An accurate representation of reality in numerical variably-saturated flow models requires reliable estimates of necessary model parameters. Inverse modeling seeks to estimate parameters such as the saturated and residual water contents, the saturated hydraulic conductivity, the shape parameters of the soil hydraulic functions, using easily attainable observations of actual or cumulative water fluxes, pressure heads, water contents, and concentrations. The inverse procedure usually combines the nonlinear leastsquares-based (SSQ) parameter optimization method with a numerical solution of the variably-saturated flow and transport equations. The SSQ-based inverse method is however sensitive to outliers. A novel Squared ε-Insensitive Loss Function (SILF) approach is introduced in this study. The SILF approach is inspired by the ε-insensitive loss function proposed by Vapnik (1995). The objective function used in the SILF approach is similar to the least-squares objective function, except that it penalizes only for errors greater than a certain predefined acceptable error term ε. The SILF approach shows an improved performance over the SSQ approach in estimating the soil hydraulic parameters. Apart from providing robust estimates of the soil hydraulic parameters, the SILF approach also gives an approximation of the relative measurement error during sampling. and Presná reprezentácia skutočností v numerických modeloch prúdenia vo vodou nenasýtenej pôde vyžaduje spoľahlivé určenie potrebných parametrov modelu. Inverzným modelovaním sa snažíme o určenie takých parametrov, ako sú reziduálna vlhkosť pôdy, nasýtená hydraulická vodivosť, tvarové parametre hydraulických funkcií pôdy, využijúc ľahko realizovateľné pozorovania momentálnych alebo kumulatívnych tokov vody, tlakových výšok, vlhkostí pôdy a koncentrácií rozpustených látok. Inverzná procedúra obyčajne kombinuje nelineárnu optimalizáciu parametrov založenú na metóde najmenších štvorcov (SSQ) s numerickým riešením transportných rovníc vo vodou nenasýtenej pôde. Táto metóda (SSQ) je však citlivá na náhodné chyby. Nová, necitlivostná stratová funkcia s necitlivosťou ε(SILF), použitá v tejto štúdii, bola inšpirovaná návrhom publikovaným Vapnikom (1995). Optimalizovaná funkcia použitá v prístupe SILF je podobná tej, ktorá sa používa v metóde najmenších štvorcov s tou výnimkou, že táto penalizuje len chyby väčšie ako je určitá preddefinovaná akceptovateľná chyba ε. Pri určovaní hydraulických parametrov pôdy táto metóda SILF preukázala svoje prednosti pred prístupom SSQ. Okrem toho, že metóda SILF dáva robustné odhady hydraulických parametrov pôdy, umožňuje tiež aproximáciu relatívnych chýb merania počas odberu vzoriek.
We study the existence and the uniqueness of the weak solution of an inverse problem for a semilinear higher order ultraparabolic equation with Lipschitz nonlinearity. The main aim is to determine the weak solution of the equation and some functions that depend on the time variable, appearing on the right-hand side of the equation. The overdetermination conditions introduced are of integral type. In order to prove the solvability of this problem in Sobolev spaces we use the Galerkin method and the method of successive approximations.
We prove some optimal logarithmic estimates in the Hardy space ${H}^{\infty }(G)$ with Hölder regularity, where $G$ is the open unit disk or an annular domain of $\mathbb {C}$. These estimates extend the results established by S. Chaabane and I. Feki in the Hardy-Sobolev space $H^{k,\infty }$ of the unit disk and those of I. Feki in the case of an annular domain. The proofs are based on a variant of Hardy-Landau-Littlewood inequality for Hölder functions. As an application of these estimates, we study the stability of both the Cauchy problem for the Laplace operator and the Robin inverse problem.
The paper examines sources of brain activity, contributing to EEG patterns which correspond to motor imagery during training to control brain-computer interface. To identify individual source contribution into electroencephalogram recorded during the training Independent Component Analysis was used. Then those independent components for which the BCI system classification accuracy was at maximum were treated as relevant to performing the motor imagery tasks, since they demonstrated well exposed event related de-synchronization and event related synchronization of the sensorimotor μ-rhythm during imagining of contra- and ipsilateral hand movements. To reveal neurophysiological nature of these components we have solved the inverse EEG problem to locate the sources of brain activity causing these components to appear in EEG. The sources were located in hand representation areas of the primary sensorimotor cortex. Their positions practically coincide with the regions of brain activity during the motor imagination obtained in fMRI study. Individual geometry of brain and its covers provided by anatomical MR images was taken into account when localizing the sources.
In the article a theoretical basis and some practical results of treatment with the inverse task as solution of the problem of free boundary are presented. This solution originates from the hydrodynamic theory of boundaries, see Kosorin (2005; 1993). Its main product is the method for transformation of Ndimensional hydrodynamic task into N-1 dimesional one which allows to formulate and solve an inverse task, where the seepage velocity field has to be determined in the domain below the given free water surface. In this case the free surface is assumed to be given by means of contour lines. and V štúdii sú uvedené teoretické východiská a praktické ukážky riešenia inverznej úlohy ako problému voľnej hranice pri sledovaní pohybu podzemnej vody. Toto riešenie vychádza z hydrodynamickej teórie hraníc, pozri Kosorin (2005; 1993). Hlavný produkt teórie je metóda transformácie N-rozmernej hydrodynamickej hraničnej úlohy na N-1 rozmernú hranicu pôvodnej oblasti. To dovoľuje formulovať a riešiť aj tie inverzné úlohy, kde sa rýchlostné pole podzemnej vody určuje v oblasti pod zadanou voľnou hladinou na základe informácií o tejto hranici a geológii prostredia. V tomto prípade ide o voľnú hladinu, zadanú vrstevnicami.