33 research outputs found
On the Convergence of Adaptive Iterative Linearized Galerkin Methods
A wide variety of different (fixed-point) iterative methods for the solution
of nonlinear equations exists. In this work we will revisit a unified iteration
scheme in Hilbert spaces from our previous work that covers some prominent
procedures (including the Zarantonello, Ka\v{c}anov and Newton iteration
methods). In combination with appropriate discretization methods so-called
(adaptive) iterative linearized Galerkin (ILG) schemes are obtained. The main
purpose of this paper is the derivation of an abstract convergence theory for
the unified ILG approach (based on general adaptive Galerkin discretization
methods) proposed in our previous work. The theoretical results will be tested
and compared for the aforementioned three iterative linearization schemes in
the context of adaptive finite element discretizations of strongly monotone
stationary conservation laws
A link between the steepest descent method and fixed-point iterations
We will make a link between the steepest descent method for an unconstrained
minimisation problem and fixed-point iterations for its Euler-Lagrange
equation. In this context, we shall rediscover the preconditioned nonlinear
conjugate gradient method for the discretised problem. The benefit of the link
between the two methods will be illustrated by a numerical experiment
On the convergence rate of the Ka\v{c}anov scheme for shear-thinning fluids
We explore the convergence rate of the Ka\v{c}anov iteration scheme for
different models of shear-thinning fluids, including Carreau and power-law type
explicit quasi-Newtonian constitutive laws. It is shown that the energy
difference contracts along the sequence generated by the iteration. In
addition, an a posteriori computable contraction factor is proposed, which
improves previously derived bounds on the contraction factor in the context of
the power-law model. Significantly, this factor is shown to be independent of
the choice of the cut-off parameters whose use was proposed in the literature
for the Ka\v{c}anov iteration applied to the power-law model. Our analytical
findings are confirmed by a series of numerical experiments
Adaptive local minimax Galerkin methods for variational problems
In many applications of practical interest, solutions of partial differential
equation models arise as critical points of an underlying (energy) functional.
If such solutions are saddle points, rather than being maxima or minima, then
the theoretical framework is non-standard, and the development of suitable
numerical approximation procedures turns out to be highly challenging. In this
paper, our aim is to present an iterative discretization methodology for the
numerical solution of nonlinear variational problems with multiple (saddle
point) solutions. In contrast to traditional numerical approximation schemes,
which typically fail in such situations, the key idea of the current work is to
employ a simultaneous interplay of a previously developed local minimax
approach and adaptive Galerkin discretizations. We thereby derive an adaptive
local minimax Galerkin (LMMG) method, which combines the search for saddle
point solutions and their approximation in finite-dimensional spaces in a
highly effective way. Under certain assumptions, we will prove that the
generated sequence of approximate solutions converges to the solution set of
the variational problem. This general framework will be applied to the specific
context of finite element discretizations of (singularly perturbed) semilinear
elliptic boundary value problems, and a series of numerical experiments will be
presented
Adaptive iterative linearization Galerkin methods for nonlinear problems
A wide variety of (fixed-point) iterative methods for the solution of nonlinear equations (in Hilbert spaces) exists. In many cases, such schemes can be interpreted as iterative local linearization methods, which, as will be shown, can be obtained by applying a suitable preconditioning operator to the original (nonlinear) equation. Based on this observation, we will derive a unified abstract framework which recovers some prominent iterative schemes. In particular, for Lipschitz continuous and strongly monotone operators, we derive a general convergence analysis. Furthermore, in the context of numerical solution schemes for nonlinear partial differential equations, we propose a combination of the iterative linearization approach and the classical Galerkin discretization method, thereby giving rise to the so-called iterative linearization Galerkin (ILG) methodology. Moreover, still on an abstract level, based on two different elliptic reconstruction techniques, we derive a posteriori error estimates which separately take into account the discretization and linearization errors. Furthermore, we propose an adaptive algorithm, which provides an efficient interplay between these two effects. In addition, the ILG approach will be applied to the specific context of finite element discretizations of quasilinear elliptic equations, and some numerical experiments will be performed
Energy contraction and optimal convergence of adaptive iterative linearized finite element methods
We revisit a unified methodology for the iterative solution of nonlinear
equations in Hilbert spaces. Our key observation is that the general approach
from [Heid & Wihler, Math. Comp. 89 (2020), Calcolo 57 (2020)] satisfies an
energy contraction property in the context of (abstract) strongly monotone
problems. This property, in turn, is the crucial ingredient in the recent
convergence analysis in [Gantner et al., arXiv:2003.10785]. In particular, we
deduce that adaptive iterative linearized finite element methods (AILFEMs) lead
to full linear convergence with optimal algebraic rates with respect to the
degrees of freedom as well as the total computational time
Gradient Flow Finite Element Discretizations with Energy-Based Adaptivity for the Gross-Pitaevskii Equation
We present an effective adaptive procedure for the numerical approximation of
the steady-state Gross-Pitaevskii equation. Our approach is solely based on
energy minimization, and consists of a combination of gradient flow iterations
and adaptive finite element mesh refinements. Numerical tests show that this
strategy is able to provide highly accurate results, with optimal convergence
rates with respect to the number of freedom
Discovery and prioritization of variants and genes for kidney function in >1.2 million individuals
Genes underneath signals from genome-wide association studies (GWAS) for kidney function are promising targets for functional studies, but prioritizing variants and genes is challenging. By GWAS meta-analysis for creatinine-based estimated glomerular filtration rate (eGFR) from the Chronic Kidney Disease Genetics Consortium and UK Biobank (n = 1,201,909), we expand the number of eGFRcrea loci (424 loci, 201 novel; 9.8% eGFRcrea variance explained by 634 independent signal variants). Our increased sample size in fine-mapping (n = 1,004,040, European) more than doubles the number of signals with resolved fine-mapping (99% credible sets down to 1 variant for 44 signals, ≤5 variants for 138 signals). Cystatin-based eGFR and/or blood urea nitrogen association support 348 loci (n = 460,826 and 852,678, respectively). Our customizable tool for Gene PrioritiSation reveals 23 compelling genes including mechanistic insights and enables navigation through genes and variants likely relevant for kidney function in human to help select targets for experimental follow-up