, we discuss optimal control problems governed by FP equations on the characterization of their solution by optimality systems. We then investigate the Hermite spectral discretization of FP optimality systems in unbounded domains.
Within the framework of Hermite discretization, we obtain sparse-band systems of ordinary differential equations. We analyze the accuracy of the discretization schemes by showing spectral convergence in approximating the state, the adjoint, and the control variables that appear in the FP optimality systems.
To validate our theoretical estimates, we present results of numerical experiments.
Alzheimer’s disease (AD) is the most prevalent neurodegenerative disease of the brain, which is characterized by a progressive loss of memory and spatial orientation. Only less than 5-10% of AD sufferers are familial cases due to genetic mutations in the amyloid precursor protein (APP) gene or presenilin (PS) 1 and 2 genes. The cause of sporadic AD (sAD) which covers > 95% of AD patients is still unknown. Current research found interactions between aging, diabetes and cognitive decline including dementia in general and in AD in particular. Disturbances of brain glucose uptake, glucose tolerance and utilization and impairment of the insulin/insulin receptor (IR) signaling cascade are thought to be key targets for the development of sAD.
In the brain of AD patients, neural plasticity is impaired indicated by synaptic and neuronal loss. Adult neurogenesis (AN), the generation of functional neurons in the adult brain, may be able to restore neurological function deficits through the integration of newborn neurons into existing neural networks. The dentate gyrus of the hippocampus is one out of few brain regions where life-long AN exists. However, there is a big controversy in literature regarding the involvement of AN in AD pathology. Most animal studies used transgenic mice based on the Amyloid ß (Aß) hypothesis which primarily act as models for the familial form of AD. Findings from human post mortem AN studies were also inconstistent. In this thesis, we focused on the possible involvement of AN in the pathogenesis of the sporadic form of AD. Streptozotocin intracerebroventricularily (STZ icv) treated rats, which develop an insulin-resistant brain state and learning and memory deficits preceding Aß pathology act as an appropriate animal model for sAD. We used STZ treatment for both parts of my work, for the in vivo and in vitro study.
In the first part of my thesis, my coworkers and I investigated STZ icv treatment effects on different stages of AN in an in vivo approach. Even if STZ icv treatment does not seem to considerably influence stem cell proliferation over a short-term (1 month after STZ icv treatment) as well as in a long-term (3 months after STZ icv treatment) period, it results in significantly less immature and newborn mature neurons 3 months after STZ icv treatment. This reduction detected after 3 months was specific for the septal hippocampus, discussed to be important for spatial learning. Subsequently we performed co-localization studies with antibodies detecting BrdU (applied appr. 27 days before sacrifice) and cell-type specific markers such as NeuN, and GFAP, we found that STZ treatment does not affect the differentiation fate of newly generated cells. Phenotype analysis of BrdU-positive cells in the hilus and molecular layer revealed that some of the BrdU-positive cells are newborn oligodendrocytes but not newborn microglia.
In the second part of my thesis I worked with cultured neural stem cells (NSCs) isolated from the adult rat hippocampus to reveal STZ effects on the proliferation of of NSCs, and on the survival and differentiation of their progeny. Furthermore, this in vitro approach enabled me to study cellular mechanisms underlying the observed impaired neurogenesis in the hippocampus of STZ-treated rats. In contrast to our findings of the STZ icv in vivo study we revealed that STZ supplied with the cell culture medium inhibits the proliferation of NSCs in a dose-dependent and time-dependent manner. Moreover, performing immunofluorescence studies with antibodies detecting cell-type specific markers after triggering NSCs to differentiate, we could show that STZ treatment affects the number of newly generated neurons but not of astrocytes. Analyzing newborn cells starting to differentiate and migrate I was able to demonstrate that STZ has no effect on the migration of newborn cells. Trying to reveal cellular mechanisms underlying the negative influence of STZ on hippocampal AN, we performed qRT-PCR and immunofluorescence staining and thus could show that in NSCs the expression of glucose transporter (GLUT)3 mRNA as well as IR and GLUT3 protein levels are reduced after STZ treatment. Therefore, the inhibition of the proliferation of NSCs may be (at least partially) caused by these two molecules. Interestingly, the effect of STZ on differentiating cells was shown to be different, as IR protein expression was not significantly changed but GLUT3 protein levels were decreased in consequence of STZ treatment.
In summary, this project delivered further insights into the interrelation between AN the sporadic form of sAD and thus provides a basis of new therapeutic approaches in sAD treatment through intervening AN. Discrepancies between the results of the two parts of my thesis, the in vivo and in vitro part, were certainly caused to a certain extent by the missing microenvironment in the in vitro approach with cultured NSCs. Future studies e.g. using co-culture systems could at least minimize the effect of a missing natural microenvironment of cultured NSCs, so that the use of an in vitro approach for the investigation of STZ treatment underlying cellular mechanisms can be improved.
Several aspects of the stability analysis of large-scale discrete-time systems are considered. An important feature is that the right-hand side does not have have to be continuous.
In particular, constructive approaches to compute Lyapunov functions are derived and applied to several system classes.
For large-scale systems, which are considered as an interconnection of smaller subsystems, we derive a new class of small-gain results, which do not require the subsystems to be robust in some sense. Moreover, we do not only study sufficiency of the conditions, but rather state an assumption under which these conditions are also necessary.
Moreover, gain construction methods are derived for several types of aggregation, quantifying how large a prescribed set of interconnection gains can be in order that a small-gain condition holds.
The learned helplessness phenomenon is a specific animal behavior induced by prior exposure to uncontrollable aversive stimuli. It was first found by Seligman and Maier (1967) in dogs and then has been reported in many other species, e.g. in rats (Vollmayr and Henn, 2001), in goldfishes (Padilla, 1970), in cockroaches (Brown, 1988) and also in fruit flies (Brown, 1996; Bertolucci, 2008). However, the learned helplessness effect in fruit flies (Drosophila melanogaster) has not been studied in detail. Thus, in this doctoral study, we investigated systematically learned helplessness behavior of Drosophila for the first time.
Three groups of flies were tested in heatbox. Control group was in the chambers experiencing constant, mild temperature. Second group, master flies were punished in their chambers by being heated if they stopped walking for 0.9s. The heat pulses ended as soon as they resumed walking again. A third group, the yoked fly, was in their chambers at the same time. However, their behavior didn’t affect anything: yoked flies were heated whenever master flies did, with same timing and durations. After certain amount of heating events, yoked flies associated their own behavior with the uncontrollability of the environment. They suppressed their innate responses such as reducing their walking time and walking speed; making longer escape latencies and less turning around behavior under heat pulses. Even after the conditioning phase, yoked flies showed lower activity level than master and control flies. Interestingly, we have also observed sex dimorphisms in flies. Male flies expressed learned helplessness not like female flies. Differences between master and yoked flies were smaller in male than in female flies. Another interesting finding was that prolonged or even repetition of training phases didn’t enhance learned helplessness effect in flies.
Furthermore, we investigated serotonergic and dopaminergic nervous systems in learned helplessness. Using genetic and pharmacological manipulations, we altered the levels of serotonin and dopamine in flies’ central nervous system. Female flies with reduced serotonin concentration didn’t show helpless behavior, while the learned helplessness effect in male flies seems not to be affected by a reduction of serotonin. Flies with lower dopamine level do not display the learned helplessness effect in the test phase, suggesting that with low dopamine the motivational change in learned helplessness in Drosophila may decline faster than with a normal dopamine level.
In this thesis, I present a model system for carbohydrate interactions with single-crystalline Ru surfaces. Geometric and electronic properties of copper phthalocyanine (CuPc) on top of graphene on hexagonal Ru(0001), rectangular Ru(10-10) and vicinal Ru(1,1,-2,10) surfaces have been studied. First, the Fermi surfaces and band structures of the three Ru surfaces were investigated by high-resolution angle-resolved photoemission spectroscopy. The experimental data and theoretical calculations allow to derive detailed information about the momentum-resolved electronic structure. The results can be used as a reference to understand the chemical and catalytic properties of Ru surfaces. Second, graphene layers were prepared on the three different Ru surfaces. Using low-energy electron diffraction and scanning tunneling microscopy, it was found that graphene can be grown in well-ordered structures on all three surfaces, hexagonal Ru(0001), rectangular Ru(10-10) and vicinal Ru(1,1,-2,10), although they have different surface symmetries. Evidence for a strong interaction between graphene and Ru surfaces is a 1.3-1.7e V increase in the graphene pi-bands binding energy with respect to free-standing graphene sheets. This energy variation is due to the hybridization between the graphene pi bands and the Ru 4d electrons, while the lattice mismatch does not play an important role in the bonding between graphene and Ru surfaces. Finally, the geometric and electronic structures of CuPc on Ru(10-10), graphene/Ru(10-10), and graphene/Ru(0001) have been studied in detail. CuPc molecules can be grown well-ordered on Ru(10-10) but not on Ru(0001). The growth of CuPc on graphene/Ru(10-10) and Ru(0001) is dominated by the Moire pattern of graphene. CuPc molecules form well-ordered structures with rectangular unit cells on graphene/Ru(10-10) and Ru(0001). The distance of adjacent CuPc molecules is 1.5 and 1.3 nm on graphene/Ru(0001) and 1.54 and 1.37 nm on graphene/Ru(10-10). This indicates that the molecule-substrate interaction dominates over the intermolecular interaction for CuPc molecules on graphene/Ru(10-10) and graphene/Ru(0001).
Pulsars (in short for Pulsating Stars) are magnetized, fast rotating neutron stars. The basic picture of a pulsar describes it as a neutron star which has a rotation axis that is not aligned with its magnetic field axis. The emission is assumed to be generated near the magnetic poles of the neutron star and emitted along the open magnetic field lines. Consequently, the corresponding beam of photons is emitted along the magnetic field line axis. The non-alignment of both, the rotation and the magnetic field axis, results in the effect that the emission of the pulsar is only seen if its beam points towards the observer.
The emission from a pulsar is therefore perceived as being pulsed although its generation is not. This rather simple geometrical model is commonly referred to as Lighthouse Model and has been widely accepted. However, it does not deliver an explanation of the precise mechanisms behind the emission from pulsars (see below for more details).
Nowadays more than 2000 pulsars are known. They are observed at various wavelengths. Multiwavelength studies have shown that some pulsars are visible only at certain wavelengths while the emission from others can be observed throughout large parts of the electromagnetic spectrum. An example of the latter case is the Crab pulsar which is also the main object of interest in this thesis. Originating from a supernova explosion observed in 1054 A.D. and discovered in 1968, the Crab pulsar has been the central subject of numerous studies. Its pulsed emission is visible throughout the whole electromagnetic spectrum which makes it a key figure in understanding the possible mechanisms of multiwavelength emission from pulsars.
The Crab pulsar is also well known for its radio emission strongly varying on long as well as on short time scales. While long time scale behaviour from a pulsar is usually examined through the use of its average profile (a profile resulting from averaging of a large number of individual pulses resulting from single rotations), short time scale behaviour is examined via its single pulses. The short time scale anomalous behaviour of its radio emission is commonly referred to as Giant Pulses and represents the central topic of this thesis.
While current theoretical approaches place the origin of the radio emission from a pulsar like the Crab near its magnetic poles (Polar Cap Model) as already indicated by the Lighthouse model, its emission at higher frequencies, especially its gamma-ray emission, is assumed to originate further away in the geometrical region surrounding a pulsar which is commonly referred to as a pulsar magnetosphere (Outer Gap Model). Consequently, the respective emission regions are usually assumed not to be connected. However, past observational results from the Crab pulsar represent a contradiction to this assumption.
Radio giant pulses from the Crab pulsar have been observed to emit large amounts of energy on very short time scales implying small emission regions on the surface of the pulsar. Such energetic events might also leave a trace in the gamma-ray emission of the Crab pulsar.
The aim of this thesis is to search for this connection in the form of a correlation study between radio giant pulses and gamma-photons from the Crab pulsar.
To make such a study possible, a multiwavelength observational campaign was organized for which radio observations were independently applied for, coordinated and carried out with the Effelsberg radio telescope and the Westerbork Synthesis Radio Telescope and gamma-ray observations with the Major Atmospheric Imaging Cherenkov telescopes. The corresponding radio and gamma-ray data sets were reduced and the correlation analysis thereafter consisted of three different approaches:
1) The search for a clustering in the differences of the times of arrival of radio giant pulses and gamma-photons;
2) The search for a linear correlation between radio giant pulses and gamma-photons using the Pearson correlation approach;
3) A search for an increase of the gamma-ray flux around occurring radio giant pulses.
In the last part of the correlation study an increase of the number of gamma-photons centered on a radio giant pulse by about 17% (in contrast with the number of gamma-photons when no radio giant pulse occurs in the same time window) was discovered. This finding suggests that a new theoretical approach for the emission of young pulsars like the Crab pulsar, is necessary.