## 1 Introduction  
Numerous inverse problems exist in the field of acoustics. For example, nearfield acoustic holography (NAH) is a noise source identification method that reconstructs a surface field of the source on the basis of sound pressure measured in the nearfield of the source [1; 2; 3; 4; 5]. The deconvolution approach for the mapping of acoustic sources (DAMAS) is also a method for noise source identification [6]. Another example is the source signal separation problem, where an individual source signal is to be extracted from a mixed array of signals [7]. Inverse problems can also be found in source sound field synthesis (SFS) problems, where the sound field produced by secondary sources is to be matched with a target field [8; 9]. Other examples include sound field control [10; 11], crosstalk cancellation in binaural audio rendering [12], noise reduction in speech enhancement [13], room response equalization, and dereverberation [14; 15]. In linear range acoustics, each of these problems can be formulated as a linear system (**Ax = b**). The current study focused on the solutions of farfield noise source identification problems, sound source localization and separation problems, and sound field analysis (SFA) and synthesis (SFS) problems. Although inverse solutions of acoustic problems have long been investigated by researchers, according to our review of the literature, no conclusive results that give solution strategies and parameter choice guidelines can be found in the literature. Furthermore, although audio quality is the chief concern in practical audio reproduction, most previous research has examined general numerical accuracy and stability. This study explored these problems from the perspective of reproduced signal quality. Solution strategies were compared in a unified framework, and guidelines of parameter choice are summarized herein. In general, inverse problems can be divided into two categories: overdetermined and square systems, and underdetermined systems. To solve overdetermined systems, least-squares methods, Tikhonov regularization (TIKR) [16], and truncated singular value decomposition (TSVD) are commonly used. Traditionally, [PERSON]'s discrepancy principle, generalized cross-validation (GCV), and the L-curve method can be used to choose the regularization parameter in the TIKR method [17; 18; 19; 20]. However, solution methods that are better suited to audio applications than conventional approaches are proposed in this work. In particular, golden section search (GSS) [21] is applied to find optimal regularization parameters. To solve underdetermined problems, compressive sampling (CS) [22; 23] solved by using convex optimization (CVX) [24; 25; 26] is a widely used approach that is known to be computationally expensive. In the present study, computationally efficient iterative approaches that incorporate sparsity constraints, including the focal underdetermined system solver (FOCUSS) [27], steepest descent (SD), Newton's (NT) method, and conjugate gradient (CG), were developed. These algorithms were compared for several audio application scenarios. The first scenario was sound source localization and separation using a uniform linear array (ULA) and a uniform circular array (UCA). To assess the separation quality, perceptual evaluation of audio quality (PEAQ), perceptual evaluation of speech quality (PESQ), and segmental signal-to-noise ratio (segSNR) were adopted [13; 28]. The second scenario was concerned with analyses and syntheses of sound fields. Recently, an integrated array system was developed on the basis of a freefield model for spatial audio recording and reproduction [8; 9]. This study extended the previous work to a reverberant environment; a live room was fitted with reflective walls. For the SFA, a 24-element circular microphone array was utilized to encode the sound field based on plane-wave decomposition, whereas in the SFS, a 32-element rectangular loudspeaker array was employed to decode the encoded sound field using three approaches.