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"While the prediction of observations is a forward problem, the use of actual observations to infer the properties of a model is an inverse problem. Inverse problems are difficult because they may not have a unique solution. The description of uncertainties plays a central role in the theory, which is based on probability theory. This book proposes a general approach that is valid for linear as well as for nonlinear problems. The philosophy is essentially probabilistic and allows the reader to understand the basic difficulties appearing in the resolution of inverse problems. The book attempts to explain how a method of acquisition of information can be applied to actual real-world problems, and many of the arguments are heuristic."

"Metode numerik eksplisit tahap tunggal pada umumnya hanya menyelesaikan persamaan diferensial biasa order pertama. Pada tugas akhir ini penulis mencoba menggabungkan beberapa cara substitusi, approksimasi dan diskritisasi, yang mana dengan penambahan fasilitas tersebut, metode Runge-Kutta dan metode Euler mampu menyelesaikan sistim persamaan diferensial dan persamaan diferensial parsil dengan batasan-batasan tertentu. Gabungan metode diskritisasi dan numerik ini disebut metode garis lurus Euler dan metode garis lurus Runge-Kutta."

"Inverse limits with set-valued functions are quickly becoming a popular topic of research due to their potential applications in dynamical systems and economics. This brief provides a concise introduction dedicated specifically to such inverse limits. The theory is presented along with detailed examples which form the distinguishing feature of this work. The major differences between the theory of inverse limits with mappings and the theory with set-valued functions are featured prominently in this book in a positive light. "

"This book is primarily devoted to topics related to matrix versions of classical interpolation and moment problems. The major themes include Schur analysis of nonnegative Hermitian block Hankel matrices and the construction of Schur-type algorithms. This book also covers a number of recent developments in orthogonal rational matrix functions, matrix-valued Carathéodory functions and maximal weight solutions for particular matricial moment problems on the unit circle.​"

"Frontiers in Applied MathematicsInverse problems arise in a number of important practical applications, ranging from biomedical imaging to seismic prospecting. This book provides the reader with a basic understanding of both the underlying mathematics and the computational methods used to solve inverse problems. It also addresses specialized topics like image reconstruction, parameter identification, total variation methods, nonnegativity constraints, and regularization parameter selection methods.Because inverse problems typically involve the estimation of certain quantities based on indirect measurements, the estimation process is often ill-posed. Regularization methods, which have been developed to deal with this ill-posedness, are carefully explained in the early chapters of Computational Methods for Inverse Problems. The book also integrates mathematical and statistical theory with applications and practical computational methods, including topics like maximum likelihood estimation and Bayesian estimation.Several web-based resources are available to make this monograph interactive, including a collection of MATLAB m-files used to generate many of the examples and figures. These resources enable readers to conduct their own computational experiments in order to gain insight. They also provide templates for the implementation of regularization methods and numerical solution techniques for other inverse problems. Moreover, they include some realistic test problems to be used to further develop and test various numerical methods."

"Prsoses inversi dan forward modeling merpakan proses yang umum dilakukan dalam keilmuan geofisika khususnya pada metode magnetotelurik (MT) untuk memprediksi data dan parameter fisis material batuan di bawah permukaan bumi. Data dalam konteks ini merupakan resistivitas semu dan fase sementara parameter fisis merupakan resistivitas batuan. Proses tersebut biasanya dilakukan dengan menggunakan perangkat lunak tertentu yang perlu di unduh dan di install pada perangkat keras yang memenuhi spesifikasi, hal ini menjadi tidak fleksibel karena perangkat lunak perlu di install ulang jika ingin mengganti perangkat keras dan tidak semua perangkat keras dapat digunakan. Pada penelitian ini penulis mengembangkan platform website berbasis bahasa pemrograman python untuk melakukan proses forward modeling dan inversi data MT 1-D. Forward modeling dilakukan dengan menggunakan persamaan rekursif yang menyatakan hubungan antara impedansi dua lapisan sedangkan inversi dilakukan dengan menggunakan metode Levenberg-Marquardt. Pada inversi dengan stasiun lebih dari 1 maka kurva sounding hasil inversi akan dibentuk menjadi penampang resistivitas bawah permukaan dengan cara Krigging. Secara kuantitatif hasil inversi dengan website memiliki RMS error berkisar 0.08-0.21 % untuk data sintetik dan 0.84-1.84 % untuk data lapangan, secara kualitatif kurva inversi website memiliki trend yang mirip dengan software konvensional sehingga inversi website dinilai cukup valid. Dengan dikembangkannya platform web ini proses inversi dan forward modeling data MT 1-D dapat dilakukan di mana saja melalui perangkat apapun selama terdapat koneksi internet dan browser. Namun karena terdapat perbedan antara penampang hsil inversi website dengan software konvensional yang menyebabkan detil struktur tidak terlihat, maka website ini lebih cocok untuk digunakan pada keperluan akademis untuk melihat pola resistivitas bawah permukan secara umum dan bukan secara detail.

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Invers and forward modeling is a common process or technique done in the field of geophysics, particularly on magnetotelluric method (MT) in order to predict data and physical parameters of rock materials underneath the surface. Data in this context is apparent resistivity and phase, while physical parameters is the conductivity of the rock materials. This process usually conducted using a certain software that has to be installed on a hardware that meets the minimum requirement, this is not flexible in modern age because the software would need to be re-installed if we wish to use other device, moreover not all device is compatible. In this research, writer is developing a web platform based on python programming language for the purpose of forward and invers modeling of 1-D MT data. The forward modeling algorithm is done by utilizing the recursive equation which states the relationship between impedance of two sequential layers while inverse modeling is done using Levenberg-Marquardt method. In a case where the inverted data has more than 1 station, the sounding curve as the output of the inversion will be made into a resistivity crossection using Krigging. Quantitatively the inversion result using this website has an RMS error of 0.08 - 0.21 % for syntethic data and 0.84 – 1.84 % for real data, qualitatively the inversion curve resulting from this website has similar trend to an inversion curve done by a conventional software, thus the inversion using this website is valid. With the development of this web platform the inverse and forward modeling process of 1-D MT data can be conducted anywhere through all kinds of device as long as internet conection and browser is available. However due to the difference between cross section resulting from inversion using website and using conventional software that has lead to certain structure can not be seen, therefore this website is more suitable to be used in an academic purposes to see subsurface resistivity pattern in general and not in detail."

"Prsoses inversi dan forward modeling merpakan proses yang umum dilakukan dalam keilmuan geofisika khususnya pada metode magnetotelurik (MT) untuk memprediksi data dan parameter fisis material batuan di bawah permukaan bumi. Data dalam konteks ini merupakan resistivitas semu dan fase sementara parameter fisis merupakan resistivitas batuan. Proses tersebut biasanya dilakukan dengan menggunakan perangkat lunak tertentu yang perlu di unduh dan di install pada perangkat keras yang memenuhi spesifikasi, hal ini menjadi tidak fleksibel karena perangkat lunak perlu di install ulang jika ingin mengganti perangkat keras dan tidak semua perangkat keras dapat digunakan. Pada penelitian ini penulis mengembangkan platform website berbasis bahasa pemrograman python untuk melakukan proses forward modeling dan inversi data MT 1-D. Forward modeling dilakukan dengan menggunakan persamaan rekursif yang menyatakan hubungan antara impedansi dua lapisan sedangkan inversi dilakukan dengan menggunakan metode Levenberg-Marquardt. Pada inversi dengan stasiun lebih dari 1 maka kurva sounding hasil inversi akan dibentuk menjadi penampang resistivitas bawah permukaan dengan cara Krigging. Secara kuantitatif hasil inversi dengan website memiliki RMS error berkisar 0.08-0.21 % untuk data sintetik dan 0.84-1.84 % untuk data lapangan, secara kualitatif kurva inversi website memiliki trend yang mirip dengan software konvensional sehingga inversi website dinilai cukup valid. Dengan dikembangkannya platform web ini proses inversi dan forward modeling data MT 1-D dapat dilakukan di mana saja melalui perangkat apapun selama terdapat koneksi internet dan browser. Namun karena terdapat perbedan antara penampang hsil inversi website dengan software konvensional yang menyebabkan detil struktur tidak terlihat, maka website ini lebih cocok untuk digunakan pada keperluan akademis untuk melihat pola resistivitas bawah permukan secara umum dan bukan secara detail.

---

Invers and forward modeling is a common process or technique done in the field of geophysics, particularly on magnetotelluric method (MT) in order to predict data and physical parameters of rock materials underneath the surface. Data in this context is apparent resistivity and phase, while physical parameters is the conductivity of the rock materials. This process usually conducted using a certain software that has to be installed on a hardware that meets the minimum requirement, this is not flexible in modern age because the software would need to be re-installed if we wish to use other device, moreover not all device is compatible. In this research, writer is developing a web platform based on python programming language for the purpose of forward and invers modeling of 1-D MT data. The forward modeling algorithm is done by utilizing the recursive equation which states the relationship between impedance of two sequential layers while inverse modeling is done using Levenberg-Marquardt method. In a case where the inverted data has more than 1 station, the sounding curve as the output of the inversion will be made into a resistivity crossection using Krigging. Quantitatively the inversion result using this website has an RMS error of 0.08 - 0.21 % for syntethic data and 0.84 – 1.84 % for real data, qualitatively the inversion curve resulting from this website has similar trend to an inversion curve done by a conventional software, thus the inversion using this website is valid. With the development of this web platform the inverse and forward modeling process of 1-D MT data can be conducted anywhere through all kinds of device as long as internet conection and browser is available. However due to the difference between cross section resulting from inversion using website and using conventional software that has lead to certain structure can not be seen, therefore this website is more suitable to be used in an academic purposes to see subsurface resistivity pattern in general and not in detail."

"Approximate global convergence and adaptivity for coefficient inverse problems is the first book in which two new concepts of numerical solutions of multidimensional Coefficient Inverse Problems (CIPs) for a hyperbolic Partial Differential Equation (PDE) are presented. Two central questions for CIPs are addressed, how to obtain a good approximations for the exact solution without any knowledge of a small neighborhood of this solution, and how to refine it given the approximation. The book also combines analytical convergence results with recipes for various numerical implementations of developed algorithms. The developed technique is applied to two types of blind experimental data, which are collected both in a laboratory and in the field. The result for the blind backscattering experimental data collected in the field addresses a real world problem of imaging of shallow explosives."

"This volume presents a mathematical development of a recent approach to the modeling and simulation of turbulent flows based on methods for the approximate solution of inverse problems. The resulting approximate deconvolution models or ADMs have some advantages over more commonly used turbulence models, as well as some disadvantages. Present the analytical theory of ADMs, along with its connections, motivations and complements in the phenomenology of and algorithms for ADMs."