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To counteract the performance degradation in electromagnetic devices due to mechanical tolerances, assembly inaccuracies, poor knowledge of material parameters, etc., many robust design techniques have been proposed. A first class of methods takes advantage from statistical analysis to improve the average performance taking into account actual, randomly distributed values rather than the nominal figure of the parameters design values. A second class, making use of the "sensitivity" of device performance with respect to different parameters, looks for solutions as insensitive as possible with respect to possible modifications of the design parameters. In this paper, with the aim of presenting a synthetic selection of existing methods, a brief survey of the different approaches is presented. A simple test case is considered for demonstration purposes.

The mathematical description of 3D magnetic surfaces is of paramount importance in several applications, but the solution can be hard in terms of the computing effort or unsatisfactory in terms of accuracy. The paper proposes an effective methodology based on the tracking of a limited portion of the magnetic field line and on the fitting of the line onto a 3D surface described in a suitable set of 3D shape functions. In this way, the field line plays the role of a non-uniform sampling of the magnetic surface and the minimization of a suitable error allows the determination of the expansion coefficients. The computational cost of the line tracking can be limited according to the actual precision needs. The proposed approach allows the introduction of a number of indexes suited to estimate the accuracy and, if the case, to indicate how to upgrade the fitting parameters to meet the actual requirements.

Due to unavoidable tolerances in components construction and assembly, the actual magnetic fi eld in Tokamak devices differs from the nominal one. The experimental studies nowadays available indicate a strong impact of the error fields on the plasma stability in next generation, large size, devices. Therefore, a careful preliminary analysis aimed at evaluating the impact of possible error field sources, to define tolerance ranges and assess the effect of current feeders, was envisaged before realization of ITER reactor. The paper discusses the tools used to perform ITER tolerance analysis, from the viewpoints of two possible approaches to tolerancing, which are statistical analysis and worst case analysis. The two methods aim to get different characterizations of devices performance, and a critical comparison of drawbacks and advantages of both methods is reported. 1. Error fields in tokamaks In magnetic confinement controlled thermonuclear fusion devices (Tokamaks), the confinement ca- pability of the machine is quite sensitive to discrepancies between the nominal magnetic field and the actual one (called "error fields"), resulting from several causes including non axi-symmetric sources (e.g. current leads for superconducting magnets) and mechanical inaccuracies in the manufacturing and assembly of magnets. The error fields are by their intrinsic nature quite complex to describe; an effective tool for their quantitative analysis is the Fourier toroidal and poloidal decomposition of normal field component on the nominal plasma surface at the start of flat top, when the impact of error fields is the highest. Experimental studies (1) have shown that only some specific Fourier components have the highest impact on plasma confinement. With reference to ITER Tokamak, the Three Modes Error Index (TMEI) hasbeenintroducedasaneffectiveindexto describetheamountoferrorfieldsinthethreemodesrelevant for confinement, namely field harmonic m = 2, n = 1 - with toroidal periodicity index n equal to one and poloidal periodicity index m equal to two - and the two side bands m = 1, n = 1a ndm = 3, n = 1. TMEI is based on the weighted average of normalized mode coefficients, that must remain below the fixed threshold of 5 × 10 −5 to guarantee expected performance (2)

Human brain electric activity related to important physiological functions can be pictured from bio electric and/or bio magnetic measurements through inverse problems approach. In this paper an iterative strategy is presented. The strategy is optimized to reconstruct brain activity confined in a sub region of the brain volume by jointly processing both bioelectric and biomagnetic signals. The procedure takes advantage from intermediate evaluations to improve the resolution, while limiting the number of unknowns. The advantages of the proposed approach is a higher robustness against noise and uncertainty and better effectiveness. In the last few decades, important contributions in the neuroscience research came from the use of imaging techniques. In particular, new investigation approaches on the anatomy of the brain, based on structuralimagingtechniques,suchascomputeraidedtomography(CT)andmagneticresonanceimaging (MRI), have been proposed. A major field of research in neuroscience is aimed to understand metabolic processes taking place in the brain; this calls for several imaging techniques such as positron emission tomography (PET), single-photon emission tomography (SPECT) and functional magnetic resonance imaging (fMRI) measuring changes of the blood flow or oxygenation in the brain. Such techniques are abletomeasurebrainactivitywithsatisfactoryspatialresolution;unfortunatelythetimeresolutionisquite poor becauseit is limited within the range 0.1 s-1 s. Therefore further techniquescharacterizedby better time resolution, are looked for in the scientific community gathering contributions from mathematical, physicalandengineeringdisciplines. Thesetechniques(1-6)includeelectroencephalography(EEG)and magnetoencephalography(MEG). Electrical currents in the brain can be identified by processingvoltage patterns measured on the scalp (EEG) and/or by measuring the extremely weak magnetic flux density close to the scalp (MEG). The MEG has been recently introduced thanks to the development of the SQUID (Superconducting Quantum Interference Devices) sensors, able to detect the very low magnetic flux density in the extracranial region, in the order of a few fT. Compared with fMRI techniques, EEG and MEG represent an appealing alternative thanks to the time resolution (in the order of 1 ms), and to the lack of radiations (1). Unfortunately their spatial resolution could be not satisfactory (1). In addition, due to the ill posedness of the model, the uncertainty affecting the knowledge of some of the electrical

Magnets for Nuclear Magnetic Resonance applications are designed to provide high level of magnetic flux density in a testing volume with the greatest level of field homogeneity. Anyway, the manufacturing and assembly tolerances cause unavoidable uniformity degradation, that is usually counteracted by correction coils. The measurement of the magnetic field is the final verification of the complex design and fabrication process of a magnetic system, aimed also at defining currents in correction coils to improve uniformity. Anyway, when seeking high accuracy, it is necessary to assess the effect on final uniformity of uncertainties in the measurement technique. In this paper, an approach is presented to assess impact of measurement error bars on final device performance, based on Worst Case Analysis, guaranteeing quick computational time.