Approximation of Bayesian inverse problems for PDEs

Inverse problems are often ill posed, with solutions that depend sensitively on data. In any numerical approach to the solution of such problems, regularization of some form is needed to counteract the resulting instability. This paper is based on an approach to regularization, employing a Bayesian formulation of the problem, which leads to a notion of well posedness for inverse problems, at the level of probability measures. The stability which results from this well posedness may be used as the basis for quantifying the approximation, in finite dimensional spaces, of inverse problems for functions. This paper contains a theory which utilizes this stability property to estimate the distance between the true and approximate posterior distributions, in the Hellinger metric, in terms of error estimates for approximation of the underlying forward problem. This is potentially useful as it allows for the transfer of estimates from the numerical analysis of forward problems into estimates for the solution of the related inverse problem. It is noteworthy that, when the prior is a Gaussian random field model, controlling differences in the Hellinger metric leads to control on the differences between expected values of polynomially bounded functions and operators, including the mean and covariance operator. The ideas are applied to some non-Gaussian inverse problems where the goal is determination of the initial condition for the Stokes or Navier–Stokes equation from Lagrangian and Eulerian observations, respectively

The US current account deficit : how did it come about and what are the policy implications

One of the most remarkable characteristics of the world economy today is the enormous, ever worsening US balance of payments current account deficit, which reached a record level of 5.7 p.c. of GDP in 2004. This has given rise to concerns in academic and political circles regarding the sustainability of the current situation and the potential dangers for the global economy of a sudden, disorderly adjustment. The size of the US current account deficit is not only unprecedented in American post-war history, but it also seems to be exceptional from an international perspective. Moreover, the US deficit contrasts with a surplus in virtually every other region and the problem has consequently taken on a global dimension. The increase in the US current account deficit recorded in the nineties reflects an internal American shortfall in savings. Whereas the private savings-investment equilibrium was restored in 2002 and 2003, the same period saw a huge deficit in the public sector budget. The start of the new millennium brought notable changes in the way the US current account deficit was financed since investments by Asian public authorities in American government debt instruments largely took over the position previously occupied by European private foreign direct investments and investments in equities. It is sometimes put forward that the US, unlike other countries facing similar circumstances, is safeguarded from an attack on its currency because of its prominent role in the international financial system. According to an influential school of thought in economic literature, the current international system can even be seen as a “revived” Bretton Woods system. Indeed, a number of East-Asian countries, including China, use a fixed or quasi-fixed exchange rate against the dollar, which brings to mind an informal dollar standard. Although this set of circumstances has undoubtedly offered various regions in the world a number of mutual benefits during recent years, these exchange rate relations may nevertheless have caused some distortions in US spending, whereas Asian countries have to deal with a growing exchange rate risk on their official reserves. Different scenarios are conceivable to deal with the global imbalances. The results of model simulations show the huge effort required to significantly reduce the US current account deficit which highlights the scale of the problem, emphasising the need for simultaneous economic policy measures in the different economies involved. The concern over global imbalances and the development of exchange rates also feature prominently on the agenda of international forums such as the G7 or G20 meetings. In the statements issued at those meetings, the need for a common approach to tackle the global imbalances is given priority and the belief that excessive exchange rate volatility is not desirable is underlined.current account imbalances, United States current account, financial flows into the United States, international monetary system

Local detection of three-dimensional inclusions in electrical impedance tomography

journal articl

Thermally latent water in a polymer matrix.

Possible phase transition of water in polymer-water systems has been understood (assumed) to be only either freezing/melting or nothing, without relation to whether the polymer was soluble in water or whether its water content was high or low. This general understanding of water structure has been structured on the basis of calorimetric analyses such as differential scanning calorimetry, DSC. DSC is one of the most frequently used methods to analyze the water structure in polymer-water systems because the data obtained are relatively easy to interpret. This easiness of interpretation, however, can be accepted only if based on the understanding stated above. Unexpectedly, results of the infrared spectroscopic analysis presented here completely denied the general understanding and provided definite evidence of the existence of all phase transitions among the three states of water in a polymer solid. Furthermore, the impossibility of detection of condensation, deposition, sublimation, and vaporization by calorimetric analysis was revealed

Data assimilation insights on selecting the most valuable atmospheric measurements

We discuss how objective guidance on selecting the most valuable atmospheric measurements on future Mars spacecraft missions can be provided through already developed Martian atmospheric data assimilation systems, and in particular through Observing System Simulation Experiments (OSSEs) which are widely used to design instruments for the Earth’s atmosphere

Random acyclic networks

Directed acyclic graphs are a fundamental class of networks that includes citation networks, food webs, and family trees, among others. Here we define a random graph model for directed acyclic graphs and give solutions for a number of the model's properties, including connection probabilities and component sizes, as well as a fast algorithm for simulating the model on a computer. We compare the predictions of the model to a real-world network of citations between physics papers and find surprisingly good agreement, suggesting that the structure of the real network may be quite well described by the random graph.Comment: 4 pages, 2 figure

The optical depth of the Universe to ultrahigh energy cosmic ray scattering in the magnetized large scale structure

This paper provides an analytical description of the transport of ultrahigh energy cosmic rays in an inhomogeneously magnetized intergalactic medium. This latter is modeled as a collection of magnetized scattering centers such as radio cocoons, magnetized galactic winds, clusters or magnetized filaments of large scale structure, with negligible magnetic fields in between. Magnetic deflection is no longer a continuous process, it is rather dominated by scattering events. We study the interaction between high energy cosmic rays and the scattering agents. We then compute the optical depth of the Universe to cosmic ray scattering and discuss the phenomological consequences for various source scenarios. For typical parameters of the scattering centers, the optical depth is greater than unity at 5x10^{19}eV, but the total angular deflection is smaller than unity. One important consequence of this scenario is the possibility that the last scattering center encountered by a cosmic ray be mistaken with the source of this cosmic ray. In particular, we suggest that part of the correlation recently reported by the Pierre Auger Observatory may be affected by such delusion: this experiment may be observing in part the last scattering surface of ultrahigh energy cosmic rays rather than their source population. Since the optical depth falls rapidly with increasing energy, one should probe the arrival directions of the highest energy events beyond 10^{20}eV on an event by event basis to circumvent this effect.Comment: version to appear in PRD; substantial improvements: extended introduction, sections added on angular images and on direction dependent effects with sky maps of optical depth, enlarged discussion of Auger results (conclusions unchanged); 27 pages, 9 figure

Phase diagram and critical properties of the frustrated Kondo necklace model in a magnetic field

The critical properties of the frustrated Kondo necklace model with a half saturation magnetization ($m=1/2$) have been studied by means of an exact-diagonalization method. It is shown from bosonization technique that the model can be effectively expressed as a quantum sine-Gordom model. Thus it may show three (dimer plateau, N{\'e}el plateau and Tomonaga-Luttinger liquid) phases due to competitions among the Ising anisotropy $\Delta$, and the nearest- and next-nearest-neighbor exchange interactions $J_1$ and $J_2$. The boundary lines on the $\Delta-J_2/J_1$ phase diagram separating the three phases are determined by the method of level spectroscopy based on the conformal field theory.Comment: 5 pages, 5 figure