Two aspects of solving the incompressible Navier-Stokes equations are described in the thesis. The preconditioning of the algebraic systems arising from the Finite Element Method discretization of the Navier-Stokes equations is complex due to the saddle point structure of the resulting algebraic problems. The Pressure Convection Diffusion Reaction and the Least Squares Commutator preconditioners constitute two possible choices studied in the thesis. Solving the flow problems in time-dependent domains requires special numerical methods, such as the Fictitious Boundary method and the Arbitrary Lagrangian Eulerian formulation of Navier-Stokes equations which are used in the thesis. The problems examined in the thesis are simulations of experiments conducted in liquid Helium at low temperatures. These simulations can be used to establish a relationship between vorticity and new quantity pseudovorticity in an experiment-like setting.

Title: Krylov Subspace Methods - Analysis and Application Author: Tomáš Gergelits Department: Department of Numerical Mathematics Supervisor: prof. Ing. Zdeněk Strakoš, DrSc., Department of Numerical Mathematics Abstract: Convergence behavior of Krylov subspace methods is often studied for linear algebraic systems with symmetric positive definite matrices in terms of the condition number of the system matrix. As recalled in the first part of this thesis, their actual convergence behavior (that can be in practice also substantially affected by rounding errors) is however determined by the whole spectrum of the system matrix, and by the projections of the initial residual to the associated invariant subspaces. The core part of this thesis investigates the spectra of infinite dimensional operators −∇ · (k(x)∇) and −∇ · (K(x)∇), where k(x) is a scalar coefficient function and K(x) is a symmetric tensor function, preconditioned by the Laplace operator. Subsequently, the focus is on the eigenvalues of the matrices that arise from the discretization using conforming finite elements. Assuming continuity of K(x), it is proved that the spectrum of the preconditi- oned infinite dimensional operator is equal to the convex hull of the ranges of the diagonal function entries of Λ(x) from the spectral decomposition K(x) =...

Problém nejmenších čtverc· (dále jen LS problém) je aproximační úloha řešení soustav lineárních algebraických rovnic, které jsou z nějakého d·vodu za- tíženy chybami. Existence a jednoznačnost řešení a metody řešení jsou známé pro r·zné typy matic, kterými tyto soustavy reprezentujeme. Typicky jsou ma- tice řídké a obrovských dimenzí, ale velmi často dostáváme z praxe i úlohy s maticemi o proměnlivé hustotě nenulových prvk·. Těmi se myslí řídké matice s jedním nebo více hustými řádky. Zde rozebíráme metody řešení tohoto LS pro- blému. Obvykle jsou založeny na rozdělení úlohy na hustou a řídkou část, které řeší odděleně. Tak pro řídkou část m·že přestat platit předpoklad plné sloupcové hodnosti, který je potřebný pro většinu metod. Proto se zde speciálně zabýváme postupy, které tento problém řeší. 1

Indiana University-Purdue University Indianapolis (IUPUI) / Acute Kidney Injury (AKI) is a prevalent and significant problem whose primary treatment is supportive care. Ischemic preconditioning is a strategy used to protect organs from ischemic injury via a prior injury. Ischemic preconditioning in the kidneys has been shown to confer protection onto kidneys from subsequent ischemic insults with attenuated serum creatinine values in treated rats. In the preconditioned kidneys, the enzyme IDH2 was discovered to be upregulated in the mitochondria. Hydrodynamic fluid delivery to the kidney was found to be a viable technique for delivering this gene to the kidney, resulting in artificially upregulated expression of IDH2. Via a two-pronged effort to discern the functional significance of ischemic preconditioning and hydrodynamic IDH2 fluid injections, we performed mitochondrial oxygen respiration assays on both preconditioned and injected kidneys. We found that renal ischemic preconditioning resulted in no significant difference between sham and preconditioned, subsequently injured kidneys, which is similar to the results from the serum creatinine studies. Hydrodynamically IDH2-injected, and subsequently injured kidneys respire significantly better than vehicle injected, and subsequently injured kidneys, which shows that hydrodynamic injections of IDH2 protects kidneys against injury, and partially mimics the effects of preconditioning.

Calcium tolerant pig and rabbit cardiomyocytes were isolated using retrograde aortic perfusion of nominally calcium-free collagenase. Preconditioning protocols used 1 or 3 x l0-min episodes of ischemic pelleting or pre-incubation with 100 μM adenosine, followed by a 15-min post-incubation and 180-240-min ischemic pelleting. Control cells were incubated and washed in parallel with the experimental groups. Injury was assessed by determination of cell morphology, trypan blue permeability following osmotic swelling, lactate and HPLC analysis of adenine nucleotides. Preconditioned pig cardiomyocytes had a reduced rate of ischemic contracture, but protection occurred without conservation of ATP. Preconditioned rabbit cardiomyocytes were protected without significant changes in rates of ischemic contracture or ATP depletion. Incubation of ischemic cells with the protein phosphatase inhibitor, fostriecin, at PP2A-selective concentrations (0.1-10 μM), mimicked preconditioning in both rabbit and pig cardiomyocytes. In rabbits, the K(ATP) channel blocker, 5-hydroxydecanoate (5-HD), did not block preconditioning or fostriecin protection. In the pig, 5-HD blocked both preconditioning and fostriecin protection, with return of the rates of ischemic contracture to control. However, 5-HD was an effective blocker of protection only in early ischemia. Fostriecin mimicked preconditioning in the rabbit and the early responses of the preconditioned pig. Preconditioning appears associated with protein phosphorylation in both the rabbit and the pig, but major pathways leading to protection may differ in the two species.

Esophagectomy often presents anastomotic leaks (AL), due to tenuous perfusion of gastric conduit fundus (GCF). Hybrid (endovascular/surgical) ischemic gastric preconditioning (IGP), might improve GCF perfusion. Sixteen pigs undergoing IGP were randomized: (1) Max-IGP (n = 6): embolization of left gastric artery (LGA), right gastric artery (RGA), left gastroepiploic artery (LGEA), and laparoscopic division (LapD) of short gastric arteries (SGA); (2) Min-IGP (n = 5): LGA-embolization, SGA-LapD; (3) Sham (n = 5): angiography, laparoscopy. At day 21 gastric tubulation occurred and GCF perfusion was assessed as: (A) Serosal-tissue-oxygenation (StO2) by hyperspectral-imaging; (B) Serosal time-to-peak (TTP) by fluorescence-imaging; (C) Mucosal functional-capillary-density-area (FCD-A) index by confocal-laser-endomicroscopy. Local capillary lactates (LCL) were sampled. Neovascularization was assessed (histology/immunohistochemistry). Sham presented lower StO2 and FCD-A index (41 ± 10.6%; 0.03 ± 0.03 respectively) than min-IGP (66.2 ± 10.2%, p-value = 0.004; 0.22 ± 0.02, p-value < 0.0001 respectively) and max-IGP (63.8 ± 9.4%, p-value = 0.006; 0.2 ± 0.02, p-value < 0.0001 respectively). Sham had higher LCL (9.6 ± 4.8 mL/mol) than min-IGP (4 ± 3.1, p-value = 0.04) and max-IGP (3.4 ± 1.5, p-value = 0.02). For StO2, FCD-A, LCL, max- and min-IGP did not differ. Sham had higher TTP (24.4 ± 4.9 s) than max-IGP (10 ± 1.5 s, p-value = 0.0008) and min-IGP (14 ± 1.7 s, non-significant). Max- and min-IGP did not differ. Neovascularization was confirmed in both IGP groups. Hybrid IGP improves GCF perfusion, potentially reducing post-esophagectomy AL.

Background: Systemic inflammation can occur after transcatheter aortic valve replacement (TAVR) and correlates with adverse outcome. The impact of remote ischemic preconditioning (RIPC) on TAVR associated systemic inflammation is unknown and was focus of this study. Methods: We performed a prospective controlled trial at a single center and included 66 patients treated with remote ischemic preconditioning (RIPC) prior to TAVR, who were matched to a control group by propensity score. RIPC was applied to the upper extremity using a conventional tourniquet. Definition of systemic inflammation was based on leucocyte count, C-reactive protein (CRP), procalcitonin (PCT) and interleukin-6 (IL-6), assessed in the first 5 days following the TAVR procedure. Mortality was determined within 6 months after TAVR. RIPC group and matched control group showed comparable baseline characteristics. Results: Systemic inflammation occurred in 66% of all patients after TAVR. Overall, survival after 6 months was significantly reduced in patients with systemic inflammation. RIPC, in comparison to control, did not significantly alter the plasma levels of leucocyte count, CRP, PCT or IL-6 within the first 5 days after TAVR. Furthermore, inflammation associated survival after 6 months was not improved by RIPC. Of all peri-interventional variables assessed, only the amount of the applied contrast agent was connected to the occurrence of systemic inflammation. Conclusions: Systemic inflammation frequently occurs after TAVR and leads to increased mortality after 6 months. RIPC neither reduces the incidence of systemic inflammation nor improves inflammation associated patient survival within 6 months.

The main theme of this work is effectiveness and efficiency of Krylov subspace methods and Krylov subspace recycling. While solving long, slowly changing sequences of large linear systems, such as the ones that arise in engineering, there are many issues we need to consider if we want to make the process reliable (converging to a correct solution) and as fast as possible. This thesis is built on three main components. At first, we target bilinear and quadratic form estimation. Bilinear form $c^TA^{-1}b$ is often associated with long sequences of linear systems, especially in optimization problems. Thus, we devise algorithms that adapt cheap bilinear and quadratic form estimates for Krylov subspace recycling. In the second part, we develop a hybrid recycling method that is inspired by a complex CFD application. We aim to make the method robust and cheap at the same time. In the third part of the thesis, we optimize the implementation of Krylov subspace methods on Graphic Processing Units (GPUs). Since preconditioners based on incomplete matrix factorization (ILU, Cholesky) are very slow on the GPUs, we develop a preconditioner that is effective but well suited for GPU implementation. / Ph. D. / In many applications we encounter the repeated solution of a large number of slowly changing large linear systems. The cost of solving these systems typically dominates the computation. This is often the case in medical imaging, or more generally inverse problems, and optimization of designs. Because of the size of the matrices, Gaussian elimination is infeasible. Instead, we find a sufficiently accurate solution using iterative methods, so-called Krylov subspace methods, that improve the solution with every iteration computing a sequence of approximations spanning a Krylov subspace. However, these methods often take many iterations to construct a good solution, and these iterations can be expensive. Hence, we consider methods to reduce the number of iterations while keeping the iterations cheap. One such approach is Krylov subspace recycling, in which we recycle judiciously selected subspaces from previous linear solves to improve the rate of convergence and get a good initial guess. In this thesis, we focus on improving efficiency (runtimes) and effectiveness (number of iterations) of Krylov subspace methods. The thesis has three parts. In the first part, we focus on efficiently estimating sequences of bilinear forms, c^TA⁻¹b. We approximate the bilinear forms using the properties of Krylov subspaces and Krylov subspace solvers. We devise an algorithm that allows us to use Krylov subspace recycling methods to efficiently estimate bilinear forms, and we test our approach on three applications: topology optimization for the optimal design of structures, diffuse optical tomography, and error estimation and grid adaptation in computational fluid dynamics. In the second part, we focus on finding the best strategy for Krylov subspace recycling for two large computational fluid dynamics problems. We also present a new approach, which lets us reduce the computational cost of Krylov subspace recycling. In the third part, we investigate Krylov subspace methods on Graphics Processing Units. We use a lid driven cavity problem from computational fluid dynamics to perform a thorough analysis of how the choice of the Krylov subspace solver and preconditioner influences runtimes. We propose a new preconditioner, which is designed to work well on Graphics Processing Units.

Ischämietoleranz bezeichnet das Phänomen, dass ein kurzer ischämischer, metabolischer oder physikalischer Stimulus das Gehirn paradoxerweise "resistent" macht gegenüber einer darauffolgenden, längerdauernden Ischämie. In einer retrospektiven Studie versuchten wir die Hypothese zu untermauern, dass transiente ischämische Attacken (als kurzdauernde ischämische Stimuli) vor einem Infarkt (prodromale TIAs) protektiv sind gegen eine nachfolgende zerebrale Ischämie. Es zeigte sich dabei, dass Patienten mit prodromalen TIAs ein geringeres Defizit und einen günstigeren Verlauf zeigten und im CT seltener Infarktfrühzeichen hatten. Somit könnten transiente ischämische Attacken, vor einem Schlaganfall, analog zu der Situation am Herzen und wie in zahlreichen in vivo Modellen gezeigt, ein klinisches Korrelat zur hypoxischen Präkonditionierung darstellen. Im experimentellen Teil der vorliegenden Arbeit wird gezeigt, dass sich hypoxische Präkonditionierung in vitro in neuronalen Kulturen modellieren lässt. Eine kurzzeitige Sauerstoff-Glucose-Deprivation (OGD) 1-3 Tage vor einer längeren OGD führt zu einem signifikanten Schutz von Neuronen, bis zu 90%. Hypoxietoleranz kann auch durch andere metabolische Stimuli, wie Inhibition von Atmungskettenenzymen durch 3-NPA im gleichen Zeitrahmen simuliert werden. Eine genaue Kenntnis der endogenen Neuroprotektion durch Ischämietoleranz könnte in Zukunft helfen, den Schaden durch ischämische Infarkte und ischämische Enzephalopathien zu minimieren. / Ischemic tolerance is a phenomenon where a brief episode of ischemia renders the brain resistant against a subsequent, longerlasting ischemic event. In a retrospective study we tested the hypothesis that transient ischemic attacks (as brief ischemic stimuli) before cerebral ischemia (prodromal TIA's) may have a protective effect. Here we show that patients with prodromal TIA's have less severe neurologic impairment, a better clinical course and have less early infarct signs. Therefore we siggest that TIA's, before stroke could represent a clinical correlate to hypoxic preconditioning, as shown in the heart. Experimentally we were able to model hypoxic preconditioning in vitro using neuronal cultures. Brief oxygen-glucose deprivation (OGD) 1-3 days before longer lasting OGD protects neurons, up to 90%. Hypoxic tolerance was also simulated by metabolic stimuli like inhibition of the respiratory chain by 3-NPA. Increasing knowledge of this endogenous neuroprotection by ischemic tolerance might help to minimize neuronal damage following ischemic strokes and hypoxic encephalopathy.

The aim of this thesis is to describe the variability of cardiac rhythm and familiarity with the methods of the analysis, ie by monitoring changes in heart rhythm electrogram signal recording and using the methods in the time domain using recurrent diagram. The work describes the quantification of the methods and possibilities of quantifiers in the evaluation of heart rate variability analysis. It also describes the clinical significance of heart rate variability and diagnostic capabilities changes of heart rate variability caused by ischemic heart disease. The practical part describes how to create applications in Matlab to calculate the quantifiers analysis of heart rate variability in the time domain using recurrent diagram. The calculation was made of the positions R wave elektrogram signal isolated rabbit hearts. The calculated values of quantifiers both methods were statistically evaluated and discussed.