In this paper, finite element discretizations of the degenerated operator -ω²(y) u_xx-ω²(x)u_yy=g in the unit square are investigated, where the weight function satisfies ω(ξ)=ξ^α with α ≥ 0. We propose two multi-level methods in order to solve the resulting system of linear algebraic equations. The first method is a multi-grid algorithm with line-smoother. A proof of the smoothing property is given. The second method is a BPX-like preconditioner which we call MTS-BPX preconditioner. We show that the upper eigenvalue bound of the MTS-BPX preconditioned system matrix grows proportionally to the level number.

This paper proposes new iterative methods for the efficient computation of the smallest eigenvalue of the symmetric nonlinear matrix eigenvalue problems of large order with a monotone dependence on the spectral parameter. Monotone nonlinear eigenvalue problems for differential equations have important applications in mechanics and physics. The discretization of these eigenvalue problems leads to ill-conditioned nonlinear eigenvalue problems with very large sparse matrices monotone depending on the spectral parameter. To compute the smallest eigenvalue of large matrix nonlinear eigenvalue problem, we suggest preconditioned iterative methods: preconditioned simple iteration method, preconditioned steepest descent method, and preconditioned conjugate gradient method. These methods use only matrix-vector multiplications, preconditioner-vector multiplications, linear operations with vectors and inner products of vectors. We investigate the convergence and derive grid-independent error estimates of these methods for computing eigenvalues. Numerical experiments demonstrate practical effectiveness of the proposed methods for a class of mechanical problems.

In this paper, a uniformly elliptic second order boundary value problem in 2D is discretized by the p-version of the finite element method. An inexact Dirichlet-Dirichlet domain decomposition pre-conditioner for the system of linear algebraic equations is investigated. The solver for the problem in the sub-domains and a pre-conditioner for the Schur-complement are proposed as ingredients for the inexact DD-pre-conditioner. Finally, several numerical experiments are given.

The boundary concentrated finite element method is a variant of the hp-version of the finite element method that is particularly suited for the numerical treatment of elliptic boundary value problems with smooth coefficients and low regularity boundary conditions. For this method we present two multilevel preconditioners that lead to preconditioned stiffness matrices with condition numbers that are bounded uniformly in the problem size N. The cost of applying the preconditioners is O(N). Numerical examples illustrate the efficiency of the algorithms.

Accumulation of osmotically active metabolites, which create an osmotic gradient estimated at ∼60 mOsM, and cell swelling are prominent features of ischemic myocardial cell death. This study tests the hypothesis that reduction of ischemic swelling by enhanced cell volume regulation is a key mechanism in the delay of ischemic myocardial cell death by ischemic preconditioning (IPC). Experimental protocols address whether: (i) IPC triggers a cell volume regulation mechanism that reduces cardiomyocyte swelling during subsequent index ischemia; (ii) this reduction in ischemic cell swelling is sufficient in magnitude to account for the IPC protection; (iii) the molecular mechanism that mediates IPC also mediates cell volume regulation. Two experimental models with rabbit ventricular myocytes were studied: freshly isolated pelleted myocytes and 48-h cultured myocytes. Myocytes were preconditioned either by distinct short simulated ischemia (SI)/simulated reperfusion protocols (IPC), or by subjecting myocytes to a pharmacological preconditioning (PPC) protocol (1 μM calyculin A, or 1 μM N6-2-(4-aminophenyl)ethyladenosine (APNEA), prior to subjecting them to either different durations of long SI or 30 min hypo-osmotic stress. Cell death (percent blue square myocytes) was monitored by trypan blue staining. Cell swelling was determined by either the bromododecane cell flotation assay (qualitative) or video/confocal microscopy (quantitative). Simulated ischemia induced myocyte swelling in both the models. In pelleted myocytes, IPC or PPC with either calyculin A or APNEA produced a marked reduction of ischemic cell swelling as determined by the cell floatation assay. In cultured myocytes, IPC substantially reduced ischemic cell swelling (P < 0.001). This IPC effect on ischemic cell swelling was related to an IPC and PPC (with APNEA) mediated triggering of cell volume regulatory decrease (RVD). IPC and APNEA also significantly (P < 0.001) reduced hypo-osmotic cell swelling. This IPC and APNEA effect was blocked by either adenosine receptor, PKC or Cl- channel inhibition. The osmolar equivalent for IPC protection approximated 50-60 mOsM, an osmotic gradient similar to the estimated ischemic osmotic load for preconditioned and non-preconditioned myocytes. The results suggest that cell volume regulation is a key mechanism that accounts for most of the IPC protection in cardiomyocytes.

Objective: The aim was to determine if isolated rabbit cardiomyocytes could be preconditioned. Methods: Cardiomyocytes isolated from rabbit hearts were subjected to 15 min oxygenated preincubation, with and without substrate, prior to concentration into an ischaemic slurry, with or without glucose present. The effects of an adenosine agonist (CCPA), an adenosine receptor blocker (SPT), and the protein kinase C blocker, calphostin C, on rates of ischaemic contracture and survival of the myocytes were determined after various times of ischaemia, following resuspension of the cells in hypotonic media. Results: A glucose-free preincubation period protected myocytes from subsequent ischaemic injury, with a 40% reduction of cell death at 90-120 min and 1-2 h delay in cell death. CCPA added during preincubation and during the ischaemic period also tended to protect from injury, but the differences were not significant and protection was less than with a glucose-free preincubation. Although preincubation with CCPA did not precondition, SPT added to the preincubation medium only, or to both the preincubation medium and the ischaemic pellet, inhibited the preconditioning effect of a glucose-free preincubation period. Calphostin C, added only into the ischaemic pellet, inhibited the preconditioning effect of glucose-free preincubation. Conclusions: Glucose-free preincubation protects ischaemic isolated myocytes from subsequent ischaemia. The degree of protection is great enough to account for protection seen in intact hearts, following preconditioning protocols. Protection is blocked by SPT and a highly specific protein kinase C inhibitor, calphostin C. Protection from ischaemic injury that seems to mimic ischaemic preconditioning can be induced in isolated cardiomyocytes, and appears dependent on adenosine receptors and activation of protein kinase C.Cardiovascular Research 1994;28:72-77.

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Thèse numérisée par la Direction des bibliothèques de l'Université de Montréal. / This thesis provides evidence to support the hypothesis that synaptic plasticity in the primary somatosensory cortex is a cellular correlate of associative learning, that the process depends upon acetylcholine and that only certain cortical neurons display this plasticity. In a first series of experiments, single-imit recordings were carried out in the barrel cortex of awake, adult rats subjected to whisker pairing, an associative learning paradigm where deflections of the recorded neuron's principle vibrissa were repeatedly paired with those of a non-adjacent one. On average, this form of sensory preconditioning increased the responses of a recorded unit to the stimulation of the non-adjacent vibrissa. In contrast, following explicitly unpaired control experiments, neuronal responsiveness decreased. The effect of pairing was further enhanced by local, microiontophoretic delivery of NMDA and the nitric oxide synthase inhibitor L-NAME and reduced by the NMDA receptor competitive antagonist AP5. These results and the fact that the influence of the pharmacological agents on neuronal excitability were either transient (liinited to the delivery period) or simply absent indicated that the somatosensory cerebral cortex is one site where plasticity emerges following whisker pairing. In subsequent experiments, using a similar conditioning paradigm that relied on evoked potential rather than single-unit recordings, increases in the responses of cortical neurons to the non-adjacent whisker were blocked by atropine sulfate, an antagonist of muscarinic cholinoreceptors. Administration of norn-ial saline or atropine methyl nitrate, a muscarinic antagonist that did not cross the blood-brain barrier, instead of atropine sulfate, did not affect plasticity. Analysis of the behavioral state of the animal showed that the changes observed in the evoked potential could not be attributed to fluctuations m the behavioral state of the animal. By combining the results described in this thesis with data foimd in related literature, the author hypothesizes that whisker pairing induces an acetylcholine-dependent form of plasticity within the somatosensory cortex through Hebbian mechanisms.

Science and engineering problems frequently require solving a sequence of single linear systems or a sequence of dual linear systems. We develop algorithms that recycle Krylov subspaces and preconditioners from one system (or pair of systems) in the sequence to the next, leading to efficient solutions. Besides the benefit of only having to store few Lanczos vectors, using BiConjugate Gradients (BiCG) to solve dual linear systems may have application-specific advantages. For example, using BiCG to solve the dual linear systems arising in interpolatory model reduction provides a backward error formulation in the model reduction framework. Using BiCG to evaluate bilinear forms -- for example, in the variational Monte Carlo (VMC) algorithm for electronic structure calculations -- leads to a quadratic error bound. Since one of our focus areas is sequences of dual linear systems, we introduce recycling BiCG, a BiCG method that recycles two Krylov subspaces from one pair of dual linear systems to the next pair. The derivation of recycling BiCG also builds the foundation for developing recycling variants of other bi-Lanczos based methods like CGS, BiCGSTAB, BiCGSTAB2, BiCGSTAB(l), QMR, and TFQMR. We develop a generalized bi-Lanczos algorithm, where the two matrices of the bi-Lanczos procedure are not each other's conjugate transpose but satisfy this relation over the generated Krylov subspaces. This is sufficient for a short term recurrence. Next, we derive an augmented bi-Lanczos algorithm with recycling and show that this algorithm is a special case of generalized bi-Lanczos. The Petrov-Galerkin approximation that includes recycling in the iteration leads to modified two-term recurrences for the solution and residual updates. We generalize and extend the framework of our recycling BiCG to CGS, BiCGSTAB and BiCGSTAB2. We perform extensive numerical experiments and analyze the generated recycle space. We test all of our recycling algorithms on a discretized partial differential equation (PDE) of convection-diffusion type. This PDE problem provides well-known test cases that are easy to analyze further. We use recycling BiCG in the Iterative Rational Krylov Algorithm (IRKA) for interpolatory model reduction and in the VMC algorithm. For a model reduction problem, we show up to 70% savings in iterations, and we also demonstrate that solving the problem without recycling leads to (about) a 50% increase in runtime. Experiments with recycling BiCG for VMC gives promising results. We also present an algorithm that recycles preconditioners, leading to a dramatic reduction in the cost of VMC for large(r) systems. The main cost of the VMC method is in constructing a sequence of Slater matrices and computing the ratios of determinants for successive Slater matrices. Recent work has improved the scaling of constructing Slater matrices for insulators, so that the cost of constructing Slater matrices in these systems is now linear in the number of particles. However, the cost of computing determinant ratios remains cubic in the number of particles. With the long term aim of simulating much larger systems, we improve the scaling of computing determinant ratios in the VMC method for simulating insulators by using preconditioned iterative solvers. The main contribution here is the development of a method to efficiently compute for the Slater matrices a sequence of preconditioners that make the iterative solver converge rapidly. This involves cheap preconditioner updates, an effective reordering strategy, and a cheap method to monitor instability of ILUTP preconditioners. Using the resulting preconditioned iterative solvers to compute determinant ratios of consecutive Slater matrices reduces the scaling of the VMC algorithm from O(n^3) per sweep to roughly O(n^2), where n is the number of particles, and a sweep is a sequence of n steps, each attempting to move a distinct particle. We demonstrate experimentally that we can achieve the improved scaling without increasing statistical errors. / Ph. D.

<p dir="ltr">A novel computational electromagnetic (CEM) solver using potential-based formulations and discrete exterior calculus (DEC) is proposed. The proposed solver consists of two parts: the DEC A-Phi solver and the DEC F-Psi solver. A and Phi are the magnetic vector potential and electric scalar potential of the electromagnetic (EM) field, respectively; F and Psi are the electric vector potential and magnetic scalar potential, respectively. The two solvers are dual to each other, and most research is carried out with respect to the DEC A-Phi solver.</p><p dir="ltr">Systematical approach for constructing the DEC A-Phi matrix equations is provided in this thesis, including the construction of incidence matrices, Hodge star operators and different boundary conditions. The DEC A-Phi solver is proved to be broadband stable from DC to optics, while classical CEM solvers suffer from stability issues at low frequencies (also known as the low-frequency breakdown). The proposed solver is ideal for broadband and multi-scale analysis, which is of great importance in modern industry.</p><p dir="ltr">To empower the proposed solver with the ability to solve industry problems with large number of unknowns, iterative solvers are preferred. The error-minimization mechanism buried in iterative solvers allows user to control the effect of numerical error accumulation to the solution vector. Proper preconditioners are almost always needed to accelerate the convergence of iterative solvers in large scale problems. In this thesis, preconditioning schemes for the proposed solver are studied.</p><p dir="ltr">In the DEC A-Phi solver, current sources can be applied easily, but it is difficult to implement voltage sources. To incorporate voltage sources in the potential-based solver, the DEC F-Psi solver is proposed. The DEC A-Phi and F-Psi solvers are dual formulations to each other, and the construction of the F-Psi solver can be generalized from the A-Phi solver straightforward.</p>