Abstract:
Most problems in gravitational lensing require numerical solutions. The most frequent types of problems are (1) finding multiple images of a single source and classifying the images according to their properties like magnification or distortion; (2) propagating light rays through large cosmological simulations; and (3) reconstructing mass distributions from their tidal field. This lecture describes methods for solving such problems. Emphasis is put on using adaptive-grid methods for finding images, issues of spatial resolution and reliability of statistics for weak lensing by large-scale structures, and methodical questions related to shear-inversion techniques.

Abstract:
The reliability of cluster lens reconstruction techniques based on weak lensing is studied in terms of the accuracy of their reproduction of the total cluster mass as a function of distance from the cluster center. To do so, a variety of reconstruction algorithms is applied to synthetic lensing data created using a sample of 60 numerically modeled clusters, and the mass reconstruction is compared to the known deflector mass. The results can be summarized as follows: (1) Reconstruction algorithms which require integrations extending over the entire real plane yield unreliable results, because they give rise to boundary effects which are hard to control; mass overestimates are more likely and more substantial in this case than underestimates. (2) Reconstruction techniques which avoid these boundary effects yield reliable lower bounds to the cluster mass. The tightness of such bounds depends on the size of the field, which can be extended synthetically to improve the results considerably. For the sample of numerical cluster models, the best lower bounds, achieved by combining synthetic field extension with non-linear, finite-field reconstruction, decline from 100% to 80% of the true cluster mass going from the cluster center to an angular distance of 2.5'. The 80% error bars of the lower mass bounds are +/- 10% to 15%.

Abstract:
For a few years now, cosmology has a standard model. By this term, we mean a consistent theoretical background which is at the same time simple and broad enough to offer coherent explanations for the vast majority of cosmological phenomena. This review will briefly summarise the cosmological model, then proceed to discuss what we know from observations about the evolution of the Universe and its contents, and what we conclude about the origin and the future of the Universe and the structures it contains.

Abstract:
It is shown in this paper that deviations of galaxy cluster lenses from spherical symmetry can render mass estimates for galaxy clusters based on the formation of large arcs systematically to high. Numerical models show that the mass needed for producing large arcs in clusters can be notably smaller than expected from simple spherically symmetric lens models. The reason is that the enhanced tidal effect in asymmetric and substructured lenses can compensate for part of the convergence necessary for strong lensing effects. An analytic argument is given to explain why deviations from radial symmetry will in general decrease the required lens mass. Mass estimates assuming radially symmetric lenses are on average too high by a factor of $\simeq1.6$, and with a probability of $\simeq20\%$ by a factor of $\simeq2$.

Abstract:
The angular two-point correlation function between background QSOs and foreground galaxies induced by gravitational lensing is derived. It is shown that the shape of this correlation function depends sensitively on the spectrum of the density fluctuations in the Universe, thus providing a possibility to distinguish between different models for the spectrum. Using numerical large-scale structure simulations, I estimate that the QSO-galaxy correlation function can be measured from galaxy counts down to $\simeq21^{\rm st}\ldots22^{\rm nd}$ magnitude in fields with radius $\la25'$ around $50\ldots100$ QSOs with redshift $z\ge1$. Since the QSO-galaxy correlation function is proportional to $(-a-1)b$, where $b$ is the biasing factor of galaxy formation and $a$ is the effective slope of the QSO number counts, steep number counts are favorable for this kind of analysis. I show that $a=-4\ldots-5$ can be achieved with the 1-Jansky sample of radio-loud QSOs when the double-waveband magnification bias is employed. Moreover, the cross-correlation analysis allows to determine the galaxy-formation bias factor $b$.

Abstract:
Navarro, Frenk, \& White have recently found numerically that the density profile of dark-matter halos can be described by a universal two-parameter function over a broad range of halo masses. The profile is singular, approaching the halo center with $\rho\propto r^{-1}$. It had been argued previously that radially distorted, gravitationally lensed images of background sources in galaxy clusters, so-called radial arcs, required a flat core in the cluster density profile. Such radial arcs have so far been detected in two galaxy clusters, in apparent contradiction with a singular density profile. I show here that the profile suggested by Navarro et al. can produce radial arcs despite its central singularity, and describe how the two parameters of the profile can be determined in clusters where radial and tangential arcs are observed. I then apply this analysis to the two clusters where radial arcs were detected. In both cases, the redshifts of the radial arcs are yet unknown, hence definitive conclusions on the profile parameters cannot yet be drawn. Numerically determined values for the parameters of cluster-sized halos can, however, be used to predict the range of the unknown arc redshifts, thus providing a direct observational test for the proposed density profile. A potential difficulty with the profile is that the radial magnification of tangential arcs is large, hence tangential arcs should be thick or their sources should be very thin in the radial direction.

Abstract:
I present an overview of strong and weak gravitational lensing by galaxy clusters. After briefly introducing the principles of gravitational lensing, I discuss the main lessons learned from lensing on the mass distribution in clusters and their relation to cosmology.

Abstract:
Given its extraordinary spatial resolution and sensitivity, the projected Next Generation Space Telescope (NGST) is likely to detect a large number of high-redshift QSOs lensed by spiral galaxies. Using realistic models for the QSO and spiral populations, we calculate the expected number density of detectable QSOs multiply imaged by spirals, and investigate the influence of various evolution effects on that number density. It is shown that NGST will probably find of order ten lensed QSOs per square degree at 26th magnitude in the V and L bands, and that various observable quantities like the total number density of lensed QSOs in these two bands, the ratio between the number densities of lensed QSOs in the V and L bands, the fraction of QSOs with more than two images and so forth can be used to constrain the evolution of the QSOs, the spirals, the dust in spirals, and the masses of spiral disks.

Abstract:
Full-sky microwave surveys like the upcoming Planck satellite mission will detect of order 10^4 galaxy clusters through their thermal Sunyaev-Zel'dovich effect. I investigate the properties of the gravitationally lensing subsample of these clusters. The main results are: (1) The combined sample comprises >~70% of the complete sample. (2) It is confined to redshifts 0.2+-0.1, and to masses (5+-3) x 10^14 solar masses. (3) Using a particular measure for the weak lensing effect, viz. the aperture mass, cluster masses can be determined with a relative accuracy of ~20% if their density profile is known. Consequently, the mass function of the combined sample can accurately be measured. (4) For low-density universes, I predict a sharp peak in the measured (aperture) mass function near 5 x 10^14 solar masses and explain its origin, showing that the peak will be absent in high-density universes. (5) The location of the peak and the exponential decrease of the mass function on its high-mass side will allow the determination of the amplitude of the dark-matter power spectrum on the cluster scale and the baryon fraction in clusters, and constrain the thermal history of the intracluster gas.

Abstract:
Gravitational lensing has developed into one of the most powerful tools for the analysis of the dark universe. This review summarises the theory of gravitational lensing, its main current applications and representative results achieved so far. It has two parts. In the first, starting from the equation of geodesic deviation, the equations of thin and extended gravitational lensing are derived. In the second, gravitational lensing by stars and planets, galaxies, galaxy clusters and large-scale structures is discussed and summarised.