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

Variational data assimilation using targetted random walks

The variational approach to data assimilation is a widely used methodology for both online prediction and for reanalysis (offline hindcasting). In either of these scenarios it can be important to assess uncertainties in the assimilated state. Ideally it would be desirable to have complete information concerning the Bayesian posterior distribution for unknown state, given data. The purpose of this paper is to show that complete computational probing of this posterior distribution is now within reach in the offline situation. In this paper we will introduce an MCMC method which enables us to directly sample from the Bayesian\ud posterior distribution on the unknown functions of interest, given observations. Since we are aware that these\ud methods are currently too computationally expensive to consider using in an online filtering scenario, we frame this in the context of offline reanalysis. Using a simple random walk-type MCMC method, we are able to characterize the posterior distribution using only evaluations of the forward model of the problem, and of the model and data mismatch. No adjoint model is required for the method we use; however more sophisticated MCMC methods are available\ud which do exploit derivative information. For simplicity of exposition we consider the problem of assimilating data, either Eulerian or Lagrangian, into a low Reynolds number (Stokes flow) scenario in a two dimensional periodic geometry. We will show that in many cases it is possible to recover the initial condition and model error (which we describe as unknown forcing to the model) from data, and that with increasing amounts of informative data, the uncertainty in our estimations reduces

Direct Displacement Based Design of Regular Steel Moment Resisting Frames

AbstractDisplacement Based Design method represents a new approach to performance-based design. This research tries to assess the Direct Displacement Based Design (DDBD) method for regular steel moment resisting frames and develop a reliable design method for them so that they withstand various seismic levels within certain performance levels. For this purpose, regular steel frames with 4, 8, 12, 16 stories are designed based on DDBD approach utilizing displacement spectrum of the Iranian Code of Practice for Seismic Resistant Design of Buildings (Standard No. 2800). In order to evaluate seismic response of the designed structures, a series of non-linear time-history analyses have been performed under different records compatible with Standard No. 2800. All the non-linear analyses were carried out using the fiber-element models developed in Seismostruct computer program. According to the results, inter-story drift profile of the structure which is corresponding to its damage was less than the allowable value in most cases. Also, Maximum displacement profile of the structure along its height is completely matched with the primary assumed design profile. The structures have mostly experienced similar residual drift values under different records. In summary, the method performed quite satisfactorily in terms of story maximum displacements, maximum interstory drifts and story ductility demands, even for tall models

Effective Symbolic Dynamics

AbstractWe investigate computable subshifts and the connection with effective symbolic dynamics. It is shown that a decidable Π10 class P is a subshift if and only if there is a computable function F mapping 2N to 2N such that P is the set of itineraries of elements of 2N. A Π10 subshift is constructed which has no computable element. We also consider the symbolic dynamics of maps on the unit interval

Satisfaction with hearing aids based on technology and style among hearing impaired persons

Introduction: Hearing loss is one of the most disabling impairments. Using a hearing aid as an attempt to improve the hearing problem can positively affect the quality of life for these people. This research was aimed to assess satisfaction of hearing impaired patients with their hearing aids regarding the employed technology and style. Materials and Methods: This descriptive-analytic cross-sectional research was conducted on 187 subjects with hearing loss who were using a hearing aid. The subjects were over 18 years of age and were using a hearing aid for at least 6 months. The Persian version of Satisfaction with Amplification in Daily Life (SADL) questionnaire was the instrument which was used for assessing satisfaction with the hearing aid. Cronbach's alpha was calculated to be 0.80 for instrument reliability. Results: A significant difference was observed among satisfaction subscales' mean scores with hearing aid technology. Also a significant difference was observed between the total satisfaction score and the hearing aid model. With respect to the analysis of satisfaction with the hearing aid and its style, cost and services was the only subscale which showed a significant difference (P=0.005). Conclusion: Respondents using hearing aids with different technology and style were estimated to be quite satisfied. Training audiologists in using more appropriate and fitting hearing aids in addition to using self-reporting questionnaires like SADL for estimating patients' social condition and participation in their life can essentially change their disability condition and countervail their hearing loss

The Effect of Ascorbic Acid on Growth and Some Biochemical Properties of Bell Pepper (Capsicum annuum) Seedling under Salinity Stress

Introduction  Plants are sometimes exposed to several stresses during their growth and development. Salinity stress is one of the most common abiotic stresses that significantly reduces the growth and yield of most plant species. A large part of the soils and a significant volume of all water resources in Iran are affected by salinity to varying degrees. Due to the rapid population growth and the need for more food and the water crisis, the use of unconventional water such as salt water or wells and treated wastewater in agriculture has received more attention. Soil and water salinity is one of the integrative problems in agriculture and is one of the most important barriers to crop production in the world. Salinity stress reduces the ratio of carbon dioxide to oxygen in the leaves by closing the leaves stomata and prevents the stabilization of carbon dioxide. Under these conditions, the formation rate of reactive oxygen species in chloroplasts and mitochondria increases, the electrons produced react with oxygen by photosynthesis and lead to the production of oxygen free radicals. These affects the growth, yield and quality of agricultural products. The use of antioxidant compounds can moderate the effects of this stress. Many compounds have been used to reduce the harmful effects of salinity stress. Ascorbic acid is a water soluble antioxidant and has some effects on plant resistance against environmental stresses by neutralizing free radicals. Researches showed that ascorbic acid increased plant resistance on various stresses such as salinity and drought. It has been widely used to control the effects of salinity stress. The aim of this study was to investigate the effect of ascorbic acid on resistance to salinity stress in bell pepper seedlings.   Materials and Methods  This study was done as a factorial experiment in a completely randomized design including ascorbic acid treatment (0, 1, 3 and 5 mM) and salinity stress of sodium chloride (0 and 100 mM) in 3 replications. The seeds of bell pepper Cv. California Wonder were planted in a seedling tray containing cocopeat. After emergence of cotyledons, they were fertilized with complete fertilizer (N20, P20, K20). At the stage of three true leaves the  salinity treatment were started. One week before the beginning of salinity treatment, ascorbic acid was sprayed on plants and repeated two more times by seven days interval. Salinity treatment was applied by irigation for three weeks. After producing 5 leaves, a few growth properties (plant fresh and dry weight, root fresh and dry weight, leaf number and leaf area), biochemical characteristics (amounts of total chlorophyll, proline, soluble proteins, ion leakage and malondialdehyde and activity of Catalase an Proxidase enzymes) and amounts of some elements (K, P, Na, Fe, Zn and Cu) were measured.   Results and Discussion  Based on the results, the simple effect of salinity stress, ascorbic acid and the combined effect of them on ion leakage and amounts of malondialdehyde were significant at the level of 1% probability. Salinity stress increased the amounts of ion leakage and malondialdehyde and application of 5 mM ascorbic acid reduced the amount of these traits to 41.01% and 46.58% compared to the control respectively. The effect of salinity stress, ascorbic acid and the combined effect of them on the concentration of proline and chlorophyll in the leaves of bell pepper were significant at the level of 1% probability. Salinity stress increased the amount of proline but decreased chlorophyll and leaf protein of pepper seedlings and application of 5 mM ascorbic acid increased them to 79.42% and 46.57% compare to control respectively. Under salinity stress the uptake of iron, zinc, copper and phosphorus decreased; using of 5 mM ascorbic acid increasing accumulation of these elements in leaves .On the other hand ascorbic asid could decrease the amounts of sodium accumulation in seedling,s leaves. The results showed that, salinity decreased growth indices (number of leaves, leaf surface, fresh weight of roots and plants) and ascorbic acid reduced the effect of salinity stress on growth parameters. Under salinity stress, 5 mM ascorbic acid treatment increased the leaf surface to 36.30% compared to salinity without ascorbic acid tretment. The effects of salinity stress and the combined effect of salinity stress and ascorbic acid on the amount of catalase and peroxidase enzymes were significant at the level of 1% probability. The effect of ascorbic acid was significant for catalase at 5% probability level and peroxidase at 1% probability level. Although salinity increased the amounts of the antioxidant catalase and peroxidase enzymes but ascorbic acid could increased the activity of them in saline condition.   Conclusion  It seems that application of ascorbic acid as an antioxidant could reduce the effects of salinity stress by increasing the activity of catalase and peroxidase enzymes, reducing sodium uptake and increasing the uptake of phosphorus, potassium, iron, zinc and copper. These could have positive effects on the growth parameters of bell pepper seedlings in salinity conditions. Generally, on based of obtained results, spraying 5 mM Ascorbic acid on bell pepper seedlings modified the resistance of them in salinity stress conditions

A TV-Gaussian prior for infinite-dimensional Bayesian inverse problems and its numerical implementations

Many scientific and engineering problems require to perform Bayesian inferences in function spaces, in which the unknowns are of infinite dimension. In such problems, choosing an appropriate prior distribution is an important task. In particular we consider problems where the function to infer is subject to sharp jumps which render the commonly used Gaussian measures unsuitable. On the other hand, the so-called total variation (TV) prior can only be defined in a finite dimensional setting, and does not lead to a well-defined posterior measure in function spaces. In this work we present a TV-Gaussian (TG) prior to address such problems, where the TV term is used to detect sharp jumps of the function, and the Gaussian distribution is used as a reference measure so that it results in a well-defined posterior measure in the function space. We also present an efficient Markov Chain Monte Carlo (MCMC) algorithm to draw samples from the posterior distribution of the TG prior. With numerical examples we demonstrate the performance of the TG prior and the efficiency of the proposed MCMC algorithm

Besov priors for Bayesian inverse problems

We consider the inverse problem of estimating a function $u$ from noisy, possibly nonlinear, observations. We adopt a Bayesian approach to the problem. This approach has a long history for inversion, dating back to 1970, and has, over the last decade, gained importance as a practical tool. However most of the existing theory has been developed for Gaussian prior measures. Recently Lassas, Saksman and Siltanen (Inv. Prob. Imag. 2009) showed how to construct Besov prior measures, based on wavelet expansions with random coefficients, and used these prior measures to study linear inverse problems. In this paper we build on this development of Besov priors to include the case of nonlinear measurements. In doing so a key technical tool, established here, is a Fernique-like theorem for Besov measures. This theorem enables us to identify appropriate conditions on the forward solution operator which, when matched to properties of the prior Besov measure, imply the well-definedness and well-posedness of the posterior measure. We then consider the application of these results to the inverse problem of finding the diffusion coefficient of an elliptic partial differential equation, given noisy measurements of its solution.Comment: 18 page